Recent zbMATH articles in MSC 20D https://zbmath.org/atom/cc/20D 2023-12-07T16:00:11.105023Z Werkzeug The non-$$\mathfrak{F}$$ graph of a finite group https://zbmath.org/1522.05192 2023-12-07T16:00:11.105023Z "Lucchini, Andrea" https://zbmath.org/authors/?q=ai:lucchini.andrea "Nemmi, Daniele" https://zbmath.org/authors/?q=ai:nemmi.daniele Summary: Given a formation $$\mathfrak{F}$$, we consider the graph whose vertices are the elements of $$G$$ and where two vertices $$g, h \in G$$ are adjacent if and only if $$\langle g, h \rangle \notin \mathfrak{F}$$. We are interested in the two following questions. Is the set of the isolated vertices of this graph a subgroup of $$G$$? Is the subgraph obtained by deleting the isolated vertices a connected graph? {{\copyright} 2021 Wiley-VCH GmbH} Cayley graphs that have a quantum ergodic eigenbasis https://zbmath.org/1522.05193 2023-12-07T16:00:11.105023Z "Naor, Assaf" https://zbmath.org/authors/?q=ai:naor.assaf "Sah, Ashwin" https://zbmath.org/authors/?q=ai:sah.ashwin "Sawhney, Mehtaab" https://zbmath.org/authors/?q=ai:sawhney.mehtaab-s "Zhao, Yufei" https://zbmath.org/authors/?q=ai:zhao.yufei Summary: We investigate which finite Cayley graphs admit a quantum ergodic eigenbasis, proving that this holds for any Cayley graph on a group of size $$n$$ for which the sum of the dimensions of its irreducible representations is $$o(n)$$, yet there exist Cayley graphs that do not have any quantum ergodic eigenbasis. Proof of the elliptic expansion moonshine conjecture of Căldăraru, He, and Huang https://zbmath.org/1522.11031 2023-12-07T16:00:11.105023Z "Hong, Letong" https://zbmath.org/authors/?q=ai:hong.letong "Mertens, Michael H." https://zbmath.org/authors/?q=ai:mertens.michael-h "Ono, Ken" https://zbmath.org/authors/?q=ai:ono.ken "Zhang, Shengtong" https://zbmath.org/authors/?q=ai:zhang.shengtong Summary: Using predictions in mirror symmetry, \textit{A. Căldăraru} et al. recently formulated a Moonshine conjecture at Landau-Ginzburg points'' [Preprint, \url{arXiv:2107.12405}] for Klein's modular $$j$$-function at $$j=0$$ and $$j=1728.$$ The conjecture asserts that the $$j$$-function, when specialized at specific flat coordinates on the moduli spaces of versal deformations of the corresponding CM elliptic curves, yields simple rational functions. We prove this conjecture, and show that these rational functions arise from classical $$_2F_1$$-hypergeometric inversion formulae for the $$j$$-function. On medium-rank Lie primitive and maximal subgroups of exceptional groups of Lie type https://zbmath.org/1522.20001 2023-12-07T16:00:11.105023Z "Craven, David A." https://zbmath.org/authors/?q=ai:craven.david-a This monograph is dedicated to the study of some topics of the theory of finite simple groups (maximal subgroups of finite simple groups of Lie type, structure of the simple algebraic groups, the morphism extension problem and generalize Steinberg's restriction theorem for $$\mathrm{GL}_{n}$$ to arbitrary semisimple algebraic groups). Let $$\mathbf{G}$$ be a finite exceptional group of Lie type and a potential maximal subgroup that is the normalizer of a finite simple group of Lie type $$H$$ in the same characteristic as $$\mathbf{G}$$, but is not of type $$\mathrm{PSL}_{2}$$. A subgroup of $$\mathbf{G}$$ is Lie primitive if it is not contained in any proper, positive-dimensional subgroup of $$\mathbf{G}$$. The author proves that, with a few possible exceptions, there are no Lie primitive subgroups $$H$$ in $$\mathbf{G}$$, with the conditions on $$H$$ and $$\mathbf{G}$$ given above. The exceptions are for $$H$$ one of $$\mathrm{PSL}_{3}(3)$$, $$\mathrm{PSU}_{3}(3)$$, $$\mathrm{PSL}_{3}(4)$$, $$\mathrm{PSU}_{3}(4)$$, $$\mathrm{PSU}_{3}(8)$$, $$\mathrm{PSU}_{4}(2)$$, $$\mathrm{PSp}_{4}(2)'$$ and $$\mathrm{Sz}(8)$$, and $$\mathbf{G}$$ of type $$\mathsf{E}_{8}$$. The first five chapters are devoted to notations and to preliminaries. Chapters 6, 7 and 8 deal with groups of rank 4, 3 and 2 (respectively) for $$\mathsf{E}_{8}$$. Chapters 9, 10 and 11 deal with subgroups of $$\mathsf{E}_{7}$$, $$\mathsf{E}_{6}$$ and $$\mathsf{F}_{4}$$. Chapter 12 deals with difficult cases $$\mathrm{PSL}_{5}(2) \leq \mathsf{E}_{8}$$, $$\mathrm{PSL}_{3}(5) \leq \mathsf{E}_{8}$$ and $$\mathrm{PSU}_{3}(4) \leq \mathsf{E}_{8}$$. In the last chapter the author investigates trilinear forms for $$\mathsf{E}_{6}$$. Reviewer: Egle Bettio (Venezia) Irredundant bases for finite groups of Lie type https://zbmath.org/1522.20005 2023-12-07T16:00:11.105023Z "Gill, Nick" https://zbmath.org/authors/?q=ai:gill.nick "Liebeck, Martin W." https://zbmath.org/authors/?q=ai:liebeck.martin-w Let $$G$$ be a finite group acting on a set $$\Omega$$ and for any list $$\Lambda :=[\omega _{1},\omega _{2},\dots,\omega _{t}]$$ of points from $$\Omega$$ let $$G_{(\Lambda )}=G_{\omega_{1},\omega _{2},\dots,\omega_{t}}$$ be the pointwise stabilizer of $$\Lambda$$. Then $$\Lambda$$ is a base for $$G\$$if $$G_{(\Lambda)}=1$$ and $$\Lambda$$ is an irredundant base if no proper sublist is a base. Let $$b(G,\Omega )$$ denote the length of the shortest base for $$(G,\Omega )$$ and let $$I(G,\Omega )$$ denote the length of the longest possible irredundant base. A theorem proved by \textit{M. W. Liebeck} and \textit{A. Shalev} [J. Am. Math. Soc. 12, No. 2, 497--520 (1999; Zbl 0916.20003)] (conjectured by Cameron and Kantor) is that $$b(G,\Omega )$$ is universally bounded over the set of almost simple primitive nonstandard permutation groups. The main theorem of the present paper is: If $$G$$ is a finite simple group of Lie type of rank $$r$$ acting primitively on $$\Omega$$, then $$I(G,\Omega )\leq 174r^{8}$$. In this estimate, the constant $$174$$ is not sharp and the exponent $$8$$ is probably far from sharp but examples show it must be at least $$2$$. The lower bound on $$I(G,\Omega )$$ is trivial; for example, if we take $$G:=\mathrm{SL}_{r}(2)$$ with $$r$$ an odd prime and $$\Omega$$ the set of cosets of the normalizer of a Singer cycle, then $$I(G,\Omega )\leq 3$$. Reviewer: John D. Dixon (Ottawa) Cherlin's conjecture for sporadic simple groups https://zbmath.org/1522.20010 2023-12-07T16:00:11.105023Z "Dalla Volta, Francesca" https://zbmath.org/authors/?q=ai:dalla-volta.francesca "Gill, Nick" https://zbmath.org/authors/?q=ai:gill.nick "Spiga, Pablo" https://zbmath.org/authors/?q=ai:spiga.pablo Summary: We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to a sporadic simple group. On counting double centralizers of symmetric groups https://zbmath.org/1522.20019 2023-12-07T16:00:11.105023Z "Lu, Zhipeng" https://zbmath.org/authors/?q=ai:lu.zhipeng Let $$S_n$$ be the symmetric group on the symbols $$\{1, 2, \dots, n\}$$ and let $$h\in S_n$$. A question of interest is, given $$h$$, how easily can we find elements $$g\in S_n$$ that can make the centralizer $$C(\langle h, ghg^{-1} \rangle )$$ small (size in polynomial of $$n$$)? To make the notion of being small'' more precise, we need some notation. Let $$H$$ be the subgroup of $$S_{2m}$$ consisting of permutations preserving the partition $$\{1,2\},\{3,4\},\ldots, \{2m-1,2m\}$$, we call $$g\in S_{2m}$$ \textit{good} if $$|H\cap gHg^{-1}|=m^{O(1)}$$ and call it \textit{bad} otherwise. The author of this paper proves that the good elements of $$S_{2m}$$ have density zero and the bad elements $$g\in S_{2m}$$ with $$|H\cap gHg^{-1}|\gg m^{\log m}$$ have zero density. Reviewer: Manjil Pratim Saikia (Ahmedabad) Huppert's conjecture and almost simple groups https://zbmath.org/1522.20030 2023-12-07T16:00:11.105023Z "Daneshkhah, Ashraf" https://zbmath.org/authors/?q=ai:daneshkhah.ashraf For a finite group $$G$$, let $$\operatorname{cd}(G)$$ denote the set of degrees of the complex irreducible characters of $$G$$. \textit{B. Huppert} [Illinois J. Math. 44, No. 4, 828--842 (2000; Zbl 0972.20006)] conjectured that when $$\operatorname{cd}(G) = \operatorname{cd}(H)$$ and $$H$$ is nonabelian simple, then $$G \cong H \times A$$ with $$A$$ abelian. This is not necessarily true when $$H$$ is an almost simple group, as examples with $$H =S_5$$ show. The main result of the paper under review is: Let $$H$$ be an almost simple group with socle $$\operatorname{PSL}(2,2^f)$$, where $$f$$ is a prime number, and let $$G$$ be a finite group with $$\operatorname{cd}(G) = \operatorname{cd}(H)$$. Then $$G/ \operatorname{Z}(G)$$ is isomorphic to $$H$$. The proof in the case $$H = S_5$$, which has socle $$A_5\cong \operatorname{PSL}(2,4)$$, uses the classification of nonsolvable groups with 4 character degrees by \textit{G. Malle} and \textit{A. Moretó} [J. Algebra 294, No. 1, 117--126 (2005; Zbl 1098.20007)]. Reviewer: Frieder Ladisch (Rostock) Non-solvable groups whose character degree graph has a cut-vertex. I https://zbmath.org/1522.20032 2023-12-07T16:00:11.105023Z "Dolfi, Silvio" https://zbmath.org/authors/?q=ai:dolfi.silvio "Pacifici, Emanuele" https://zbmath.org/authors/?q=ai:pacifici.emanuele "Sanus, Lucia" https://zbmath.org/authors/?q=ai:sanus.lucia "Sotomayor, Victor" https://zbmath.org/authors/?q=ai:sotomayor.victor Let $$\mathrm{cd}(G)$$ be the set of all irreducible character degrees of a finite group $$G$$. A simple graph $$\Delta(G)$$ is said to be the \textit{character degree graph} of $$G$$ if its vertices are the prime divisors of the numbers in $$\mathrm{cd}(G)$$, and two distinct vertices $$p, q$$ are adjacent if and only if $$pq$$ divides some number in $$\mathrm{cd}(G)$$. In this paper, the authors show that $$\Delta(G)$$ has at most one cut-vertex, a vertex whose removal increases the number of connected components of the graph. Moreover, they study the structure of the non-solvable groups whose character degree graph has a cut-vertex. Reviewer: Mahmood Robati Sajjad (Qazvin) Strongly monolithic characters of finite groups https://zbmath.org/1522.20033 2023-12-07T16:00:11.105023Z "Erkoç, Temha" https://zbmath.org/authors/?q=ai:erkoc.temha "Güngör, Sultan Bozkurt" https://zbmath.org/authors/?q=ai:gungor.sultan-bozkurt "Özkan, Jülide Miray" https://zbmath.org/authors/?q=ai:ozkan.julide-miray Let $$X$$ be an irreducible character of a finite group $$G$$. If $$G/\ker X$$ has only one minimal normal subgroup, then $$X$$ is called a monolithic character. Furthermore, if $$X$$ is a monolithic character and either $$Z(X)=\ker X$$ or $$G/\ker X$$ is a $$p$$-group whose commutator subgroup is its unique minimal normal subgroup, then $$X$$ is called a strongly monolithic character. In the paper under review, the authors prove the following: Let $$G$$ be a finite group. (1) If the degrees of strongly monolithic characters of $G$ whose kernels are the same are distinct, then $$G$$ is solvable. (2) If the degrees of all strongly monolithic characters of a non-abelian $p$-group are distinct, then $G$ is either an extra-special $$2$$-group or a maximal class $$2$$-group of order greater than 8. (3) If $G$ is non-abelian all of whose non-linear irreducible characters are monolithic, then $G$ has only one strongly monolithic characters if and only if $G$ is a Seitz group, $G$ is a maximal class $$2$$-group of order greater than $$8$$ or $G$ is isomorphic to $$\mathrm{SL}(2,3)$$. Here, a group is called Seitz group if it has only one non-linear irreducible character. Reviewer: Mohammad-Reza Darafsheh (Tehran) Codegrees and element orders of almost simple groups https://zbmath.org/1522.20036 2023-12-07T16:00:11.105023Z "Madanha, Sesuai Y." https://zbmath.org/authors/?q=ai:madanha.sesuai-yash \textit{G. Qian} conjectured [Arch. Math. 97, No. 2, 99--103 (2011; Zbl 1232.20014)] that, for every finite group $$G$$ and every element $$g \in G$$, there exists an irreducible character $$\chi$$ of $$G$$ such that the codegree $$|G:\mathrm{Ker}(\chi)|/\chi(1)$$ of $$\chi$$ is divisible by the order of $$g$$. He also proved the conjecture for solvable groups. In the paper under review, the conjecture is verified for almost simple groups. Reviewer: Burkhard Külshammer (Jena) Characterization of some almost simple groups with socle $$\mathrm{PSp}_4(q)$$ by their character degrees https://zbmath.org/1522.20040 2023-12-07T16:00:11.105023Z "Moghadam, Safoura Madady" https://zbmath.org/authors/?q=ai:moghadam.safoura-madady "Iranmanesh, Ali" https://zbmath.org/authors/?q=ai:iranmanesh.ali The set of character degrees of a finite group $$G$$ is defined to be $\mathrm{cd}(G) := \{n \in \mathbb{N} \mid n = \chi(1) \text{ for some irreducible character } \chi \in \operatorname{Irr}(G)\}.$ The set $$\mathrm{cd}(G)$$ can not characterize the structure of $$G$$. But Huppert's conjecture [\textit{B. Huppert}, Ill. J. Math. 44, No. 4, 828--842 (2000; Zbl 0972.20006)] predicts that $$\mathrm{cd}(G)$$ is capable to determine a finite non-abelian simple group up to a direct product by an abelian factor. Following the line proposed in [\textit{S. H. Alavi} et al., Bull. Aust. Math. Soc. 94, No. 2, 254--265 (2016; Zbl 1398.20009); Rend. Semin. Mat. Univ. Padova 138, 115--127 (2017; Zbl 1380.20007)], and using F. Lübeck's list [\url{http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/DegMult/index.html}] of relating character degrees, the paper under review proves the following generalization of Huppert's conjecture for projective conformal symplectic groups $$\mathrm{PCS}p_4(q)$$. Theorem 1.3. Let $$G$$ be a finite group and $$H$$ be a projective conformal symplectic group $$\mathrm{PCS}p_4(q)$$, which is extended from $$\mathrm{PS}p_4(q)$$ by its diagonal automorphism $$d$$. If $$\mathrm{cd}(G) = \mathrm{cd}(H)$$, then $$G / Z(G)$$ is isomorphic to $$H$$. Reviewer: Hongsheng Hu (Beijing) Character degrees of normally monomial $$p$$-groups of maximal class https://zbmath.org/1522.20045 2023-12-07T16:00:11.105023Z "Yang, Dongfang" https://zbmath.org/authors/?q=ai:yang.dongfang "Lv, Heng" https://zbmath.org/authors/?q=ai:lv.heng In this paper, the authors study the largest irreducible character degree and the maximal abelian normal subgroup of normally monomial $$p$$-groups of maximal class in terms of $$p$$. In particular, they determine all possible irreducible character degree sets of normally monomial $$5$$-groups of maximal class. Reviewer: Hu Jun (Beijing) Principal blocks for different primes. II https://zbmath.org/1522.20049 2023-12-07T16:00:11.105023Z "Navarro, Gabriel" https://zbmath.org/authors/?q=ai:navarro.gabriel.1 "Rizo, Noelia" https://zbmath.org/authors/?q=ai:rizo.noelia "Schaeffer Fry, A. A." https://zbmath.org/authors/?q=ai:schaeffer-fry.amanda-a In [J. Algebra 610, 632--654 (2022; Zbl 1511.20044)], the authors proposed new conjectures about the relationship between the principal blocks of finite groups for different primes. In particular: Conjecture B. Let $$G$$ be a finite group and let $$p,q \in \pi(G)$$. If $$\mathrm{Irr}_{p'}(B_{p}(G)) = \mathrm{Irr}_{q'}(B_{q}(G))$$, then $$p=q$$. Conjecture C. Let $$G$$ be a finite group, and let $$p$$ and $$q$$ be different primes. Then $$q$$ does not divide $$\chi(1)$$ for all $$\chi \in \mathrm{Irr}_{p'}(B_{p}(G))$$ and $$p$$ does not divide $$\chi(1)$$ for all $$\chi \in \mathrm{Irr}_{q'}(B_{q}(G))$$ if and only if there are a Sylow $$p$$-subgroup $$P$$ of $$G$$ and a Sylow $$q$$-subgroup $$Q$$ of $$G$$ such that $$[P,Q]=1$$. In the paper under review, the authors (using the classification of finite simple groups) prove the following two theorems. Theorem D: Conjecture C implies Conjecture B. Theorem E: Conjecture C holds for finite simple groups. Reviewer: Egle Bettio (Venezia) Reduction theorems for generalised block fusion systems https://zbmath.org/1522.20051 2023-12-07T16:00:11.105023Z "Serwene, Patrick" https://zbmath.org/authors/?q=ai:serwene.patrick In modular representation theory of finite groups, the notion fusion system' plays a very important role though the original motivation for it is just conjugation of elements in a finite group. The paper under review extends three well-known classical results, namely, Brauer's third main theorem, Fong's (Fong-Reynold's) first reduction theorem and Fong's second reduction theorem, to generalized block fusion systems, and it does also extend \textit{M. Cabanes}'s relatively new result on non-exotic fusion systems of unipotent blocks of finite groups of Lie type [in: Local representation theory and simple groups. Extended versions of short lecture courses given during a semester programme on Local representation theory and simple groups'' held at the Centre Interfacultaire Bernoulli of the EPF Lausanne, Switzerland, 2016. Zürich: European Mathematical Society (EMS). 179--265 (2018; Zbl 1430.20009)]. Let $$k$$ be an algebraically closed field of prime characteristic $$p$$. The key tool, which plays the most important role in the paper under review, is defined by \textit{R. Kessar} and \textit{R. Stancu} [J. Algebra 319, No. 2, 806--823 (2008; Zbl 1193.20006)]. In their paper, for a finite group $$\widetilde G$$ containing a normal subgroup $$G$$ and $$b$$ a $$\widetilde G$$-stable block of $$kG$$, they define a $$(b,\widetilde G)$$-Brauer pair $$(\widetilde Q,e_{\widetilde Q})$$ where $$\widetilde Q$$ is a $$p$$-subgroup of $$\widetilde G$$ and $$e_{\widetilde Q}$$ is a block of $$k\,C_G(\widetilde Q)$$ with $${\mathrm{Br}}_{\widetilde Q}^{kG}(b){\cdot}e_{\widetilde Q}\,{\not=}\,0$$. Recall that $${\mathrm{Br}}_{\widetilde Q}^{kG}$$ is the Brauer homomorphism from $$(kG)^{\widetilde Q}\rightarrow (kG)(\widetilde Q)$$ and that $$kG$$ is considered as a $$\widetilde G$$-algebra over $$k$$ by conjugation and $$(kG)(\widetilde Q)$$ is the Brauer construction with respect to $$\widetilde Q$$. Note that a $$(b, G)$$-Brauer pair is nothing but a (usual) $$b$$-Brauer pair. More precisely, the author first generalizes Brauer's third main theorem. Namely, if $$G$$ is a normal subgroup of a finite group $$\widetilde G$$ and $$b$$ is the principal block of $$kG$$ with a maximal $$(b,\widetilde G)$$-Brauer pair $$(\widetilde P,e_{\widetilde P})$$, then $$\widetilde P$$ is a Sylow $$p$$-subgroup of $$\widetilde G$$, $$e_{\widetilde P}$$ is the principal block of $$kC_G({\widetilde P})$$ and $$\mathcal F_{\widetilde P}(\widetilde G)\cong\mathcal F_{(\widetilde P,e_{\widetilde P})}(\widetilde G,G,b)$$. Here, $$\mathcal F_{\widetilde P}(\widetilde G)$$ is the (usual) fusion system of $$\widetilde G$$ on $$\widetilde P$$, and $$\mathcal F_{(\widetilde P,e_{\widetilde P})}(\widetilde G,G,b)$$ is a category whose objects are all subgroups of $$\widetilde P$$, and all whose morphisms $$\phi: \widetilde Q\rightarrow\widetilde R$$ are given such that there is an element $$\widetilde g\in\widetilde G$$ satisfying that $$\phi(u)=\widetilde gu{\widetilde g}^{-1}$$ for every $$u\in\widetilde Q$$ and that $$\widetilde g{\cdot}(\widetilde Q,e_{\widetilde Q}){\cdot}{\widetilde g}^{-1}\leq (\widetilde R,e_{\widetilde R})$$, see Definitions 2.7, 2.17 and 2.21 of [\textit{M. Aschbacher} et al., Fusion systems in algebra and topology. Cambridge: Cambridge University Press (2011; Zbl 1255.20001), Part IV]. As the second and third main results, the author extends Fong's first and second reduction theorems. It is known that the so-called Fong-Reynolds correspondence induces a splendid Morita equivalence (Puig equivalence, source algebra isomorphism equivalence) between the corresponding blocks, see [the reviewer et al., J. Algebra 279, No. 2, 638--666 (2004: Zbl 1065.20021), Theorem 1.5]. Then Fong's second reduction theorem is considered. The correspondence there realizes a Morita equivalence but not necessarily a splendid Morita equivalence. Nevertheless, it has been used quite a few times especially for $$p$$-solvable groups in order to reduce problems to a situation such as $$O_p(G)\,{\not=}\,1$$, and hence sometimes the induction on the orders of groups does work. As the fourth main theorem, Cabane's result [loc. cit.] is extended. Finally, the reviewer wants to know the following: (1) What about the converse of the first result? It is just because Brauer's original third main theorem says that for two blocks $$b$$ and $$c$$ corresponding via the Brauer homomorphism, $$b$$ is the principal block if and only if so is $$c$$. (2) Isn't it possible to extend the second result on Fong's first reduction theorem to prove existence not only of the equality between the corresponding generalized fusion systems also to akind of splendid Morita equivalence (that would probably be even stronger)? Reviewer: Shigeo Koshitani (Chiba) The Lie algebra structure of the degree one Hochschild cohomology of the blocks of the sporadic Mathieu groups https://zbmath.org/1522.20055 2023-12-07T16:00:11.105023Z "Murphy, William" https://zbmath.org/authors/?q=ai:murphy.william-s-jun|murphy.william-d Gerstenhaber showed that the Hochschild cohomology of an algebra carries a Lie algebra structure which reduces in degree $$1$$ to the more classical Lie algebra structure of derivations modulo inner derivations. In recent years this Lie structure was subject to a lot of research and spectacular progress. One of the main questions was to determine when the Lie algebra in degree $$1$$ is solvable. The paper under review determines the dimension of the Lie algebra for the blocks of the modular group algebras of each of the sporadic simple Mathieu groups in all possible characteristics. The paper uses a lot of the abstract theory known for the representation theory of these groups. For the more specific results on the problem, the article builds namely on work of \textit{F. Eisele} and \textit{T. Raedschelders} [Trans. Am. Math. Soc. 373, No. 11, 7607--7638 (2020; Zbl 1476.16006)], \textit{B. Briggs} and \textit{L. Rubio y Degrassi} [Pac. J. Math. 321, No. 1, 45--71 (2022; Zbl 1511.13015)], as well as \textit{L. Rubio y Degrassi} et al. [Quaest. Math. 46, No. 9, 1955--1980 (2023; Zbl 07740716)]. The main tool is a result due to \textit{B. Külshammer} and \textit{G. R. Robinson} [J. Algebra 249, No. 1, 220--225 (2002; Zbl 1009.20012)]. They proved that an alternating sum of dimensions of Hochschild cohomology vanishes for blocks of groups defined by a rather sophisticated construction with topological flavour. The main result then proceeds by a case by case analysis and gives a complete list for the dimensions of the degree $$1$$ Hochschild cohomology for the blocks and determines which of them are solvable or not. Reviewer: Alexander Zimmermann (Amiens) OD-characterization of almost simple groups related to $$L=\mathrm{PSL}(2,p^2)$$ except $$\Aut(L)$$ https://zbmath.org/1522.20056 2023-12-07T16:00:11.105023Z "Sajjadi, Masoumeh" https://zbmath.org/authors/?q=ai:sajjadi.masoumeh Let $$G$$ be a finite group and $$\pi(G):=\{p_1,\dots,p_k\}$$ denote the set of prime divisors of $$|G|$$. The Gruenberg-Kegel graph (prime graph) of $$G$$ denoted by $$\mathrm{GK}(G)$$ has vertex set $$\pi(G)$$ and two vertices $$p_i$$ and $$p_j$$ are adjacent iff there exists an element of order $$p_ip_j$$ in $$G$$. For $$p\in \pi(G)$$, $$\deg(p)$$ is the number of vertices adjacent to $$p$$ in $$\mathrm{GK}(G)$$. Further, $$D(G):=\{\deg(p_1),\dots \deg(p_k)\}$$ is called the degree pattern of $$G$$. A finite group $$G$$ is called \textit{OD-characterizable} if for every finite group $$H$$ such that $$|H|=|G|$$ and $$D(G)=D(H)$$, $$H\cong G$$. The main theorem of this article shows that if $$G$$ is an almost simple group related to $$P:=\mathrm{PSL}(2,p^2)$$ and $$G\neq \Aut(P)$$ then $$G$$ is \textit{OD-characterizable}. Note that a group $$M$$ is called an almost simple group related to a simple group $$S$$ if $$S\unlhd M\leq \Aut(S)$$. Reviewer: Rijubrata Kundu (S.A.S. Nagar) On the existence of $$G$$-permutable subgroups in alternating groups https://zbmath.org/1522.20057 2023-12-07T16:00:11.105023Z "Yang, N." https://zbmath.org/authors/?q=ai:yang.nana|yang.nianhua|yang.ningguang|yang.nanhai|yang.ni|yang.nathan|yang.ningning|yang.naiding|yang.nijing|yang.na|yang.nanying|yang.nachuan|yang.nansheng|yang.nanping|yang.ningxue|yang.ninghui|yang.nianwan|yang.nian|yang.nanfang|yang.nanjun|yang.nannan|yang.ningwei|yang.nan|yang.ningman|yang.ning "Galt, A." https://zbmath.org/authors/?q=ai:galt.aleksei-albertovich|galt.alexey-albertovich Summary: Recall that a subgroup $$A$$ of a group $$G$$ is called $$G$$-permutable in $$G$$ if for every subgroup $$B$$ of $$G$$ there exists an element $$x\in G$$ such that $$A$$ and $$B^x$$ commute. The following question was posed in the Kourovka Notebook: is there an integer $$n$$ such that for all $$m>n$$ the alternating group $$\mathrm{A}_m$$ has no non-trivial $$\mathrm{A}_m$$-permutable subgroups? We give a positive answer to this question. Moreover, in the case of prime $$p$$ we prove that $$\mathrm{A}_p$$ has no non-trivial $$\mathrm{A}_p$$-permutable subgroups except $$p=5$$. The 2-fusion system of the Monster https://zbmath.org/1522.20058 2023-12-07T16:00:11.105023Z "Aschbacher, Michael" https://zbmath.org/authors/?q=ai:aschbacher.michael-george The theory of fusion systems is an important tool in the study and classification of finite simple groups. For nomenclature, terminology and properties of fusion systems see the monograph of the author et al. [Fusion systems in algebra and topology. Cambridge: Cambridge University Press (2011; Zbl 1255.20001)]. The paper under review is dedicated to the study of $$2$$-fusion systems of the sporadic group of Fischer-Griess (now known by everyone as Monster). The main result proved by the author is Theorem 1: Assume that $$\mathcal{F}$$ is a saturated fusion system on a finite $$2$$-group $$S$$, $$t$$ is a fully centralized involution in $$S$$, and $$\mathcal{C}= \mathcal{C}_{\mathcal{F}}(t) =\mathcal{F}_{T}(K)$$, where $$T \in \mathcal{F}^{f}$$ is Sylow in $$\mathcal{C}$$ and $$K$$ is a universal covering group of a copy of the Baby Monster. Assume that each member of $$\mathfrak{C}(\mathcal{F})$$ is known. Then either $$\mathcal{F}=\mathcal{C}$$ or $$\mathcal{F}$$ is the $$2$$-fusion system of the monster. Reviewer: Enrico Jabara (Venezia) Closed Majorana representations of $$\{3, 4\}^+$$-transposition groups https://zbmath.org/1522.20059 2023-12-07T16:00:11.105023Z "Ivanov, Alexander A." https://zbmath.org/authors/?q=ai:ivanov.alexander-a|ivanov.aleksandr-aleksandrovich.1|ivanov.aleksander Majorana theory has been introduced by the author in [The Monster group and Majorana involutions. Cambridge: Cambridge University Press (2009; Zbl 1205.20014)] as an axiomatic background for studying the 196 884-dimensional Conway-Griess-Norton algebra of the Monster group, which is the largest sporadic simple group. This theory suggests a procedure one can follow in an attempt to realize a group $$G$$ as a group generated by Majorana involutions in the automorphism group of a Majorana algebra. The paper contributes to Majorana theory. Among the eight non-trivial Norton-Sakuma algebras, four algebras are closed on the set of Majorana generators. These algebras are $$2A$$, $$2B$$, $$3C$$ and $$4B$$. The classification of Majorana representations restricted to the closed shapes was anticipated for a long time. In the present article, the classification is achieved for shapes restricted to $$2A$$, $$3C$$ and $$4B$$ and for the set of generating involutions in the target group forming a single conjugacy class. \textit{F. G. Timmesfeld}'s classification of $$\{3, 4\}^+$$-transposition groups [Geom. Dedicata 1, 269--321 (1973; Zbl 0262.20015)] reduces to the consideration of just three groups: $$L_3(2)$$, $$G_2(2)'$$ and $${^3}D_4(2)$$. Each of these groups possesses a unique Majorana representation of the required shape. Only the representation of $$L_3(2)$$, known before, is based on an embedding into the Monster. Reviewer: Anatoli Kondrat'ev (Ekaterinburg) Fusion systems realizing certain Todd modules https://zbmath.org/1522.20060 2023-12-07T16:00:11.105023Z "Oliver, Bob" https://zbmath.org/authors/?q=ai:oliver.bob The main result proved in the paper under review is Theorem A: Let $$\mathcal{F}$$ be a saturated fusion system over a finite 3-group $$S$$ with an elementary abelian subgroup $$A \leq S$$ such that $$C_{S}(A)=A$$ and such that either (i) $$\mathrm{rk}(A)=6$$ and $$O^{3'}(\Aut_{\mathcal{F}}(A)) \simeq 2M_{12}$$; or (ii) $$\mathrm{rk}(A)=5$$ and $$O^{3'}(\Aut_{\mathcal{F}}(A)) \simeq M_{11}$$; or (iii) $$\mathrm{rk}(A)=4$$ and $$O^{3'}(\Aut_{\mathcal{F}}(A)) \simeq A_{6}$$. Assume also that $$A \not \trianglelefteq \mathcal{F}$$. Then $$A \trianglelefteq S$$, $$S$$ splits over $$A$$, and $$O^{3'}(\mathcal{F})$$ is simple and isomorphic to the 3-fusion system of $$\mathrm{Co}_{1}$$ in case (i), to that of $$\mathrm{Suz}$$, $$\mathrm{Ly}$$, or $$\mathrm{Co}_{3}$$ in case (ii), or to that of $$U_{4}(3)$$, $$U_{6}(2)$$, $$\mathrm{McL}$$, or $$\mathrm{Co}_{2}$$ in case (iii). All three cases of Theorem A have already been shown in earlier papers using different methods. In [Exotic fusion systems related to sporadic simple groups'', Preprint, \url{arXiv:2201.01790}], \textit{M. van Beek} determined all fusion systems $$\mathcal{F}$$ over a Sylow 3-subgroup of $$\mathrm{Co}_{1}$$ with $$O_{3}(\mathcal{F})=1$$. In [J. Group Theory 22, No. 4, 689--711 (2019; Zbl 1468.20039)], \textit{E. Baccanelli} et al. listed all saturated fusion systems $$\mathcal{F}$$ with $$O_{3}(\mathcal{F})=1$$ over a Sylow 3-subgroup of the split extension $$E_{81} \rtimes A_{6}$$, and this includes the four systems that appear in case (iii) of the above theorem. In [Math. Comput. 90, No. 331, 2415--2461 (2021; Zbl 1478.20016)], \textit{C. Parker} and \textit{J. Semeraro} list all saturated fusion systems $$\mathcal{F}$$ over 3-groups of order at most $$3^{7}$$ and $$O^{3}(\mathcal{F}) =\mathcal{F}$$. However, the goals of the author are different from those in the earlier papers, in that as he wants to develop tools which can be used in other situations within the framework of the general problem described above, and are using these Todd modules as test cases. Reviewer: Enrico Jabara (Venezia) On the solubilizer of an element in a finite group https://zbmath.org/1522.20061 2023-12-07T16:00:11.105023Z "Akbari, B." https://zbmath.org/authors/?q=ai:akbari.banafsheh|akbari.behzad "Delizia, C." https://zbmath.org/authors/?q=ai:delizia.costantino "Monetta, C." https://zbmath.org/authors/?q=ai:monetta.carmine The solvability graph $$\Gamma_{S}(G)$$ associated with a finite group $$G$$ is a simple graph whose vertices are the elements of $$G$$, and there is an edge between two distinct elements $$x$$ and $$y$$ if and only if $$\langle x, y\rangle$$ is a solvable subgroup of $$G$$. For $$x\in G$$, the neighborhood of $$x$$ in $$\Gamma_{S}(G)$$ is called the solvabilizer of $$x$$ in $$G$$, and it is denoted by $$\mathrm{Sol}_{G}(x)$$. Hence, $$\mathrm{Sol}_{G}(x)=\{y\in G \mid \langle x, y\rangle \text{ is solvable}\}$$. \textit{J. G. Thompson} [Bull. Am. Math. Soc. 74, 383--437 (1968; Zbl 0159.30804)] proved that a finite group $$G$$ is solvable if and only if for all $$x\in G$$, $$\mathrm{Sol}_{G}(x)=G$$. Let $$R(G)$$ be the solvable radical of a finite group $$G$$, that is, the largest solvable normal subgroup of $$G$$. Then \textit{R. Guralnick} et al. [J. Algebra 300, No. 1, 363--375 (2006; Zbl 1118.20021)] extends Thompson's theorem and proved that $$x\in R(G)$$ if and only if $$\mathrm{Sol}_{G}(x)=G$$. In the paper under review, the authors prove that a finite group $$G$$ is nilpotent of class at most 2 if and only if there exists an element $$x\in G$$ such that $$[u, v, w]=1$$ for every $$u, v, w\in \mathrm{Sol}_{G}(x)$$. Also, they show that if $$G$$ is a non-solvable group, then for all $$x\in G$$, $$|\mathrm{Sol}_{G}(x)|\neq p^{2}$$ for any prime $$p$$. Reviewer: Kamal Aziziheris (Tabriz) Large characteristically simple sections of finite groups https://zbmath.org/1522.20062 2023-12-07T16:00:11.105023Z "Ballester-Bolinches, A." https://zbmath.org/authors/?q=ai:ballester-bolinches.adolfo "Esteban-Romero, R." https://zbmath.org/authors/?q=ai:esteban-romero.ramon "Jiménez-Seral, P." https://zbmath.org/authors/?q=ai:jimenez-seral.paz Summary: In this paper we prove that if $$G$$ is a group for which there are $$k$$ non-Frattini chief factors isomorphic to a characteristically simple group $$A$$, then $$G$$ has a normal section $$C/R$$ that is the direct product of $$k$$ minimal normal subgroups of $$G/R$$ isomorphic to $$A$$. This is a significant extension of the notion of crown for isomorphic chief factors. On $$SS$$-supplemented subgroups of some Sylow subgroups of finite groups https://zbmath.org/1522.20063 2023-12-07T16:00:11.105023Z "He, Xuanli" https://zbmath.org/authors/?q=ai:he.xuanli "Guo, Qinghong" https://zbmath.org/authors/?q=ai:guo.qinghong "Huang, Muhong" https://zbmath.org/authors/?q=ai:huang.muhong In the last decades, many scholars have been interested in characterizing the structure of a finite group with some embedding properties of its subgroups. In the paper under review, the following embedding property of subgroups is introduced: Let $$G$$ be a finite group. A subgroup $$H$$ of $$G$$ is said to be $$SS$$-supplemented in $$G$$ if there exists a subgroup $$K$$ of $$G$$ such that $$G = HK$$ and $$H\cap K$$ is permutes with all Sylow subgroups of $$K$$. Some new results about $$p$$-nilpotency of a finite group are given and the discussion is extended to the universe of saturated formations. Reviewer: Yangming Li (Guangzhou) Centralizers in finite solvable groups https://zbmath.org/1522.20064 2023-12-07T16:00:11.105023Z "Jabara, Enrico" https://zbmath.org/authors/?q=ai:jabara.enrico Denote by $$\pi(G)$$ the set of prime divisors of the order of a finite group $$G$$. Let $$k^*_G=\max\{\pi(C_G(x))\mid x \notin G\setminus Z(G)\}$$ and $$k_G=\max\{\pi(C_G(x))\mid 1\neq x \in G\}.$$ The author proves the following results: Theorem A. If $$G$$ is a finite non-abelian solvable group, then $$\pi(G)\leq 2k^*_G$$, with equality only if $$Z(G)=1.$$ Theorem B. If a finite solvable group $$G$$ satisfies the equality $$\pi(G)=2k_G$$ then $$G$$ is a Frobenius or a $$2$$-Frobenius group. Reviewer: Andrea Lucchini (Padova) Finite groups whose maximal subgroups have only soluble proper subgroups https://zbmath.org/1522.20065 2023-12-07T16:00:11.105023Z "Lytkina, Daria Viktorovna" https://zbmath.org/authors/?q=ai:lytkina.daria-viktorovna "Zhurtov, Archil Khazeshovich" https://zbmath.org/authors/?q=ai:zhurtov.archil-khazeshevich Authors' abstract: We give a description of a finite group whose maximal subgroups possess only soluble proper subgroups, which implies the answer to the well-known question on composition factors of finite groups, whose second maximal subgroups are soluble. Reviewer: Alexander Ivanovich Budkin (Barnaul) Sylow intersections and control of fusion https://zbmath.org/1522.20066 2023-12-07T16:00:11.105023Z "Meng, Hangyang" https://zbmath.org/authors/?q=ai:meng.hangyang "Tian, Jinyue" https://zbmath.org/authors/?q=ai:tian.jinyue "Guo, Xiuyun" https://zbmath.org/authors/?q=ai:guo.xiuyun In a finite group $$G$$, with subgroups $$K \le H \le G$$, we say that $$H$$ controls strong $$G$$-fusion in $$K$$ if and only if the following holds: For all subsets $$X \subseteq H$$ and all $$g \in G$$ such that $$X^g \subseteq K$$, there is some $$c \in C_G(X)$$ and some $$h \in H$$ such that $$g=ch$$. This article investigates criteria for a subgroup of $$G$$ to control strong $$G$$-fusion in a Sylow $$p$$-subgroup of $$G$$, where $$p$$ is a prime. Hence, let $$G$$ be a finite group, let $$p$$ be a prime and let $$P$$ be a Sylow $$p$$-subgroup of $$G$$. For each subgroup $$H$$ of $$G$$, we use the notation $$\Aut_G(H)$$ for the automizer of $$H$$ in $$G$$, which is the subgroup of $$\Aut(H)$$ of elements induced by inner automorphisms of $$N_G(H)$$. Here are two of the main results of the article, and we recall that $$P$$ is a Sylow $$p$$-subgroup of $$G$$. \begin{itemize} \item[1.] Suppose that $$N_G(P) \le T \le G$$ and that, for all $$g \in G \setminus T$$, it is true that $$\Aut_G(P \cap P^g)$$ is a $$p$$-group. Then $$T$$ controls strong $$G$$-fusion in $$P$$. \item[2.] If $$\Aut_G(P \cap P^g)$$ is a $$p$$-group for all $$g \in G$$, then $$G$$ is $$p$$-nilpotent. \end{itemize} The authors also give an alternative proof for a result by \textit{I. M. Isaacs} and \textit{M. Y. Kızmaz} [Arch. Math. 113, No. 6, 561--563 (2019; Zbl 1515.20099)]: If $$P$$ is not cyclic and if $$P \le H \le G$$ is such that, for all $$g \in G \setminus N_G(H)$$, the Sylow $$p$$-subgroups of $$H \cap H^g$$ are cyclic, then $$N_G(H)$$ controls strong $$G$$-fusion in $$P$$. Reviewer: Rebecca Waldecker (Halle) On properties of the lattice of all $$\tau$$-closed $$n$$-multiply $$\sigma$$-local formations https://zbmath.org/1522.20067 2023-12-07T16:00:11.105023Z "Safonova, Inna N." https://zbmath.org/authors/?q=ai:safonova.inna-n Let $$G$$ be a finite group, $$\sigma = \{\sigma_{i} \mid i \in I\}$$ be a partition of the set of all prime numbers, $$\sigma(G) = \{\sigma_{i} \mid \sigma_{i} \cap \pi(G)\not = \emptyset\}$$, $$\mathfrak{F}$$ a class of finite groups, and $$\sigma(\mathfrak{F}) = \bigcup_{G \in \mathfrak{F}}\sigma(G)$$. A function of $$\sigma$$ to the class of formations of groups is called a formation $$\sigma$$-function. For any formation $$\sigma$$-function $$f$$, the class $$LF_{\sigma} (f)$$ is defined as $$LF_{\sigma}(f) = \{G \mid G = 1 \mbox{ or } G \not = 1 \mbox{ and } G/O_{\sigma_{i}',\sigma_{i}} \in f(\sigma_{i}) \mbox{ for all } \sigma_{i} \in \sigma(G)\}$$. If for some formation $$\sigma$$-function $$f$$ we have $$\mathfrak{F}= LF_{\sigma}(f)$$, then the $$\mathfrak{F}$$ is called $$\sigma$$-local and $$f$$ is called a $$\sigma$$-local definition of $$\mathfrak{F}$$. A set $$\tau(G)$$ of subgroups of $$G$$ such that $$G \in \tau(G)$$ is called a subgroup functor if for every epimorphism $$\varphi: A \rightarrow B$$ and any groups $$H \in \tau(A)$$ and $$T \in \tau(B)$$ we have $$H^{\varphi} \in \tau(B)$$ and $$T^{\varphi^{-1}} \in \tau(A)$$. A class $$\mathfrak{F}$$ is called $$\tau$$-closed if $$\tau(G) \subseteq \mathfrak{F}$$ for all $$G \in \mathfrak{F}$$. In the paper under review, the author proves that the lattice of all $$\tau$$-closed $$n$$-multiply $$\sigma$$-local formations is an algebraic modular lattice of formations. Reviewer: Enrico Jabara (Venezia) On Lockett's conjecture for $$\sigma$$-local Fitting classes https://zbmath.org/1522.20068 2023-12-07T16:00:11.105023Z "Vorob'ev, N. T." https://zbmath.org/authors/?q=ai:vorobev.nikolai-timofeevich "Volkova, E. D." https://zbmath.org/authors/?q=ai:volkova.e-d All groups appearing in this review will be finite. A class of groups $$\mathcal F$$ is called a Fitting class if it is closed under normal subgroups and products of normal $$\mathcal F$$-subgroups. It is well known that many problems related to Fitting classes can be studied by using the operators $$^{*}"$$ and $$_{*}"$$ defined by \textit{F. P. Lockett} [Math. Z. 137, 131--136 (1974; Zbl 0286.20017)]. Every non-empty Fitting class $$\mathcal F$$ can be compared with the Fitting classes $$\mathcal F^{*}$$ and $$\mathcal F_{*}$$, where $$\mathcal F^{*}$$ is the smallest Fitting class containing $$\mathcal F$$ such that the $$\mathcal F^{*}$$-radical of the direct product $$G \times H$$ of any two groups $$G$$ and $$H$$ is equal to the direct product of the $$\mathcal F^{*}$$-radical of $$G$$ and the $$\mathcal F^{*}$$-radical of $$H$$, and $$\mathcal F_{*}$$ is the intersection of all Fitting classes $$X$$ such that $$\mathcal X^{*}$$ = $$\mathcal F^{*}$$. A Fitting class $$\mathcal F$$ is a Lockett class if $$\mathcal F =\mathcal F^{*}$$. Any non-empty Fitting class satifies the inclusions $$\mathcal F_{*} \subseteq\mathcal F\subseteq\mathcal F^{*}$$. A Fitting class $$\mathcal F$$ is called normal in the class of all solvable groups if $$\mathcal F\subseteq\mathcal S$$ and for any group $$G \in\mathcal S$$, its $$\mathcal F$$-radical is the largest of the subgroups $$G$$ belonging to $$\mathcal F$$. Lockett [loc. cit.] conjectured that for a Fitting class $$\mathcal F$$, there exists a normal Fitting class $$\mathcal X$$ such that $$\mathcal F=\mathcal F^{*} \cap \mathcal X$$. Every solvable normal Fitting class $$\mathcal F$$ satisfies Lockett's conjecture. This leads to the study by many authors of the problem of descriptions of families of Fitting classes of groups, generally nonsolvable, for which Lockett's conjecture is valid. \textit{A. N. Skiba} [Probl. Fiz. Mat. Tekh. 2018, No. 1(34), 79--82 (2018; Zbl 1388.20030)] proposed a $$\sigma$$-method to study the structure of groups and formations, which was dualized by \textit{W. Guo} et al. [J. Algebra 542, 116--129 (2020; Zbl 1480.20048)] for Fitting classes by definining $$\sigma$$-local Fitting classes, where $$\sigma$$ is a partition of the set $$\mathbb{P}$$, the set of all primes. The authors analyse the behaviour of $$\sigma$$-local Fitting classes. The main goal of the paper is to prove that every $$\sigma$$-local Fitting class satisfies Lockett's conjecture. Reviewer: M. Carmen Pedraza-Aguilera (València) On the modularity and algebraicity of the lattice of multiply $$\omega$$-composition Fitting classes https://zbmath.org/1522.20069 2023-12-07T16:00:11.105023Z "Yang, N." https://zbmath.org/authors/?q=ai:yang.nianwan|yang.nian|yang.naiding|yang.ningguang|yang.ni|yang.ningwei|yang.nan|yang.nannan|yang.nanjun|yang.ningman|yang.nianhua|yang.ningning|yang.nana|yang.nachuan|yang.nanping|yang.nanying|yang.nanhai|yang.na|yang.ninghui|yang.ningxue|yang.ning|yang.nansheng|yang.nijing|yang.nanfang|yang.nathan "Vorob'ev, N. N." https://zbmath.org/authors/?q=ai:vorobev.nikolai-nikolaevich|vorobev.nikolay-n.1|vorobev.nikolai-n-jun "Staselka, I. I." https://zbmath.org/authors/?q=ai:staselka.i-i Authors' abstract: In this paper, the sufficient conditions for the modularity equality for collections of $$\varpi$$-multiply $$\varpi$$-composition Fitting classes ($$n>0$$) are found. It is proved that the lattice of all $$\varpi$$-multiply $$\varpi$$-composition Fitting classes is algebraic ($$n \geq 0$$). Reviewer: Egle Bettio (Venezia) Some density results involving the average order of a finite group https://zbmath.org/1522.20070 2023-12-07T16:00:11.105023Z "Lazorec, Mihai-Silviu" https://zbmath.org/authors/?q=ai:lazorec.mihai-silviu If $$G$$ is a finite group, then the \textit{average order} of $$G$$ is the quantity $o(G)=\frac{1}{|G|}\sum_{x\in G}|x|,$ which provides a lower bound for the number of conjugacy classes of $$G$$; cf.\ [\textit{A. Jaikin-Zapirain}, Adv. Math. 227, No. 3, 1129--1143 (2011; Zbl 1227.20014), Cor.~2.10]. In [loc. cit,] Jaikin-Zapirain shows that $$o(G)\geq o(\mathrm{Z}(G))$$ and poses the following question: \begin{center} If $$N$$ is a normal subgroup of $$G$$, is it true that $$o(G)\geq o(N)^{1/2}$$?\end{center} \textit{E. I. Khukhro} et al. [J. Algebra 569, 1--11 (2021; Zbl 1462.20010)] construct, for each prime number $$p\geq 5$$, a $$p$$-group $$G$$ with a normal abelian subgroup $$N$$ providing a negative answer to Jaikin-Zapirain's question. The family from [Khukhro et al., loc. cit.] is used in this paper to show that the set $O_{\mathcal{N}}=\left\{\frac{o(G)}{o(H)}\ :\ G \text{ is nilpotent, } H \text{ is a subgroup of }G\right\}$ is dense in the interval $$[0,\infty)$$ considered with the usual topology; cf.\ Theorem 1.2. More precisely, the author \begin{itemize} \item first shows that, the subset of $$O_{\mathcal{N}}$$ of those relative average orders $$o(G)/o(H)$$ where $$G$$ is abelian is dense in $$[1,\infty)$$, \item then demonstrates that the interval $$[0,1)$$ is covered by a subfamily of $$O_{\mathcal{N}}$$ where $$G$$ and $$H$$ are $$p$$-groups constructed exploiting the examples of Khukhro, Moretó, and Zarrin [loc. cit.]. \end{itemize} The paper ends with the open question of whether Theorem 1.2 is true when $$G$$ is only allowed to range over finite $$p$$-groups. Reviewer: Mima Stanojkovski (Trento) The maximum number of triangles in a graph and its relation to the size of the Schur multiplier of special $$p$$-groups https://zbmath.org/1522.20071 2023-12-07T16:00:11.105023Z "Mavely, Tony Nixon" https://zbmath.org/authors/?q=ai:mavely.tony-nixon "Zachariah Thomas, Viji" https://zbmath.org/authors/?q=ai:thomas.viji-zachariah In this article, novel findings in extremal graph theory and the Schur multiplier of $$p$$-groups are presented. The authors establish a precise limit on the number of triangles that can exist in a graph with a given number of edges, and elucidate the characteristics of graphs that achieve the maximum triangle count. Additionally, they leverage this upper bound to provide a sharp limitation on the size of the Schur multiplier for special $$p$$-groups across all ranks, leading to an enhancement of existing bounds for the size of the Schur multiplier of $$p$$-groups. The paper contains comprehensive proofs and notation, making it a valuable resource for researchers engaged in algebra, graph theory, and related domains. Reviewer: Peyman Niroomand (Dāmghān) Finite groups with some $$S$$-permutably embedded subgroups https://zbmath.org/1522.20072 2023-12-07T16:00:11.105023Z "Qiu, Z." https://zbmath.org/authors/?q=ai:qiu.zhengtian "Chen, G." https://zbmath.org/authors/?q=ai:chen.guiyun "Liu, J." https://zbmath.org/authors/?q=ai:liu.jianjun A subgroup $$H$$ of a finite group $$G$$ is $$S$$-permutably embedded in $$G$$ if each Sylow subgroup of $$H$$ is a Sylow subgroup of some $$S$$-permutable subgroup of $$G$$. The authors investigate the structure of the finite groups some of whose subgroups are $$S$$-permutably embedded. Their results improve and generalize many available results. Reviewer: Erich W. Ellers (Toronto) Bounds on the order of the Schur multiplier of $$p$$-groups https://zbmath.org/1522.20073 2023-12-07T16:00:11.105023Z "Rai, Pradeep K." https://zbmath.org/authors/?q=ai:rai.pradeep-kumar Let $$G$$ be a finite group. The Schur multiplier $$M(G)$$ of $$G$$, defined as $$H^{2}(G,\mathbb{C}^{\times})$$, the second cohomology group of $$G$$ with coefficients in $$\mathbb{C}^{\times}$$, plays an important role in the theory of extensions of groups. If $$|G|=p^{n}$$ ($$p$$ a prime), then \textit{J. A. Green} in [Proc. R. Soc. Lond., Ser, A 237, 574--581 (1956; Zbl 0071.02301)] proved that $$|M(G)| \leq p^{\frac{1}{2}n(n-1)}$$ (and this bound can not be improved if $$G$$ is elementary abelian). After that result the bound has been strengthened in many ways. For example, if $$|G'|=p^{k}$$, then $$|M(G)| \leq p^{\frac{1}{2}(n-k-1)(n-k-2)+1}$$ [\textit{P. Niroomand}, J. Algebra 322, No. 12, 4479--4482 (2009; Zbl 1186.20013)] and if $$|G/\Phi(G)|=p^{d}$$ (so $$d$$ is the minimal number of generators of $$G$$), then $$|M(G)| \leq p^{\frac{1}{2}(d-1)(n+k-2)+1}$$ [\textit{G. Ellis} and \textit{J. Wiegold}, Bull. Aust. Math. Soc. 60, No. 2, 191--196 (1999; Zbl 0940.20017)]. As the author points out in [Int. J. Algebra Comput. 27, No. 5, 495--500 (2017; Zbl 1371.20014)], this bound is better than the one found by Niroomand [loc. cit.]. The paper under review is intended to demonstrate further limitations to the order of $$M(G)$$ when $$c$$, the nilpotency class of $$G$$, is added to the parameters $$n$$ and $$d$$. The formulas found are too complicated to be reported here. Another interesting result proved by the author is Theorem 1.7: Let $$p$$ be an odd prime and $$G$$ be a finite $$p$$-group of maximal class with $$|G| = p^{n}$$ and $$n \geq 4$$. Then $$|M(G)| \leq p^{\frac{1}{2}n}$$. Reviewer: Enrico Jabara (Venezia) Finite symmetries of surfaces of $$p$$-groups of co-class 1 https://zbmath.org/1522.20074 2023-12-07T16:00:11.105023Z "Sarkar, Siddhartha" https://zbmath.org/authors/?q=ai:sarkar.siddhartha Summary: The genus spectrum of a finite group $$G$$ is a set of integers $$g\geq 2$$ such that $$G$$ acts on a closed orientable compact surface $$\Sigma_g$$ of genus $$g$$ preserving the orientation. In this paper, we complete the full classification of spectrum sets of finite $$p$$-groups of co-class $$1$$, where $$p$$ is an odd prime. As a consequence, it follows that for any prime $$p$$ and a finite $$p$$-group of co-class $$1$$ of order $$p^n$$ and exponent $$p^e$$, there are at the most seven genus spectra despite the infinite growth of their isomorphism types along with $$(n,e)$$. On finite Sylow tower and $$\sigma$$-tower groups https://zbmath.org/1522.20075 2023-12-07T16:00:11.105023Z "Cai, Jinzhuan" https://zbmath.org/authors/?q=ai:cai.jinzhuan "Safonova, Inna N." https://zbmath.org/authors/?q=ai:safonova.inna-n "Skiba, Alexander N." https://zbmath.org/authors/?q=ai:skiba.alexander-n "Wang, Zhigang" https://zbmath.org/authors/?q=ai:wang.zhigang.1 Let $$G$$ be a finite group, $$\sigma=\{\sigma_{i}\}_{i \in I}$$ a partition of the set of all prime numbers and let $$\sigma(G)=\{\sigma_{i} \mid \sigma_{i} \cap \pi(G) \not = \emptyset \}$$. $$G$$ is a $$\sigma$$-tower group if either $$G = 1$$ or $$G$$ has a normal series $$1=G_{0} < G_{1} < \dots < G_{t}=G$$ such that $$G_{k}/G_{k-1}$$ is a $$\sigma_{i}$$-group and $$G/G_{k}$$ and $$G_{k-1}$$ are $$\sigma_{i}'$$-groups ($$i \in I$$). The Hawkes $$\sigma$$-graph $$\Gamma=\Gamma_{\sigma H}(G)$$ of $$G$$ is the directed graph whose vertices set is $$\sigma(G)$$ and $$\sigma_{i} \rightarrow \sigma_{j}$$ is an arc of $$\Gamma$$ if and only if $$\sigma_{j} \in \sigma(G/O_{\sigma_{i}',\sigma_{i}}(G))$$. The main result in this paper is that $$G$$ is a $$\sigma$$-tower group if and only if the Hawkes $$\sigma$$-graph of $$G$$ has no circuits. Reviewer: Enrico Jabara (Venezia) On the rank of a verbal subgroup of a finite group https://zbmath.org/1522.20076 2023-12-07T16:00:11.105023Z "Detomi, Eloisa" https://zbmath.org/authors/?q=ai:detomi.eloisa "Morigi, Marta" https://zbmath.org/authors/?q=ai:morigi.marta "Shumyatsky, Pavel" https://zbmath.org/authors/?q=ai:shumyatsky.pavel If all Sylow subgroups of a finite group $$G$$ can be generated by $$d$$ elements, then the group $$G$$ itself can be generated by $$d+1$$ elements [\textit{A. Lucchini}, Arch. Math. 53, No. 4, 313--317 (1989; Zbl 0679.20028); \textit{R. M. Guralnick}, Arch. Math. 53, No. 6, 521--523 (1989; Zbl 0675.20026)]. It follows that if all nilpotent subgroups of a finite group $$G$$ have rank at most $$r$$, then the group $$G$$ has rank at most $$r+1$$. In the paper under review, the authors show that if $$w$$ is a multilinear commutator word and $$G$$ a finite group in which every metanilpotent subgroup generated by $$w$$-values is of rank at most $$r$$, then the rank of the verbal subgroup $$w(G)$$ is bounded in terms of $$r$$ and $$w$$ only. Furthermore, they prove that if $$G$$ is soluble and every nilpotent subgroup of $$G$$ generated by $$w$$-values is of rank at most $$r$$, then the rank of $$w(G)$$ is at most $$r+1$$. Reviewer: Enrico Jabara (Venezia) Simple groups and Sylow subgroups https://zbmath.org/1522.20077 2023-12-07T16:00:11.105023Z "Harada, Koichiro" https://zbmath.org/authors/?q=ai:harada.koichiro Summary: Firstly a problem to characterize Sylow $$2$$-subgroups (of small order $$\leqslant 2^{10})$$ of finite simple groups is proposed. Next some (possibly necessery) reduction steps are discussde. The latter half of these notes is devoted to finite groups having extra-spacial $$p$$-groups of order $$p^3$$ as Sylow subgroups. Group orders that imply a nontrivial $$p$$-core https://zbmath.org/1522.20078 2023-12-07T16:00:11.105023Z "Villarroel-Flores, Rafael" https://zbmath.org/authors/?q=ai:villarroel-flores.rafael Summary: Given a prime number $$p$$ and a natural number $$m$$ not divisible by $$p$$, we propose the problem of finding the smallest number $$r_0$$ such that for $$r \geq r_0$$, every group $$G$$ of order $$p^r m$$ has a non-trivial normal $$p$$-subgroup. We prove that we can explicitly calculate the number $$r_0$$ in the case where every group of order $$p^r m$$ is solvable for all $$r$$, and we obtain the value of $$r_0$$ for a case where $$m$$ is a product of two primes. Baumann-components of finite groups of characteristic $$p$$, the $$W(B)$$-theorem https://zbmath.org/1522.20079 2023-12-07T16:00:11.105023Z "Meierfrankenfeld, U." https://zbmath.org/authors/?q=ai:meierfrankenfeld.ulrich "Parmeggiani, G." https://zbmath.org/authors/?q=ai:parmeggiani.gemma "Stellmacher, B." https://zbmath.org/authors/?q=ai:stellmacher.bernd A finite group is said to be a $$\mathcal{CK}$$-group if each of its composition factors is a cyclic group, an alternating group, a sporadic group, or a group of Lie type. In the paper under review, the authors complete the investigation of $$\mathcal{CK}$$-group of characteristic $$p$$ in terms of their Baumann components, initiated in their previous papers [J. Algebra 515, 19--51 (2018; Zbl 1403.20031); J. Algebra 561, 295--354 (2020; Zbl 1485.20058)]. If $$B$$ is a finite, non-trivial $$p$$-group, the authors define a certain non-trivial characteristic subgroup $$\mathrm{W}(B)$$ of $$B$$. Let $$\mathbb{E}_{W} (G)$$ be the normal subgroup generated by the Baumann blocks of $$G$$, for whose definition we refer to the Introduction of the paper. The authors prove in Theorem A that $$G = N_{G} (\mathrm{W}(B))\mathbb{E}_{W} (G)$$. The major part of the paper is then devoted to giving the exact structure of the Baumann blocks of $$G$$ (Theorem B); in particular it is shown that any two distinct Baumann blocks centralise each other. Reviewer: Andrea Caranti (Trento) On the w-supersolubility of a finite group factorized by mutually permutable subgroups https://zbmath.org/1522.20080 2023-12-07T16:00:11.105023Z "Artemenko, Natal'ya Vital'evich" https://zbmath.org/authors/?q=ai:artemenko.natalya-vitalevich "Trofimuk, Aleksandr Aleksandrovich" https://zbmath.org/authors/?q=ai:trofimuk.aleksandr-aleksandrovich Let $$G$$ be a finite group. A subgroup $$H$$ of $$G$$ is called $$\mathbb{P}$$-subnormal in $$G$$, if either $$H=G$$ or there is a finite chain subgroups $$H=H_{0} \leq H_{1} \leq \dots \leq H_{n}=G$$ such that $$|H_{i}: H_{i-1}| \in \mathbb{P}$$ for all $$i \in \{1,2,\ldots,n\}$$. The group $$G$$ is called $$\mathrm{w}$$-supersoluble (widely supersoluble), if every Sylow subgroup of $$G$$ is $$\mathbb{P}$$-subnormal in $$G$$. Every supersoluble group is $$\mathrm{w}$$-supersoluble, so $$\mathrm{w}$$-supersolubility is a generalization of supersolubility. Two subgroups $$A$$ and $$B$$ of $$G$$ are said to be mutually permutable if $$A$$ permutes with all subgroups of $$B$$ and $$B$$ permutes with all subgroups of $$A$$. \textit{M. Asaad} and \textit{A. Shaalan} in [Arch. Math. 53, No. 4, 318--326 (1989; Zbl 0685.20018)] established the supersolubility of a group $$G =AB$$ factorized by two mutually permutable subgroups $$A$$ and $$B$$ if $$B$$ is nilpotent or if the derived subgroup $$G'$$ is nilpotent. The main result of this paper is Theorem 1.1: If $$G=AB$$ is the mutually permutable product of $$\mathrm{w}$$-supersoluble subgroups $$A$$ and $$B$$, then $$G$$ is $$\mathrm{w}$$-supersoluble in each of the following cases: (1) $$B$$ is nilpotent; (2) $$\big ( |G: AF(G)|, |G:BF(G)| \big )=1$$; (3) $$B \trianglelefteq G$$. Reviewer: Egle Bettio (Venezia) A classification of skew morphisms of dihedral groups https://zbmath.org/1522.20081 2023-12-07T16:00:11.105023Z "Hu, Kan" https://zbmath.org/authors/?q=ai:hu.kan "Kovács, István" https://zbmath.org/authors/?q=ai:kovacs.istvan|kovacs.istvan-a|kovacs.istvan.1|kovacs.istvan.2 "Kwon, Young Soo" https://zbmath.org/authors/?q=ai:kwon.young-soo Let $$A$$ be a finite group with identity element $$1_A$$. A skew morphism of $$A$$ is a permutation $$\varphi$$ of $$A$$ such that $$\varphi(1_A) = 1_A$$, and there is a function $$\pi: A\rightarrow Z_{|\varphi|}$$ such that $$\varphi(xy) = \varphi(x)\varphi^{\pi(x)}(y)$$ for all $$x, y \in A$$. In this paper, the authors study the case when $$A=D_{2n}$$, the dihedral group of order $$2n$$. \textit{N.-E. Wang} et al. [Ars Math. Contemp. 16, No. 2, 527--547 (2019; Zbl 1416.05302)] determined all $$\varphi$$ under the condition that $$\pi(\varphi(x)) \equiv \pi(x)\pmod{|\varphi|}$$ holds for every $$x\in D_{2n}$$, and later \textit{I. Kovács} and \textit{Y. S. Kwon} [J. Comb. Theory, Ser. B 148, 84--124 (2021; Zbl 1459.05119)] characterised those $$\varphi$$ such that there exists an inverse-closed $$\langle \varphi\rangle$$-orbit, which generates $$D_{2n}$$. The authors of this paper show that these two types of skew morphisms comprise all skew morphisms of $$D_{2n}$$. Reviewer: Yangming Li (Guangzhou) Automorphisms of Zappa-Szép products https://zbmath.org/1522.20082 2023-12-07T16:00:11.105023Z "Lal, Ratan" https://zbmath.org/authors/?q=ai:lal.ratan "Kakkar, Vipul" https://zbmath.org/authors/?q=ai:kakkar.vipul Let $$G$$ be a group and let $$H$$ and $$K$$ be subgroups of $$G$$. If $$H\cap K=\{1\}$$ and $$G=HK$$, then $$G$$ is said to be the internal \textit{Zappa-Szép product} of $$H$$ and $$K$$. Analogously to what is done for semidirect products, one can define the external Zappa-Szép product $$H{\triangleright\hspace{-1.8pt}\triangleleft} K$$ of two groups $$H$$ and $$K$$ equipped with so-called \textit{matched pairs}. These are functions \begin{align*} K\times H\longrightarrow H, & \quad (k,h)\longmapsto k\cdot h, \\ K\times H\longrightarrow K, & \quad (k,h)\longmapsto k^h \end{align*} satisfying a number of requirements that make sure that the product $(h,k)(h',k')=(h(k\cdot h'),k^{h'}k')$ defines a group structure $$H{\triangleright\hspace{-1.8pt}\triangleleft} K$$ on $$H\times K$$ with the property that, if $$G$$ is the internal Zappa-Szép product of its subgroups $$H$$ and $$K$$, then $$G$$ is isomorphic to $$H{\triangleright\hspace{-1.8pt}\triangleleft} K$$, where the matched pair is naturally defined from the group structure of $$G$$ and reflects the fact that $$HK=KH$$. Identifying internal and external Zappa-Szép products, the authors show how to identify the automorphism group of such a group with $$2\times 2$$ matrices with coefficients in $$\mathrm{Hom}(H,H)$$, $$\mathrm{Hom}(H,K)$$, $$\mathrm{Hom}(K,H)$$ and $$\mathrm{Hom}(K,K)$$ again satisfying some compatibility requirements. In the paper under consideration, the authors also explicitly compute the automorphism groups of Zappa-Szép products of two finite cyclic groups where one of the factors has order equal to the square of a prime number. The results rely on the classifications given in [\textit{K. R. Yacoub}, Proc. Math. Phys. Soc. Egypt 21, 119--126 (1958; Zbl 0123.02703); Publ. Math. Debr. 6, 26--39 (1959; Zbl 0106.02102)]. Reviewer: Mima Stanojkovski (Trento) On central automorphisms of finite $$p$$-groups of class 2 https://zbmath.org/1522.20083 2023-12-07T16:00:11.105023Z "Shabani-Attar, M." https://zbmath.org/authors/?q=ai:shabani-attar.mehdi Let $$G$$ be a group, an automorphism $$\alpha \in \Aut(G)$$ is central if $$x^{-1}x^{\alpha} \in Z(G)$$ for every $$x \in G$$. For the set of central automorphisms of $$G$$, denoted by $$\Aut_{c}(G)$$, fix $$G'$$ elementwise and form a normal subgroup of $$\mathrm{Aut}(G)$$. Let $$\mathbf{C}_{\Aut_{c}}(G)(Z(G))$$ be the set of all central automorphisms of $$G$$ fixing $$Z(G)$$ elementwise. The author proved in [Arch. Math. 89, No. 4, 296--297 (2007; Zbl 1133.20014)] that $$\mathbf{C}_{\Aut_{c}}(G)(Z(G))=\mathrm{Inn}(G)$$ if and only if $$G$$ has class 2 and $$Z(G)$$ is cyclic. The first result proved in the paper under review is that there is no finite $$p$$-group of class 2 for which $$[\mathbf{C}_{\Aut_{c}}(G)(Z(G)) : \mathrm{Inn}(G)]=p$$. The second result consists in the classification of $$p$$-groups of class 2 such that $$[\mathbf{C}_{\Aut_{c}}(G)(Z(G)) : \mathrm{Inn}(G)]=p^{2}$$. Reviewer: Enrico Jabara (Venezia) On the maximum number of subgroups of a finite group https://zbmath.org/1522.20084 2023-12-07T16:00:11.105023Z "Fusari, Marco" https://zbmath.org/authors/?q=ai:fusari.marco "Spiga, Pablo" https://zbmath.org/authors/?q=ai:spiga.pablo Summary: Given a finite group $$R$$, we let $$\operatorname{Sub}(R)$$ denote the collection of all subgroups of $$R$$. We show that $$| \operatorname{Sub}(R) | < c \cdot | R |^{\frac{ \log_2 ( | R | )}{ 4}}$$, where $$c < 7.372$$ is an explicit absolute constant. This result is asymptotically best possible. Indeed, as $$| R |$$ tends to infinity and $$R$$ is an elementary abelian 2-group, the ratio $\frac{ | \operatorname{Sub} ( R ) |}{ | R |^{\frac{ \log_2 ( | R | )}{ 4}}}$ tends to $$c$$. Finite groups with many cyclic subgroups https://zbmath.org/1522.20085 2023-12-07T16:00:11.105023Z "Gao, Xiaofang" https://zbmath.org/authors/?q=ai:gao.xiaofang "Shen, Rulin" https://zbmath.org/authors/?q=ai:shen.rulin Let $$G$$ be a finite group, let $$c(G)$$ be the number of cyclic subgroups of a group $$G$$ and let $$\alpha(G)=c(G)/|G|$$. \textit{M. Garonzi} and \textit{I. Lima} in [Bull. Braz. Math. Soc. (N.S.) 49, No. 3, 515--530 (2018; Zbl 1499.20036)] completely classified groups such that $$\alpha(G) > 3/4$$ (the value 3/4 is the largest nontrivial accumulation point of the set of numbers of the form $$\alpha(G)$$). Let $$I(G)=\{g \in G \mid g^{2}=1 \}$$. The main theorem proved in the article under review is: Let $$G$$ be a group with $$\alpha(G)=3/4$$. Then $$G$$ is isomorphic to a direct product of an elementary abelian 2-group and a dihedral group $$D_{16}$$, $$D_{24}$$ or a group of exponent 4 satisfying $$I(G)=\frac{1}{2}|G|$$. Groups satisfying the relation $$I(G)=\frac{1}{2}|G|$$ were classified in [\textit{G. A. Miller}, Tôhoku Math. J. 17, 88--102 (1920; JFM 47.0095.04)]. Reviewer: Enrico Jabara (Venezia) Commuting graph of a group action with few edges https://zbmath.org/1522.20086 2023-12-07T16:00:11.105023Z "Güloğlu, İsmail Ş." https://zbmath.org/authors/?q=ai:guloglu.ismail-suayip "Ercan, Gülin" https://zbmath.org/authors/?q=ai:ercan.gulin A great deal is known on deriving information about the structure of a finite group $$G$$ from some certain properties of an associated graph. In [J. Group Theory 24, No. 3, 573--586 (2021; Zbl 1489.20009)], the authors introduced the concept of the commuting graph of a group action, i.e. the commuting graph $$\Gamma(G,A)$$ of $$A$$-orbits. In this article, the authors characterize the finite groups $$G$$ for which $$\Gamma(G,A)$$ is an $$\mathcal{F}$$-graph, that is, a connected graph which contains at most one vertex whose degree is not less than three. Reviewer: Haoran Yu (Changchun) Groups having all elements off a normal subgroup with prime power order https://zbmath.org/1522.20087 2023-12-07T16:00:11.105023Z "Lewis, Mark L." https://zbmath.org/authors/?q=ai:lewis.mark-l Let $$G$$ be a finite group and let $$N$$ be a normal subgroup of $$G$$. In the paper under review, the author proves that if there is a prime $$p$$ so that all the elements in $$G\setminus N$$ have $$p$$-power order, then either $$G$$ is a $$p$$-group or $$G = PN$$ where $$P$$ is a Sylow $$p$$-subgroup of $$G$$ and $$(G,P,P \cap N)$$ is a Frobenius-Wielandt triple. Furthermore he proves that if all the elements of $$G \setminus N$$ have prime power orders and the orders are divisible by two primes $$p$$ and $$q$$, then $$G$$ is a $$\{p, q\}$$-group and $$G/N$$ is either a Frobenius group or a $$2$$-Frobenius group. Finally, he proves that if all the elements of $$G \setminus N$$ have prime power orders and the orders are divisible by at least three primes, then all elements of $$G$$ have prime power order and $$G/N$$ is non-solvable. Reviewer: Enrico Jabara (Venezia) A note on friendly and solitary groups https://zbmath.org/1522.20088 2023-12-07T16:00:11.105023Z "Mittal, Shubham" https://zbmath.org/authors/?q=ai:mittal.shubham "Mittal, Gaurav" https://zbmath.org/authors/?q=ai:mittal.gaurav "Sharma, R. K." https://zbmath.org/authors/?q=ai:sharma.rajendra-kumar Summary: In this paper, we extend the notions of friendly and solitary numbers to group theory and define friendly and solitary groups of type-1 and type-2. We provide many examples of friendly and solitary groups and study certain properties of the type-2 friends of cyclic $$p$$-groups, where $$p$$ is a prime number. A note on the number of centralizers in finite AC-groups https://zbmath.org/1522.20089 2023-12-07T16:00:11.105023Z "Pezzott, Julio C. M." https://zbmath.org/authors/?q=ai:pezzott.julio-c-m "Nakaoka, Irene N." https://zbmath.org/authors/?q=ai:nakaoka.irene-n A finite group $$G$$ is called an AC-group if $$G$$ is non-abelian and $$C_G(x)$$ is abelian for any $$x\in G\setminus Z(G).$$ Let $$\mathrm{Cent}(G)$$ be the set $$\{C_G(x)\mid x\in G\}.$$ The authors prove that if $$G$$ is an AC-group such that $$[G : Z(G)] = 2^ns$$ with $$s$$ odd and $$|\mathrm{Cent}(G)| = 2^{n+1},$$ then $$s = 3,$$ $$n$$ is even, the set of the sizes of the conjugacy classes of $$G$$ is $$\{1, 2^n, 3\cdot 2^{n-1}\}$$, $$G/Z(G)$$ is a Frobenius group whose Frobenius kernel is an elementary abelian 2-group of order $$2^n$$. Reviewer: Andrea Lucchini (Padova) A criterion for nonsolvability of a finite group and recognition of direct squares of simple groups https://zbmath.org/1522.20090 2023-12-07T16:00:11.105023Z "Wang, Zh." https://zbmath.org/authors/?q=ai:wang.zhigang.1 "Vasil'ev, A. V." https://zbmath.org/authors/?q=ai:vasilev.andrei-viktorovich "Grechkoseeva, M. A." https://zbmath.org/authors/?q=ai:grechkoseeva.maria-aleksandrovna "Zhurtov, A. Kh." https://zbmath.org/authors/?q=ai:zhurtov.archil-khazeshevich If $$G$$ is a finite group, then the spectrum $$\varpi(G)=\{ o(g) \mid g \in G \}$$ of $$G$$ is the set of orders of elements of $$G$$. Two groups $$G$$ and $$H$$ are said to be isospectral if $$\varpi(G)=\varpi(H)$$ and the number of pairwise non-isomorphic finite groups isospectral to a group $$G$$ is denoted by $$h(G)$$. A group $$G$$ is recognizable by spectrum if $$h(G)=1$$, almost recognizable if $$h(G)$$ is finite and unrecognizable if $$h(G) = \infty$$. A first result proved in the paper under review is that if among the prime divisors of the order of $$G$$, there are four different primes such that $$\varpi(G)$$ contains all their pairwise products but not a product of any three of these numbers, then $$G$$ is nonsolvable. Using this result, the authors show that for $$q\geq 8$$ and $$q \not = 32$$, the direct square $$\mathrm{Sz}(q) \times \mathrm{Sz}(q)$$ of the simple exceptional Suzuki group $$\mathrm{Sz}(q)$$ is recognizable by spectrum. Furthermore, if $$G=\mathrm{Sz}(32) \times \mathrm{Sz}(32)$$, then $$h(G)=5$$. Reviewer: Enrico Jabara (Venezia) The set $$K_p$$ in some finite groups https://zbmath.org/1522.20091 2023-12-07T16:00:11.105023Z "Zabarina, Anna Ivanovna" https://zbmath.org/authors/?q=ai:zabarina.anna-ivanovna "Fomina, Elena Anatol'evna" https://zbmath.org/authors/?q=ai:fomina.elena-anatolevna This article continues the study of the properties of a set $$K_p$$ for groups of order $$p_1p_2\dots p_k$$, $$k\geq 3$$ and $$p^2q$$ where $$p_i$$ and $$q$$ are prime numbers. The set $$K_p$$ is the set of all elements of a group $$G$$ which are permutable with exactly $$p$$ elements of $$G$$. It is also proved that the set $$K_5$$ is not empty in the three-dimensional projective special linear group $$L_3(4)$$. This group has the same order with the group $$A_8$$ in which the set $$K_5$$ is empty. Reviewer: Alexander Ivanovich Budkin (Barnaul) Finite groups with permuted strongly generalized maximal subgroups https://zbmath.org/1522.20113 2023-12-07T16:00:11.105023Z "Gorbatova, Yuliya Vladimirovna" https://zbmath.org/authors/?q=ai:gorbatova.yuliya-vladimirovna Author's abstract: Let $$G$$ be a finite group. If there is a maximal subgroup $$M$$ in $$G$$ such that $$H\leq M$$ and $$H$$ is a maximal subgroup of $$M$$, then $$H$$ is called the $$2$$-maximal subgroup of $$G$$. The $$3$$-maximal subgroups can be defined similarly. This paper is devoted to describing the structure of the groups in which any strongly $$2$$-maximal subgroup is permutable with the arbitrary strongly $$3$$-maximal subgroup. The class of groups with this property coincides with the class of groups in which any $$2$$-maximal subgroup is permuted with the arbitrary $$3$$-maximal subgroup. As an auxiliary result, this work presents a description of groups in which any strongly $$2$$-maximal subgroup is permutable with an arbitrary maximal subgroup. Reviewer: Alexander Ivanovich Budkin (Barnaul) The reduction theorem for relatively maximal subgroups https://zbmath.org/1522.20114 2023-12-07T16:00:11.105023Z "Guo, Wenbin" https://zbmath.org/authors/?q=ai:guo.wenbin.1 "Revin, Danila O." https://zbmath.org/authors/?q=ai:revin.danila-olegovitch "Vdovin, Evgeny P." https://zbmath.org/authors/?q=ai:vdovin.evgeny-petrovitch This paper deals with the Wielandt program outlined by him in the 1960's. It deals with the so-called $$\chi$$-Reduktionssatz for a class $$\chi$$ of groups that is complete (that is, closed under taking subgroups, homomorphic images and extensions). Examples of complete classes is rich, and includes the class of finite solvable groups. The $$\chi$$-Reduktionssatz for a group $$A$$ means that the number of conjugacy classes of $$\chi$$-maximal subgroups of $$G$$ is equal to that for $$G/N$$ for every group $$G$$ containing a normal subgroup $$N$$ isomorphic to $$A$$. In particular, (considering $$N=G$$) this means that all $$\chi$$-maximal subgroups of $$G$$ are conjugate. The authors answer some questions of Wielandt including the determination of all groups satisfying the $$\chi$$-Reduktionssatz in terms of their composition factors. Using the classification of finite simple groups, they prove that a finite group $$G$$ satisfies the $$\chi$$-Reduktionssatz if, and only if, all its $$\chi$$-maximal subgroups are conjugate, and furthermore, these two assertions are equivalent to an assertion on composition factors (which is too technical to state here). The authors also answer other questions in this program and the paper includes a fairly complete survey of known and unknown results in this subject. We direct the reader to the body of the paper itself for lots of details about the Wielandt program. Reviewer: Balasubramanian Sury (Bangalore) Locally finite simple groups whose non-abelian subgroups are pronormal https://zbmath.org/1522.20115 2023-12-07T16:00:11.105023Z "Brescia, Mattia" https://zbmath.org/authors/?q=ai:brescia.mattia "Trombetti, Marco" https://zbmath.org/authors/?q=ai:trombetti.marco Let $$G$$ be a group. A subgroup $$H$$ of $$G$$ is said to be pronormal (in $$G$$) if $$H$$ and $$H^{g}$$ are conjugate in $$\langle H, H^{g} \rangle$$ for any $$g \in G$$. A group $$G$$ is prohamiltonian if every non-pronormal subgroup of $$G$$ is abelian. In the article under review, the authors determine the finite simple groups which are prohamiltonian. As a corollary, they prove that the only prohamiltonian locally finite simple groups are the finite ones. Reviewer: Enrico Jabara (Venezia) Conjugacy classes of maximal cyclic subgroups https://zbmath.org/1522.20120 2023-12-07T16:00:11.105023Z "Bianchi, Mariagrazia" https://zbmath.org/authors/?q=ai:bianchi.mariagrazia "Camina, Rachel D." https://zbmath.org/authors/?q=ai:camina.rachel-deborah "Lewis, Mark L." https://zbmath.org/authors/?q=ai:lewis.mark-l "Pacifici, Emanuele" https://zbmath.org/authors/?q=ai:pacifici.emanuele Summary: In this paper, we study the number of conjugacy classes of maximal cyclic subgroups of a finite group $$G$$, denoted $$\eta(G)$$. First we consider the properties of this invariant in relation to direct and semi-direct products, and we characterize the normal subgroups $$N$$ with $$\eta(G/N)=\eta(G)$$. In addition, by applying the classification of finite groups whose nontrivial elements have prime order, we determine the structure of $$G/\langle G^-\rangle$$, where $$G^-$$ is the set of elements of $$G$$ generating non-maximal cyclic subgroups of $$G$$. More precisely, we show that $$G/\langle G^-\rangle$$ is either trivial, elementary abelian, a Frobenius group or isomorphic to $$A_5$$. Conjugacy classes of $$\pi$$-elements and nilpotent/abelian Hall $$\pi$$-subgroups https://zbmath.org/1522.20122 2023-12-07T16:00:11.105023Z "Nguyen N., Hung" https://zbmath.org/authors/?q=ai:nguyen-n.hung "Maróti, Attila" https://zbmath.org/authors/?q=ai:maroti.attila "Martínez, Juan" https://zbmath.org/authors/?q=ai:martinez.juan-jose|sepulcre.juan-matias|martinez.juan-manuel|martinez.juan-carlos|martinez.juan-f|martinez.juan-antonio|martinez.juan-pablo The probability $$\mathrm{Pr}(G)$$ that two uniformly and randomly chosen elements of a finite group $$G$$ commute is given by $$k(G)/|G|$$ where $$k(G)$$ denotes the number of conjugacy classes of $$G$$. The implications of the value of $$\mathrm{Pr}(G)$$ on the algebraic structure of $$G$$ have long been studied. For example, \textit{W. H. Gustafson} [Am. Math. Mon. 80, 1031--1034 (1973; Zbl 0276.60013)] proved that $$\mathrm{Pr}(G) > 5/8$$ if and only if $$G$$ is abelian. More recently, \textit{R. M. Guralnick} and \textit{G. R. Robinson} [J. Algebra 300, No. 2, 509--528 (2006; Zbl 1100.20045)] proved that if $$\mathrm{Pr}(G) > 1/p$$ then $$G$$ is nilpotent, where $$p$$ is the smallest prime divisor of $$|G|$$, and \textit{T. C. Burness} et al. [`On the commuting probability of $p$-elements in a finite group'', Preprint, \url{arXiv 2112.08681}] proved that if $$\mathrm{Pr}(G) > (p^2 + p-1)/p^3$$ then $$G$$ is abelian. \textit{A. Maróti} and \textit{H. N. Nguyen} [Arch. Math. 102, No. 2, 101--108 (2014; Zbl 1295.20016)] generalised the result of Gustafson [loc. cit.] in the following sense. Let $$\pi$$ be a set of primes and call an element of $$G$$ a $$\pi$$-element if its order is a $$\pi$$-number, that is its order is only divisible by primes in $$\pi$$. Let $$k_{\pi}(G)$$ denote the number of conjugacy classes of $$\pi$$-elements in $$G$$ and let $$|G|_{\pi}$$ denote the largest $$\pi$$-number that divides $$|G|$$. Set $$d_{\pi}(G) = k_{\pi}(G)/|G|_{\pi}$$. Maróti and Nguyen [loc. cit.] proved that if $$d_{\pi}(G) > 5/8$$ then $$G$$ has an abelian Hall $$\pi$$-subgroup. In the paper under review, the authors follow this theme and generalise the previously mentioned results as follows. Suppose $$p$$ is the smallest member of $$\pi$$. If $$d_{\pi}(G) > 1/p$$, then $$G$$ has a nilpotent Hall $$\pi$$-subgroup, and if $$d_{\pi}(G) > (p^2+p-1)/p^3$$ then $$G$$ has an abelian Hall $$\pi$$-subgroup. For $$p \neq 2$$, the authors reduce the problem to that of simple groups and use the classification of finite simple groups. For $$p=2$$, the classification can be avoided. The paper concludes with examples and the following open question: for $$\pi$$ a set of odd primes what is the exact lower bound for $$d_{\pi}(G)$$ to ensure the existence of a nilpotent Hall $${\pi}$$-subgroup? Reviewer: Rachel D. Camina (Cambridge) Groups with semi-partitions https://zbmath.org/1522.20125 2023-12-07T16:00:11.105023Z "Foguel, T." https://zbmath.org/authors/?q=ai:foguel.tuval-s "Mahmoudifar, A." https://zbmath.org/authors/?q=ai:mahmoudifar.ali "Moghaddamfar, A. R." https://zbmath.org/authors/?q=ai:moghaddamfar.ali-reza "Schmidt, J." https://zbmath.org/authors/?q=ai:schmidt.jan-philip|schmidt.joe|schmidt.jorge-f|schmidt.j-william|schmidt.jonas|schmidt.jens-georg|schmidt.jamie|schmidt.j-d|schmidt.jeannot|schmidt.joachim-w|schmidt.jorn-marc|schmidt.j-a|schmidt.j-w-jun|schmidt.johannes.2|schmidt.john-r|schmidt.jorg|schmidt.j-j|schmidt.jan-cornelius|schmidt.jorn|schmidt.jacob-peter|schmidt.julian|schmidt.jack|schmidt.jacob-w|schmidt.joachim-p|schmidt.jeffrey-r|schmidt.james|schmidt.jurgen.1|schmidt.johannes|schmidt.jens-m|schmidt.jan-hendrik|schmidt.jan-henrik|schmidt.jochen.1|schmidt.e-j-p-georg|schmidt.jochen.2|schmidt.joseph-h|schmidt.julia-k|schmidt.jeanette-p A \textit{group cover} is a collection of proper subgroups whose union is the group. A \textit{finite cover} $$\Pi$$ of a group $$G$$ is a finite collection of proper subgroups of $$G$$ such that $$G$$ is equal to the union of all of the members of $$\Pi$$. Such a cover is called \textit{minimal} if it has the smallest cardinality among all finite covers of $$G$$. The \textit{covering number} of $$G$$, denoted by $$\sigma(G)$$, is the number of subgroups in a minimal cover of $$G$$. A \textit{group partition} is a group cover in which the components have trivial pairwise intersections. A classification for groups admitting a group partition has been given in 1961 by \textit{R. Baer} [Math. Z. 75, 333--372 (1960/1961; Zbl 0103.01404)], \textit{O.H. Kegel} [Arch. Math. 12, 170--175 (1961; Zbl 0123.02505)] and \textit{M. Suzuki} [Arch. Math. 12, 241--254 (1961; Zbl 0107.25902)]. A \textit{finite partition} is a group partition with finitely many components. A finite partition is called \textit{minimal} if it has the smallest cardinality among all finite partitions of that group. The \textit{partition number} of a group $$G$$, denoted by $$\rho(G)$$, is the number of subgroups in a minimal partition of $$G$$. The authors generalize the concept of group partition in the following way: A group cover $$\Pi$$ of a group $$G$$ is called a \textit{semi-partition} of $$G$$ if the intersection of any three components of $$\Pi$$ is trivial. The group $$G$$ has a \textit{proper semi-partition} if it has a semi-partition which is not a partition. A \textit{finite semi-partition} is a semi-partition with finitely many components. The \textit{semi-partition number} of $$G$$, denoted by $$\rho_\sigma(G)$$, is defined by $\rho_\sigma(G)=\min\{|\Pi|\mid\Pi\text{ is a finite semi-partition of } G\}$ The main results of the paper are the following {Theorem A}. Every group having a semi-partition by proper normal (respectively, subnormal) subgroups is abelian (respectively, nilpotent). {Theorem B}. If $$G$$ is a finite simple group with partition, then $\sigma(G)\leq\rho_\sigma(G)\leq\rho(G)$ Reviewer: Mattia Brescia (Napoli) An upper bound for the nonsolvable length of a finite group in terms of its shortest law https://zbmath.org/1522.20134 2023-12-07T16:00:11.105023Z "Fumagalli, Francesco" https://zbmath.org/authors/?q=ai:fumagalli.francesco "Leinen, Felix" https://zbmath.org/authors/?q=ai:leinen.felix "Puglisi, Orazio" https://zbmath.org/authors/?q=ai:puglisi.orazio Every finite group $$G$$ has a normal series each of whose factors is either a solvable group or a direct product of non-abelian simple groups. The minimum number of nonsolvable factors, attained on all possible such series in $$G$$, is called the nonsolvable length $$\lambda(G)$$ of $$G$$. Another invariant that can be attached to any finite group $$G$$ is the length of the shortest law that holds in $$G$$, defined as $\nu(G)=\min \big \{ |w| \; \big | \; w \text{ is a nontrivial reduced law in } G \big \}.$ A first result proved in this paper is Theorem A: Let $$G$$ be any finite group. Then $$\lambda(G)< \nu(G)$$. Theorem A answers in the affirmative to a conjecture formulated by Laresen (see Conjecture 1.6 in [\textit{A. Bors} and \textit{A. Shalev}, J. Comb. Algebra 5, No. 2, 93--122 (2021; Zbl 1511.20055)]). The authors point out that their bound to the nonsolvable length can probably be improved: they propose the conjecture that $$\lambda(G) < O(\log(\nu(G)))$$. The proof of Theorem A follows from Theorem B: Let $$G$$ be a finite group of nonsolvable length $$\lambda(G)=n$$. Then, for every nontrivial word $$w=w(x_{1},\dots,x_{k})$$ of length at most $$n$$, there exists $$(g_{1}, \dots, g_{k}) \in G^{k}$$ such that $$w(g_{1},\dots, g_{k}) \not =1$$. In turn Theorem B is a consequence of the fact that partial subwords of $$w$$ give distinct cosets of a core-free subgroup in $$G$$ (see Theorem C). Finally, it should be noticed that Theorem C gives a positive answer for $$p=2$$ to a conjecture of \textit{E. I. Khukhro} and \textit{P. Shumyatsky} (Problem 1.2 in [Isr. J. Math. 207, Part 2, 507--525 (2015; Zbl 1333.20023)]). Reviewer: Egle Bettio (Venezia) Thompson subgroups and large abelian unipotent subgroups of Lie-type groups https://zbmath.org/1522.20203 2023-12-07T16:00:11.105023Z "Levchuk, Vladimir M." https://zbmath.org/authors/?q=ai:levchuk.vladimir-mikhailovich "Suleimanova, Galina S." https://zbmath.org/authors/?q=ai:suleimanova.galina-safiullanovna Summary: Let $$U$$ be a unipotent radical of a Borel subgroup of a Lie-type group over a finite field. For the classical types the Thompson subgroups and large abelian subgroups of the group $$U$$ were found to the middle 1980's. We complete a solution of well-known problem of their description for the exceptional Lie-types. Modular quasi-Hopf algebras and groups with one involution https://zbmath.org/1522.20214 2023-12-07T16:00:11.105023Z "Mason, Geoffrey" https://zbmath.org/authors/?q=ai:mason.geoffrey "Ng, Siu-Hung" https://zbmath.org/authors/?q=ai:ng.siu-hung Summary: In a previous paper [Dev. Math. 38, 229--246 (2014; Zbl 1320.16017)] the authors constructed a class of quasi-Hopf algebras $$D^\omega(G, A)$$ associated to a finite group $$G$$, generalizing the twisted quantum double construction. We gave necessary and sufficient conditions, cohomological in nature, that the corresponding module category $$\operatorname{Rep}( D^\omega(G, A))$$ is a modular tensor category. In the present paper we verify the cohomological conditions for the class of groups $$G$$ which \textit{contain a unique involution}, and in this way we obtain an explicit construction of a new class of modular quasi-Hopf algebras. We develop the basic theory for general finite groups $$G$$, and also a parallel theory concerned with the question of when $$\operatorname{Rep}( D^\omega(G, A))$$ is super-modular rather than modular. We give some explicit examples involving binary polyhedral groups and some sporadic simple groups. Conway subgroup symmetric compactifications redux https://zbmath.org/1522.81317 2023-12-07T16:00:11.105023Z "Baykara, Zihni Kaan" https://zbmath.org/authors/?q=ai:baykara.zihni-kaan "Harvey, Jeffrey A." https://zbmath.org/authors/?q=ai:harvey.jeffrey-a Summary: We extend the investigation in [\textit{J. A. Harvey} and \textit{G. W. Moore}, J. Phys. A, Math. Theor. 51, No. 35, Article ID 354001, 35 p. (2018; Zbl 1397.81250)] of special toroidal compactifications of heterotic string theory for which the half-BPS states provide representations of subgroups of the Conway group. We also explore dual descriptions of these theories and find that they are all linked to either F-theory or type IIA string theory on $$K3$$ surfaces with symplectic automorphism groups that are the same Conway subgroups as those of the heterotic dual. The matching with type IIA $$K3$$ dual theories includes both the matching of symmetry groups and a comparison between the Narain lattice on the heterotic side and the cohomology lattice on the type IIA side. We present twelve examples where we can identify a type IIA dual $$K3$$ orbifold theory as the dual description of the heterotic theory. In addition, we include a supplementary Mathematica package that performs most of the computations required for these comparisons.