Recent zbMATH articles in MSC 20D https://zbmath.org/atom/cc/20D 2022-07-25T18:03:43.254055Z Werkzeug A survey on enhanced power graphs of finite groups https://zbmath.org/1487.05119 2022-07-25T18:03:43.254055Z "Ma, Xuanlong" https://zbmath.org/authors/?q=ai:ma.xuanlong "Kelarev, Andrei" https://zbmath.org/authors/?q=ai:kelarev.andrei-v "Lin, Yuqing" https://zbmath.org/authors/?q=ai:lin.yuqing "Wang, Kaishun" https://zbmath.org/authors/?q=ai:wang.kaishun Summary: We survey known results on enhanced power graphs of finite groups. Open problems, questions and suggestions for future work are also included. Relative $$g$$-noncommuting graph of finite groups https://zbmath.org/1487.05123 2022-07-25T18:03:43.254055Z "Sharma, Monalisha" https://zbmath.org/authors/?q=ai:sharma.monalisha "Nath, Rajat Kanti" https://zbmath.org/authors/?q=ai:nath.rajat-kanti Summary: Let $$G$$ be a finite group. For a fixed element $$g$$ in $$G$$ and a given subgroup $$H$$ of $$G$$, the relative $$g$$-noncommuting graph of $$G$$ is a simple undirected graph whose vertex set is $$G$$ and two vertices $$G$$ and $$y$$ are adjacent if $$x \in H$$ or $$y \in H$$ and $$[x, y] \neq g,g^{-1}$$. We denote this graph by $$\Gamma_{H, G}^g$$. In this paper, we obtain computing formulae for degree of any vertex in $$\Gamma_{H, G}^g$$ and characterize whether $$\Gamma_{H, G}^g$$ is a tree, star graph, lollipop or a complete graph together with some properties of $$\Gamma_{H, G}^g$$ involving isomorphism of graphs. We also present certain relations between the number of edges in $$\Gamma_{H, G}^g$$ and certain generalized commuting probabilities of $$G$$ which give some computing formulae for the number of edges in $$\Gamma_{H, G}^g$$. Finally, we conclude this paper by deriving some bounds for the number of edges in $$\Gamma_{H, G}^g$$. Generalised Beauville groups https://zbmath.org/1487.14083 2022-07-25T18:03:43.254055Z "Carta, Ludo" https://zbmath.org/authors/?q=ai:carta.ludo "Fairbairn, Ben T." https://zbmath.org/authors/?q=ai:fairbairn.ben A Beauville group'' is a finite group that can be generated from two elements in two different ways, so $$G = \langle g_1, g_2 \rangle = \langle h_1, h_2 \rangle$$, in a way that no power nor of $$g_1$$, neither of $$g_2$$ nor of their product $$g_1g_2$$ other than identity is conjugated to any non-trivial power of $$h_1$$, nor of $$h_2$$ nor of their product $$h_1h_2$$. The interest in these groups arises from algebraic geometry, in particular from works by F. Catanese who, inspired by a construction of A. Beauville, showed how to associate to each such group an algebraic surface with interesting properties such as the rigidity of its complex structure. The study of groups with such properties then gave rise to a series of interesting researches, more properly of group theory, within which this article is placed. The authors consider in this article a generalization of the concept of Beauville group, as indicated by the title, in which they ask themselves the question of finding $$d$$ sets of generators with a similar property as above, i.e. that no element induced as above by the first set of generators is conjugated to elements induced as above by ALL the other $$d-1$$ together. If $$d$$ is the minimum for which the answer is affirmative, they call it the Beauville dimension''. The Beauville groups are therefore those of dimension $$2$$. The generalized Beauville groups have $$d \geq 2$$, while $$d = 1$$ is set when the answer is negative for each $$d \geq 2$$. This generalization is also motivated by algebraic geometry, since $$d$$ such sistems of generators induce in a similar way an interesting complex manifold of dimension $$d$$. The main result is the classification of the Beauville dimension of all finite groups of order less than or equal to $$1023$$, contained in the tables at the end of the article. Reviewer: Roberto Pignatelli (Trento) Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VI. Suzuki and Ree groups https://zbmath.org/1487.16033 2022-07-25T18:03:43.254055Z "Carnovale, Giovanna" https://zbmath.org/authors/?q=ai:carnovale.giovanna "Costantini, Mauro" https://zbmath.org/authors/?q=ai:costantini.mauro The paper under review is the sixth in a series of papers published under the common title Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type''. The authors of the five previous papers were \textit{N. Andruskiewitsch} et al. [J. Algebra 442, 36--65 (2015; Zbl 1338.16034); Commun. Contemp. Math. 18, No. 4, Article ID 1550053, 35 p. (2016; Zbl 1357.16047); Rev. Mat. Iberoam. 33, No. 3, 995--1024 (2017; Zbl 1377.16024); Algebr. Represent. Theory 23, No. 3, 621--655 (2020; Zbl 1442.16031); Manuscr. Math. 166, No. 3--4, 605--647 (2021; Zbl 07428820)]. In the present paper, the authors discuss the following conjecture: Let $$G$$ be a finite simple non-abelian group. Then the Nichols algebra $$\mathcal{B}(V)$$ is infinite-dimensional for every complex Yetter-Drinfeld module $$V$$ of $$G$$. Thus the only finite-dimensional complex pointed Hopf algebra whose the group of grouplike elements is $$G$$ is the group algebra $$\mathbb{C}G$$. The authors prove that this conjecture holds if $$G$$ is a simple Suzuki or Ree group. Reviewer: Małgorzata E. Hryniewicka (Białystok) Rings of congruence preserving functions. II https://zbmath.org/1487.16045 2022-07-25T18:03:43.254055Z "Maxson, C. J." https://zbmath.org/authors/?q=ai:maxson.carlton-j "Meyer, J. H." https://zbmath.org/authors/?q=ai:meyer.johan-h Let $$G:=\langle G, +, \cdot \rangle$$ be a finite group written additively but not necessarily abelian, with neutral element $$0$$. Let $$M_0(G)$$ denote the near-ring of zero-preserving functions on $$G$$ under the operations of pointwise addition and function composition. We consider subnear-rings $$P_0(G)$$, the near-ring of polynomial functions on $$G$$, and $$C_0(G)$$, the near-ring of congruence preserving functions on $$G$$. In this paper, for several classes of special $$p$$-groups $$G$$, of exponent $$p$$, $$p > 2$$, the authors show that the near-ring, $$C_0(G)$$, of congruence preserving functions on $$G$$ is a ring if and only if $$G$$ is a $$1$$-affine complete group. For Part I see [the authors, Monatsh. Math. 187, No. 3, 531--542 (2018; Zbl 1430.16044)]. Reviewer: Bijan Davvaz (Yazd) Supercharacter theories for Sylow $$p$$-subgroups of the Ree groups https://zbmath.org/1487.20004 2022-07-25T18:03:43.254055Z "Sun, Yujiao" https://zbmath.org/authors/?q=ai:sun.yujiao Summary: We determine a supercharacter theory for Sylow $$p$$-subgroups $$^2\mathbf{G}_2^{\mathrm{syl}}(3^{2m+1})$$ of the Ree groups $$^2\mathbf{G}_2(3^{2m+1})$$, calculate the conjugacy classes of $${}^2\mathbf{G}_2^{\mathrm{syl}}(3^{2m+1})$$, and establish the character table of $${}^2\mathbf{G}_2^{\mathrm{syl}}(3)$$.'' Reviewer's remarks: In the representation theory of finite groups there recently appeared a new theory, namely that of the so-called supercharacter theory [\textit{P. Diaconis} and \textit{I. M. Isaacs}, Trans. Am. Math. Soc. 360, No. 5, 2359--2392 (2008; Zbl 1137.20008)]. This theory is of use in the study of representation theory of finite groups of Lie type, in order to determine irreducible representations and irreducible characters in the ordinary-known'' theory of groups of Lie type. In the paper under review, the author establishes (technical as the working-out has been surely) the conjugacy classes of the Sylow 3-subgroups of the simple Ree groups as well as the character tables of these Sylow 3-subgroups. The reader might also consult the reviewer's work [J. Reine Angew. Math. 309, 156--175 (1979; Zbl 0409.20011)]. In it, the character theory of the normalizes $$N_G(P)$$ of a Sylow 3-subgroup $$P$$ in a finite simple group $$G$$ of Ree type has been elaborated as well as the interrelated connections between the irreducible characters of $$N_G(P)$$ and $$P$$. That source has not been mentioned in the list of references inside the paper under review. As such, the reviewer regards the author's paper a useful expansion of, and a different view on, his 1979-paper. Reviewer: Robert W. van der Waall (Huizen) Groups that have a partition by commuting subsets https://zbmath.org/1487.20007 2022-07-25T18:03:43.254055Z "Foguel, Tuval" https://zbmath.org/authors/?q=ai:foguel.tuval-s "Hiller, Josh" https://zbmath.org/authors/?q=ai:hiller.josh "Lewis, Mark L." https://zbmath.org/authors/?q=ai:lewis.mark-l "Moghaddamfar, Alireza" https://zbmath.org/authors/?q=ai:moghaddamfar.ali-reza This paper investigates problems related to groups with abelian partitions. Let $$G$$ be a finite group. Then the group $$G$$ has an abelian partition (AP-group) if there exists a partition of $$G$$ into commuting subsets $$A_1,A_2,\dots,A_n$$ of $$G$$ such that $$|A_i| \ge 2$$ for each $$i=1,2,\dots,n.$$ If $$G$$ is not AP-group, we say $$G$$ is an NAP-group. One way to study the properties of a group is to associate a graph with the group and use the properties of the graph to understand the group. For a subset $$X$$ of $$G$$, the commuting graph on the set $$X$$ is defined as the graph which has $$X$$ as its vertex set and for $$g,h \in X$$ (distinct), $$g$$ and $$h$$ is joined by edge iff $$gh=hg.$$ For $$X=G,$$ the commuting graph is denoted by $$\Delta(G).$$ AP-groups were first introduced by \textit{A. Mahmoudifar} and \textit{A. R. Moghaddamfar} [Commun. Algebra 45, No. 7, 3159--3165 (2017; Zbl 1368.05069)], where the authors classified AP-groups that have abelian partition for $$n=2,3.$$ They introduced the concept of commuting graph while classification. Later many mathematician such as M.Akbari, A.R. Moghaddamfar, S. Dolfi, M. Herzong, E. Jabara closely studied the relationship between AP-group and commuting graph. Also it has been observed that the dihedral groups of order $$2n,$$ for $$n\ge 3$$ odd, are NAP-groups and that certain direct product of dihedral groups are NAP-groups. In this paper, the authors show that the class of AP-groups and class of NAP-groups are large'' in the sense that every finite group is isomorphic to a subgroup of a group with an abelian partition and also isomorphic to a group with no abelian partition. They also find the bounds for the minimum number of partitions for several families of groups which admit an abelian partition. Finally, they examine how the size of a partition with the minimum number of parts behaves with respect to the direct product. Their study forces us to think about the complete classification of AP-groups, which remains mystery. Also this study leave a natural question which is: are there any NAP-group that are simple? Reviewer: Eshita Mazumdar (Kolkata) Another criterion for solvability of finite groups https://zbmath.org/1487.20008 2022-07-25T18:03:43.254055Z "Herzog, Marcel" https://zbmath.org/authors/?q=ai:herzog.marcel "Longobardi, Patrizia" https://zbmath.org/authors/?q=ai:longobardi.patrizia "Maj, Mercede" https://zbmath.org/authors/?q=ai:maj.mercede Let $$o(G)$$ be the average order of the elements of a finite group $$G$$. It was conjectured by \textit{E. I. Khukhro} et al. [J. Algebra 569, 1--11 (2021; Zbl 1462.20010)] that if $$o(G)<o(A_5)$$, where $$A_5$$ is the alternating group on $$5$$ letters, then $$G$$ is solvable. The main result of this paper shows that this conjecture holds. Furthermore, if $$o(G)=o(A_5)$$, then $$G\cong A_5$$. An important part of the proof is to show that if $$G$$ is simple, then $$o(G)\geq o(A_5)$$. This is achieved in two steps. Counting involutions, the authors show that they may assume that the primes that divide $$|G|$$ do not exceed 13. The simple groups that satisfy this property are considered case by case. Reviewer: Alexander Moretó (Valencia) Centric linking systems of locally finite groups https://zbmath.org/1487.55024 2022-07-25T18:03:43.254055Z "Molinier, Rémi" https://zbmath.org/authors/?q=ai:molinier.remi The concept of a centric linking system for a finite group was introduced by \textit{C. Broto} et al. [Invent. Math. 151, No. 3, 611--664 (2003; Zbl 1042.55008)]. A centric linking system of a finite group $$G$$ allows one to associate a classifying space to the $$p$$-fusion in $$G$$. In a later paper [\textit{C. Broto} et al., J. Am. Math. Soc. 16, No. 4, 779--856 (2003; Zbl 1033.55010)] the authors generalised the concept to abstract $$p$$-fusion systems. \textit{A. Chermak} showed that to any abstract fusion system over a finite $$p$$-group there is an associated centric linking system that is unique up to isomorphism, and hence a unique classifying space [Acta Math. 211, No. 1, 47--139 (2013; Zbl 1295.20021)]. In the current paper the concept of a centric linking system is generalised to locally finite groups. The paper proposes natural definition of a Sylow $$p$$-subgroup of a locally finite group and a generalisation of the concept of $$p$$-centricity of subgroups. This allows a straightforward definition of a centric linking system $$\mathcal{L}_S^c(G)$$ similar to the finite case. The main theorem then states that if $$G$$ is a locally finite group that admits a Sylow $$p$$-subgroup $$S$$, and if the non-$$p$$-centric subgroups in $$S$$ fall into finitely many conjugacy classes, then the $$p$$-completed nerve of $$\mathcal{L}_S^c(G)$$ is homotopy equivalent to the $$p$$-completion of $$BG$$. The work here generalises some aspects of the theory of fusion and linking systems over discrete $$p$$-toral groups [\textit{C. Broto} et al., Geom. Topol. 11, 315--427 (2007; Zbl 1135.55008)]. Reviewer: Ran Levi (Aberdeen)