Recent zbMATH articles in MSC 20Dhttps://zbmath.org/atom/cc/20D2021-05-28T16:06:00+00:00WerkzeugOn \(\pi\)-solvable irreducible linear groups with a Hall TI-subgroup of odd order. II.https://zbmath.org/1459.200402021-05-28T16:06:00+00:00"Yadchenko, A. A."https://zbmath.org/authors/?q=ai:yadchenko.a-aSummary: \(\pi\)-solvable finite irreducible complex linear groups whose Hall \(\pi\)-subgroups are TI-subgroups and the degree of the group is small with respect to the order of such subgroup, are investigated. This is the second one in the series of the author's papers aimed on determining the possible values of the degree \(n\) if a Hall \(\pi\)-subgroup \(H\) is not normal, \(|H|\) is odd, and \(n<2|H|\). The proof of a theorem that yields the complete list of these values is continued, it was started in the first paper of the series [the author, Tr. Inst. Mat., Minsk 16, No. 2, 118--130 (2008; Zbl 1165.20319)]. A number of properties of a minimal counterexample to the theorem is established.Bogomolov multipliers of some groups of order \(p^6\).https://zbmath.org/1459.130072021-05-28T16:06:00+00:00"Chen, Yin"https://zbmath.org/authors/?q=ai:chen.yin.4|chen.yin.3|chen.yin.2|chen.yin|chen.yin.1|chen.yin.5"Ma, Rui"https://zbmath.org/authors/?q=ai:ma.ruiThe present article investigates the Bogomolov multiplier for the groups of order \(p^6\), where \(p\geq 3\) is a prime. The Bogomolov multiplier is a group-theoretical invariant introduced as an obstruction to the rationality problem in algebraic geometry. Let \(K\) be a field, \(G\) be a finite group and \(V\) be a faithful representation of \(G\) over \(K\). Then there is natural action of \(G\) upon the field of rational functions \(K(V)\). The rationality problem (also known as Noether's problem) asks whether the field of \(G\)-invariant functions \(K(V)^G\) is rational (purely transcendental) over \(K\). In [Invent. Math. 77, 71--84 (1984; Zbl 0546.14014)], \textit{D. J. Saltman} found examples of metabelian groups \(G\) of order \(p^9\) such that \(\mathbb C(V)^G\) is not stably rational over \(\mathbb C\). His main method was an application of the unramified cohomology group \(H_{nr}^2(\mathbb C(V)^G,\mathbb Q/\mathbb Z)\) as an obstruction. In [Math. USSR, Izv. 30, No. 3, 455--485 (1988; Zbl 0679.14025); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 3, 485--516 (1987)],\textit{F. A. Bogomolov} proved that \(H_{nr}^2(\mathbb C(V)^G,\mathbb Q/\mathbb Z)\) is canonically isomorphic to \[ B_0(G)=\bigcap_{A}\ker\{\mathrm{res}_G^A:H^2(G,\mathbb Q/\mathbb Z)\to H^2(A,\mathbb Q/\mathbb Z)\} \] where \(A\) runs over all the bicyclic subgroups of \(G\) (a group \(A\) is called bicyclic if \(A\) is either a cyclic group or a direct product of two cyclic groups). The group \(B_0(G)\) is a subgroup of the Schur multiplier \(H^2(G,\mathbb Q/\mathbb Z)\), and \textit{B. Kunyavskiĭ} [Prog. Math. 282, 209--217 (2010; Zbl 1204.14006)] called it the \(Bogomolov\) \(multiplier\) of \(G\). Thus the vanishing of the Bogomolov multiplier is an obstruction to Noether's problem.
The Bogomolov multipliers for groups of order \(p^k\) for \(k\leq 5\) have been recently calculated by various authors. The purpose of the present article is to study the vanishing property of \(B_0(G)\) for a nonabelian group \(P\) of order \(p^6\), where \(p\geq 3\). The authors use the well known classification by \textit{R. James} of these groups [Math. Comput. 34, 613--637 (1980; Zbl 0428.20013)]. The groups are distributed in the so called isoclinic families \(\Phi_2,\dots\). In [Ars Math. Contemp. 7, No. 2, 337--340 (2014; Zbl 1327.14099)], \textit{P. Moravec} proved that if two finite groups \(G\) and \(H\) are isoclinic, then their respective Bogomolov multipliers are also isomorphic. Therefore, it is enough to calculate the Bogomolov multiplier for only one group from each isoclinic family. The main result of the present article is
Theorem 1.1. Let \(p\geq 3\) be a prime and \(\{\Phi_k| 2\leq k\leq 43\}\) be the set of isoclinic families of groups of order \(p^6\). Then \(B_0(\Phi_k)\) is not zero for \(k\in\Delta :=\{10, 18, 20, 21, 36, 38, 39\}\) and \(B_0(\Phi_k)=0\) for \(k\in \{2, 3,\dots, 43\}\setminus (\Delta\cup \{15, 28, 29\})\).
The authors also make a conjecture that \(B_0(\Phi_k)=0\) for \(k=15,28,29\). They claim that computer calculations confirm it, but they have no strict proof.
Reviewer: Ivo M. Michailov (Shumen)Towards the classification of finite simple groups with exactly three or four supercharacter theories.https://zbmath.org/1459.200012021-05-28T16:06:00+00:00"Ashrafi, A. R."https://zbmath.org/authors/?q=ai:ashrafi.ali-reza"Koorepazan-Moftakhar, F."https://zbmath.org/authors/?q=ai:koorepazan-moftakhar.fatemehCentral intersections of element centralizers.https://zbmath.org/1459.200282021-05-28T16:06:00+00:00"Brough, Julian"https://zbmath.org/authors/?q=ai:brough.julian-m-aFinite groups which have 20 elements of maximal order.https://zbmath.org/1459.200252021-05-28T16:06:00+00:00"Han, Zhangjia"https://zbmath.org/authors/?q=ai:han.zhangjia"Xie, Longjiang"https://zbmath.org/authors/?q=ai:xie.longjiang"Guo, Pengfei"https://zbmath.org/authors/?q=ai:guo.pengfeiSummary: The structure of finite groups is widely used in various fields and has a great influence on various disciplines. The object of this article is to classify these groups \(G\) whose number of elements of maximal order of \(G\) is 20.Finite groups with \(\mathbb{P}\)-subnormal biprimary subgroups.https://zbmath.org/1459.200232021-05-28T16:06:00+00:00"Knyagina, V. N."https://zbmath.org/authors/?q=ai:knyagina.viktoriya-nikolaevnaSummary: In this paper we study finite groups with \(\mathbb{P}\)-subnormal biprimary dispersive subgroups. We prove that a group all of whose biprimary \(p\)-closed \(pd\)-subgroups are \(\mathbb{P}\)-subnormal is \(p\)-solvable, where \(p\) is the largest prime divisor of the order of the group. We also prove that a group with biprimary \(2\)-nilpotent \(\mathbb{P}\)-subnormal \(2d\)-subgroups is solvable.On the intersection of maximal supersoluble subgroups of a finite group.https://zbmath.org/1459.200132021-05-28T16:06:00+00:00"Guo, Wenbin"https://zbmath.org/authors/?q=ai:guo.wenbin.1"Skiba, Alexander N."https://zbmath.org/authors/?q=ai:skiba.alexander-nSummary: The hyper-generalized-center \(\mathrm{genz}^*(G)\) of a finite group \(G\) coincides with the largest term of the chain of subgroups \(1=Q_0(G)\leq Q_1(G)\leq\dots\leq Q_t(G)\leq\dots\) where \(Q_i(G)/Q_{i-1}(G)\) is the subgroup of \(G/Q_{i-1}(G)\) generated by the set of all cyclic \(S\)-quasinormal subgroups of \(G/Q_{i-1}(G)\). It is proved that for any finite group \(A\), there is a finite group \(G\) such that \(A\leq G\) and \(\mathrm{genz}^*(G)\neq\operatorname{Int}_{\mathfrak{U}}(G)\).Sylow properties of finite groups.https://zbmath.org/1459.200202021-05-28T16:06:00+00:00"Vedernikov, V. A."https://zbmath.org/authors/?q=ai:vedernikov.victor-aSummary: Let \(\mathfrak{F}\) be a non-empty class of finite groups, and \(\pi\) be some set of prime numbers. An \(S_\pi\)-subgroup of group \(G\) that belongs to the class \(\mathfrak{F}\) is called an \(S_\pi(\mathfrak{F})\)-subgroup of \(G\). \(C_{\pi}(\mathfrak{F})\) is the class of all groups \(G\) that have \(S_{\pi}(\mathfrak{F})\)-subgroups, and any two \(S_{\pi}(\mathfrak{F})\)-subgroups of \(G\) are conjugate in \(G\); \(D_\pi(\mathfrak{F})\) is the class of all \(C_\pi(\mathfrak{F})\)-groups \(G\) in which every \(\mathfrak{F}_\pi\)-subgroup is contained in some \(S_\pi(\mathfrak{F})\)-subgroup of \(G\). In this paper the new D-theorems are obtained, a number of properties of \(D_\pi(\mathfrak{F})\)-groups, and \(C_\pi(\mathfrak{F})\)-groups are established.Non-radicality of the class \(E_\pi\)-groups.https://zbmath.org/1459.200192021-05-28T16:06:00+00:00"Vdovin, E. P."https://zbmath.org/authors/?q=ai:vdovin.evgeny-petrovitch"Revin, D. O."https://zbmath.org/authors/?q=ai:revin.danila-olegovitchSummary: In the note it is proven that the class \(E_\pi\) of all finite groups possessing \(\pi\)-Hall subgroups, for given set of primes \(\pi\), is not radical (i.e., the product of normal \(E_\pi\)-subgroups of a finite group is not necessary an \(E_\pi\)-group).On subgroup lattices of subnormal type in finite groups.https://zbmath.org/1459.200222021-05-28T16:06:00+00:00"Vasil'ev, A. F."https://zbmath.org/authors/?q=ai:vasilev.alexander-fedorovich"Khalimonchik, I. N."https://zbmath.org/authors/?q=ai:khalimonchik.i-nSummary: In this paper we obtain a description of hereditary transitive subgroup Skiba functors and hereditary saturated formations which are a lattice in the class of all finite soluble groups with nilpotent length less than or equal \(k\), where \(k\ge 3\).Some finiteness questions about formations.https://zbmath.org/1459.200122021-05-28T16:06:00+00:00"Burichenko, V. P."https://zbmath.org/authors/?q=ai:burichenko.vladimir-pSummary: In this paper we give negative answers to some open ``finiteness questions'' related to local and Baer-local formations of finite groups. Relevant examples are constructed explicitly. In particular, we describe a pair of local formations \((\mathfrak{F},\mathfrak{H})\) such that the set of all local formations that are contained in \(\mathfrak{F}\) and not contained in \(\mathfrak{H}\) has no minimal element.On a problem of Praeger and Schneider.https://zbmath.org/1459.200292021-05-28T16:06:00+00:00"Bettio, Egle"https://zbmath.org/authors/?q=ai:bettio.egle"Jabara, Enrico"https://zbmath.org/authors/?q=ai:jabara.enricoIn the paper under review, the authors answer affirmatively to a question of \textit{C. E. Praeger} and \textit{C. Schneider} [Isr. J. Math. 228, No. 2, 1001--1023 (2018; Zbl 06976757)] by proving the following result. Let \(\alpha\) and \(\beta\) be automorphisms of a group \(G\). Then \(G\) can be embedded in a group \(\tilde G\) admitting automorphisms \(\tilde\alpha\) and \(\tilde\beta\) such that the position \[\tilde g\mapsto(\tilde g^{-1}\tilde g^{\tilde\alpha},\tilde g^{-1}\tilde g^{\tilde\beta})\] defines a surjective map from \(\tilde G\) onto \(\tilde G\times\tilde G\) with \(\tilde\alpha|G=\alpha\) and \(\tilde\beta|G=\beta\). Moreover, the cardinality of \(\tilde G\) is the largest between \(\aleph_0\) and the cardinality of \(G\), and if \(G\) is countable then \(\tilde G\) can be chosen to be simple.
Reviewer: Francesco de Giovanni (Napoli)Finite groups of rank two which do not involve \(\mathrm{Qd}(p)\).https://zbmath.org/1459.200182021-05-28T16:06:00+00:00"Kızmaz, Muhammet Yasir"https://zbmath.org/authors/?q=ai:kizmaz.muhammet-yasir"Yalçın, Ergün"https://zbmath.org/authors/?q=ai:yalcin.ergunA \(p\)-group is said to be of rank \(k\) if the largest possible order of an elementary abelian subgroup of the group is \(p^k\). Further, the \(p\)-rank of a finite group \(G\) is \(k\) if a Sylow \(p\)-subgroup of \(G\) is of rank \(k\). The invariant \(p\)-rank of a finite group is conjecturally related to the minimum product of spheres on which the finite group can act freely.
Let \(Qd(p) := (\mathbb{Z}/p \times \mathbb{Z}/p) \rtimes SL(2, p)\), where \(\mathrm{SL}(2, p)\) acts on \(\mathbb{Z}/p \times \mathbb{Z}/p\) in the usual manner with \(\mathbb{Z}/p \times \mathbb{Z}/p\) viewed as a two-dimensional vector space over the field of \(p\)-elements. A finite group \(G\) is said to involve \(Qd(p)\) if there exist subgroups \(K \triangleleft H \le G\) such that \(H/K \cong Qd(p)\). Further, \(G\) is said to \(p'\)-involve \(Qd(p)\) if there exist \(K \triangleleft H \le G\) such that \(K\) has order coprime to \(p\) and \(H/K \cong Qd(p)\). If a group \(p'\)-involves \(Qd(p)\), then obviously it involves \(Qd(p)\), but the converse does not hold in general.
One of the main results of the paper is that these two conditions are equivalent for finite groups with \(p\)-rank equal to two, where \(p > 3\). The result is no longer true for \(p =2, 3\). The main ingredient in the proof is the classification of \(p\)-groups of rank two. The result allows the authors to use a version of Glauberman's ZJ-theorem to give a more direct construction of finite group actions on mod-\(p\) homotopy spheres.
Reviewer: Mahender Singh (Sahibzada Ajit Singh Nagar)Towards constructing fully homomorphic encryption without ciphertext noise from group theory.https://zbmath.org/1459.941362021-05-28T16:06:00+00:00"Nuida, Koji"https://zbmath.org/authors/?q=ai:nuida.kojiSummary: \textit{R. Ostrovsky} and \textit{W. E. Skeith III} [Lect. Notes Comput. Sci. 5157, 379--396 (2008; Zbl 1183.94043)] 1 year earlier than Gentry's pioneering ``bootstrapping'' technique for the first fully homomorphic encryption (FHE) scheme, had suggested a completely different approach towards achieving FHE. They showed that the \(\mathsf{NAND}\) operator can be realized in some non-commutative groups; consequently, homomorphically encrypting the elements of the group will yield an FHE scheme, without ciphertext noise to be bootstrapped. However, no observations on how to homomorphically encrypt the group elements were presented in their paper, and there have been no follow-up studies in the literature. The aim of this paper is to exhibit more clearly what is sufficient and what seems to be effective for constructing FHE schemes based on their approach. First, we prove that it is sufficient to find a surjective homomorphism \(\pi:\widetilde{G}\rightarrow G\) between finite groups for which bit operators are realized in \(G\) and the elements of the kernel of \(\pi\) are indistinguishable from the general elements of \(\widetilde{G}\). Secondly, we propose new methodologies to realize bit operators in some groups \(G\). Thirdly, we give an observation that a naive approach using matrix groups would never yield secure FHE due to an attack utilizing the ``linearity'' of the construction. Then we propose an idea to avoid such ``linearity'' by using combinatorial group theory. Concretely realizing FHE schemes based on our proposed framework is left as a future research topic.
For the entire collection see [Zbl 1457.94002].On finite groups with generalized subnormal critical subgroups.https://zbmath.org/1459.200242021-05-28T16:06:00+00:00"Semenchuk, V. N."https://zbmath.org/authors/?q=ai:semenchuk.vladimir-n"Velesnitskiĭ, V. F."https://zbmath.org/authors/?q=ai:velesnitskii.v-fSummary: This work is devoted to studying of the structure of finite groups with critical generalized subnormal subgroups.On irreducible linear groups of prime-power degree.https://zbmath.org/1459.200392021-05-28T16:06:00+00:00"Yadchenko, A. A."https://zbmath.org/authors/?q=ai:yadchenko.a-aSummary: Let \(\Gamma=AG\) be a finite group, \(G\triangleleft\Gamma\), \((|A|,|G|)=1\), \(C_G(a)=C_G(A)\) for each element \(a\in A^{\#}\), and let the subgroup \(A\) have a nonprimary odd order and be not normal in \(\Gamma\). Assume that \(\chi\) is an irreducible complex character of \(G\) that is invariant for at least one non-unity element of \(A\) and \(\chi(1)<2|A|\). Then it is proved that \(G=O_q(G)C_G(A)\) and \(\chi(1)\) is a power of a prime \(q\). Furthermore, if \(G\) is not solvable, then \(\chi(1)=2(|A|-1)\) and \(C_G(A)/Z(\Gamma)\cong \mathrm{PSL}(2,5)\).On solvable groups whose Sylow subgroups are either abelian or extraspecial.https://zbmath.org/1459.200172021-05-28T16:06:00+00:00"Gritsuk, D. V."https://zbmath.org/authors/?q=ai:gritsuk.d-v"Monakhov, V. S."https://zbmath.org/authors/?q=ai:monakhov.victor-stepanovichSummary: A \(p\)-group \(G\) is called extraspecial if its derived subgroup, center and Frattini subgroup are groups of order \(p\). We consider the solvable groups whose Sylow subgroups are either abelian or extraspecial. It is proved that derived length is at most \(2\cdot|\pi(G)|\) and nilpotent length is at most \(2+|\pi(G)|\).On permutability of \(n\)-maximal subgroups with \(p\)-nilpotent Schmidt subgroups.https://zbmath.org/1459.200162021-05-28T16:06:00+00:00"Knyagina, V. N."https://zbmath.org/authors/?q=ai:knyagina.viktoriya-nikolaevnaSummary: A Schmidt group is a finite nonnilpotent group in which every proper subgroup is nilpotent. Fix a positive integer \(n\). Let \(G\) be a solvable group. Suppose that each \(n\)-maximal subgroup of \(G\) is permutable with every \(p\)-nilpotent Schmidt subgroup. We prove that if \(n\in\{1,2,3\},\) then \(G/F(G)\) is \(p\)-closed, where \(F(G)\) is the Fitting subgroup of \(G\).On some formations closed under taking wreath products.https://zbmath.org/1459.200142021-05-28T16:06:00+00:00"Kamornikov, S. F."https://zbmath.org/authors/?q=ai:kamornikov.sergey-fedorovichSummary: We construct some series of subgroup-closed saturated formations \(\mathfrak{F}\) satisfying the following properties: 1) \(\mathfrak{F}\) is a proper subformation of \(\mathfrak{E}_\pi\) where \(\pi=\operatorname{char}(\mathfrak{F})\); 2) if \(G\in\mathfrak{F}\), then there exists a prime \(p\) (depending on the group \(G\)) such that the wreath product \(C_p\wr G\) belongs to \(\mathfrak{F}\) where \(C_p\) is the cyclic group of order \(p\). Thus an affirmative answer is obtained to Problem 18.9 from the Kourovka Notebook.On \(p\)-locally \(N\)-closed formations of finite groups.https://zbmath.org/1459.200152021-05-28T16:06:00+00:00"Rodionov, A. A."https://zbmath.org/authors/?q=ai:rodionov.an-a|rodionov.aleksandr-a"Shemetkov, L. A."https://zbmath.org/authors/?q=ai:shemetkov.leonid-aSummary: All groups considered are finite. A formation \(\mathfrak{F}\neq\emptyset\) is called locally \(N\)-closed (\(N\)-closed) in a group class \(\mathfrak{X}\), if the following assertion holds: if \(G\in\mathfrak{X}\) and \(P\leq Z_{\mathfrak{F}}(N_G(P))\) (\(N_G(P)\in \mathfrak{F}\) respectively) for every Sylow subgroup \(P\neq1\) in \(G\), then \(G\in\mathfrak{F}\). It is proved that in the soluble universe, every hereditary saturated locally \(N\)-closed non-empty formation is \(N\)-closed. It is proved that the formation of all supersoluble groups is \(N\)-closed in the class of all soluble groups with \(p\)-length \(\leq1\) for every prime \(p,\) and is not \(N\)-closed in the class of all soluble groups with \(p\)-length \(\leq2\) for every prime \(p\). The authors also consider \(p\)-locally \(N\)-closed formations.Periodic groups with prescribed element orders.https://zbmath.org/1459.200332021-05-28T16:06:00+00:00"Mazurov, V. D."https://zbmath.org/authors/?q=ai:mazurov.victor-danilovichSummary: Some recent results obtained with the participation of the author are discussed.
1. Known results on the structure of periodic groups with given spectra.
2. On groups of period 24.
3. The local finiteness of some groups with elementary abelian centralizers of involutions.
4. Recognition of finite simple groups by orders and spectra.Filters compatible with isomorphism testing.https://zbmath.org/1459.200312021-05-28T16:06:00+00:00"Maglione, Joshua"https://zbmath.org/authors/?q=ai:maglione.joshuaThere are several general methods to associate a Lie ring to a nilpotent group. A common theme is to make group-theoretic problems easier by employing linear algebra in the context of the Lie ring.
A recent development generalizes approaches from \textit{W. Magnus} [Ann. Math. (2) 52, 111--126 (1950; Zbl 0037.30401); J. Reine Angew. Math. 182, 142--149 (1940; Zbl 0025.24201; JFM 66.1208.03)] and \textit{M. Lazard} [Ann. Sci. Éc. Norm. Supér. (3) 71, 101--190 (1954; Zbl 0055.25103)]. \textit{J. B. Wilson} [J. Group Theory 16, No. 6, 875--897 (2013; Zbl 1298.20051)] defined filters as a means to allow refinements
of the well-studied upper and lower central series associated to nilpotent groups, while still connected to a graded Lie ring. In the paper under review, the authors describe a process of lifting isomorphisms between the associated Lie rings to (potential) isomorphisms between the groups, which has applications to isomorphism testing of finite \(p\)-groups. In detail, the authors obtain the following theorem:
Theorem. If \(\Phi\) is a finite, inertia-free, faithful-filter functor, then all isomorphisms between groups \(G\) and \(H\) are lifts of \(M\)-graded isomorphisms between \(L(\Phi(G))\) and \(L(\Phi(H))\).
Reviewer: Yangming Li (Guangzhou)Sufficient conditions for the solvability of a finite group.https://zbmath.org/1459.200112021-05-28T16:06:00+00:00"Bastos, R."https://zbmath.org/authors/?q=ai:bastos.rui|bastos.rafaela|bastos.raimundo|bastos.rogerio-cid|bastos.raimundo-a|bastos.ronaldo-r"Matos, H."https://zbmath.org/authors/?q=ai:matos.helder|matos.henrique-a"Seimetz, R."https://zbmath.org/authors/?q=ai:seimetz.rA famous theorem of [\textit{H. Wielandt}, Ill. J. Math. 2, 611--618 (1958; Zbl 0084.02904)] shows that every finite group which is factorized by two nilpotent subgroups of coprime orders is soluble. Wielandt's result [loc. cit.] was later improved by \textit{O. H. Kegel} [Arch. Math. 12, 90--93 (1961; Zbl 0099.01401)],who removed the coprime order assumption on the factors.
In the paper under review, the authors provide the following interesting generalization of the theorem of Wielandt: if a finite group \(G=HK\) is the product of two subgroups \(H\) and \(K\) of coprime orders, then \(G\) is soluble, provided that \(H\) is \(2\)-nilpotent and \(K\) is nilpotent of odd order. The authors leave as an open question whether the coprimality assumption can be omitted in this case. A suitable easy example shows the \(2\)-nilpotency assumption on \(H\) cannot be replaced by \(p\)-nilpotency for a prime number \(p\) which does not divide the order of \(K\).
Reviewer: Francesco de Giovanni (Napoli)Local nearrings on finite non-abelian 2-generated \(p\)-groups.https://zbmath.org/1459.160412021-05-28T16:06:00+00:00"Raievska, I. Yu."https://zbmath.org/authors/?q=ai:raevska.iryna-y"Raievska, M. Yu."https://zbmath.org/authors/?q=ai:raevska.maryna-yA local nearring is a nearring with identity for which the subset of non-invertible elements forms a subgroup of the underlying additive group. The authors investigate the structure of the additive subgroup of such a nearrings.
Let \(G\) be a finite non-abelian non-metacyclic \(2\)-generated p-group \((p>2)\) of nilpotency class \(2\) with cyclic commutator subgroup \(G^{\prime }\) and centre \(Z(G).\) Then \(G\) can be written as \(G=G(p^{m},p^{n},p^{d})\) where the parameters \(1\leq d\leq n\leq m\) are given by the following: Let \(a\) and \(b\) be generators for \(G\) such that \(G/G^{\prime }=\left\langle aG^{\prime }\right\rangle \times \left\langle bG^{\prime }\right\rangle ,aG^{\prime }\) has order \(p^{m}\) and \(bG^{\prime }\) has order \(p^{n}\). Then \(c=[a,b]\) generates \(G^{\prime }\), \(c\) has order \(p^{d}\) and \(c\in Z(G)=\left\langle ap^{m},bp^{n},c\right\rangle .\)
The main results shows that for each prime \(p\) and positive integers \(m,n\) and \(d\) with \(1\leq d\leq n\leq m,\) there is a local nearring \(R\) whose additive group is isomorphic to the group \(G(p^{m},p^{n},p^{d}).\)
Reviewer: Stefan Veldsman (Port Elizabeth)On the algebraic and arithmetic structure of the monoid of product-one sequences.https://zbmath.org/1459.200432021-05-28T16:06:00+00:00"Oh, Jun Seok"https://zbmath.org/authors/?q=ai:oh.jun-seokSummary: Let \(G\) be a finite group. A finite unordered sequence \(S = g_1 \bullet \cdots \bullet g_\ell\) of terms from \(G\), where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals \(1_G\), the identity element of the group. As usual, we consider sequences as elements of the free abelian monoid \(\mathcal{F}(G)\) with basis \(G\), and we study the submonoid \(\mathcal{B}(G) \subset \mathcal{F}(G)\) of all product-one sequences. This is a finitely generated C-monoid, which is a Krull monoid if and only if \(G\) is abelian. In case of abelian groups, \(\mathcal{B}(G)\) is a well-studied object. In the present paper we focus on nonabelian groups, and we study the class semigroup and the arithmetic of \(\mathcal{B}(G)\).On spectra of almost simple extensions of even-dimensional orthogonal groups.https://zbmath.org/1459.200102021-05-28T16:06:00+00:00"Grechkoseeva, M. A."https://zbmath.org/authors/?q=ai:grechkoseeva.maria-aleksandrovnaSummary: The spectrum of a finite group is the set of the orders of its elements. We consider the problem that arises within the framework of recognition of finite simple groups by spectrum: Determine all finite almost simple groups having the same spectrum as its socle. This problem was solved for all almost simple groups with exception of the case that the socle is a simple even-dimensional orthogonal group over a field of odd characteristic. Here we address this remaining case and determine the almost simple groups in question.
Also we prove that there are infinitely many pairwise nonisomorphic finite groups having the same spectrum as the simple 8-dimensional symplectic group over a field of characteristic other than 7.On the pronormality of subgroups of odd index in some extensions of finite groups.https://zbmath.org/1459.200212021-05-28T16:06:00+00:00"Guo, W."https://zbmath.org/authors/?q=ai:guo.wenbin.1"Maslova, N. V."https://zbmath.org/authors/?q=ai:maslova.natalya-vladimirovna"Revin, D. O."https://zbmath.org/authors/?q=ai:revin.danila-olegovitchSummary: We study finite groups with the following property (*): All subgroups of odd index are pronormal. Suppose that \(G\) has a normal subgroup \(A\) with property (*), and the Sylow 2-subgroups of \(G/A\) are self-normalizing. We prove that \(G\) has property (*) if and only if so does \(N_G(T)/T\), where \(T\) is a Sylow 2-subgroup of \(A\). This leads to a few results that can be used for the classification of finite simple groups with property (*).Hopf-Galois structures on finite extensions with almost simple Galois group.https://zbmath.org/1459.120022021-05-28T16:06:00+00:00"Tsang, Cindy"https://zbmath.org/authors/?q=ai:tsang.cindy-sin-yiSummary: In this paper, we study the Hopf-Galois structures on a finite Galois extension whose Galois group \(G\) is an almost simple group in which the socle \(A\) has prime index \(p\). Each Hopf-Galois structure is associated to a group \(N\) of the same order as \(G\). We shall give necessary criteria on these \(N\) in terms of their group-theoretic properties, and determine the number of Hopf-Galois structures associated to \(A \times C_p\), where \(C_p\) is the cyclic group of order \(p\).