Coxeter-like groups for set-theoretic solutions of the Yang–Baxter equation

We attach with every finite, involutive, nondegenerate set-theoretic solution of the Yang–Baxter equation a finite group that plays for the associated structure group the role that a finite Coxeter group plays for the associated Artin–Tits group.     

Solvability of the conjugacy problem for finite subgroups in automorphism groups of free groups

We prove that there exists an algorithm which solves a conjugacy problem for finite subgroups in automorphism and outer automorphism groups of a free group of finite rank. Of independent interest is the construction of an algorithm of decomposing an arbitrary free-by-finite group into a fundamental group of a finite graph of finite groups, with the number of steps evaluated explicitly. In passing, we solve the conjugacy problem for finite subgroups in almost free groups. As a consequence, an algorithm is obtained computing generating sets for a group of fixed points in an arbitrary finite automorphism group of a free group of finite rank.Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 558–606, September-October, 1995.Supported by the RFFR grant No. 93-011-1508 and by the ISF (International Science Foundation) grant RPC000.     

On the generalized Davenport constant and the Noether number

Known results on the generalized Davenport constant relating zero-sum sequences over a finite abelian group are extended for the generalized Noether number relating rings of polynomial invariants of an arbitrary finite group. An improved general upper degree bound for polynomial invariants of a non-cyclic finite group that cut out the zero vector is given.     

On the second commutants of finite Alperin groups

We refer to an Alperin group as a group in which the commutant of every 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with the property are metabelian. Nevertheless, finite Alperin 2-groups may fail to be metabelian. We prove that for each finite abelian group H there exists a finite Alperin group G for which G″ is isomorphic to H.     

On the residual finiteness of outer automorphisms of relatively hyperbolic groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.     

Recognizing the Automorphism Groups of Mathieu Groups

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders and at most two irreducible character degrees of their character tables.     

On the Residual Finiteness of Outer Automorphisms of Hyperbolic Groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.     

On the permutability of sylow subgroups with Schmidt subgroups

A Schmidt group is a finite nonnilpotent group in which every proper subgroup is nilpotent. Sufficient conditions are established for the p-solvability of a finite group in which a Sylow p-subgroup is permutable with some Schmidt subgroups. Sufficient conditions for the solvability of a finite group in which some Schmidt subgroups are permutable are also obtained.     

On the lower central series of the free nilpotent groups of finite rank

In their article, “On the derived subgroup of the free nilpotent groups of finite rank” R. D. Blyth, P. Moravec, and R. F. Morse describe the structure of the derived subgroup of a free nilpotent group of finite rank n as a direct product of a nonabelian group and a free abelian group, each with a minimal generating set of cardinality that is a given function of n. They apply this result to computing the nonabelian tensor squares of the free nilpotent groups of finite rank. We generalize their main result to investigate the structure of the other terms of the lower central series of a free nilpotent group of finite rank, each again described as a direct product of a nonabelian group and a free abelian group. In order to compute the ranks of the free abelian components and the size of minimal generating sets for the nonabelian components we introduce what we call weight partitions.     

On the group with the same orders of Sylow normalizers as the finite projective special unitary group

In this paper we prove that a finite group G is isomorphic to the finite projective special unitary group Un(q) if and only if they have the same order of Sylow r-normalizer for every prime r.     

On the characteristic map of finite unitary groups

In his classic book on symmetric functions, Macdonald describes a remarkable result by Green relating the character theory of the finite general linear group to transition matrices between bases of symmetric functions. This connection allows one to analyze the character theory of the general linear group via symmetric group combinatorics. Using works of Ennola, Kawanaka, Lusztig and Srinivasan, this paper describes the analogous setting for the finite unitary group. In particular, we explain the connection between Deligne–Lusztig theory and Ennola's efforts to generalize Green's work, and from this we deduce various character theoretic results. Applications include calculating certain sums of character degrees, and giving a model of Deligne–Lusztig type for the finite unitary group, which parallels results of Klyachko and Inglis and Saxl for the finite general linear group.     

Shelling the coset poset

It is shown that the coset lattice of a finite group has shellable order complex if and only if the group is complemented. Furthermore, the coset lattice is shown to have a Cohen-Macaulay order complex in exactly the same conditions. The group theoretical tools used are relatively elementary, and avoid the classification of finite simple groups and of minimal finite simple groups.     

On the Shunkov groups acting freely on Abelian groups

We prove the existence of a locally finite periodic part in every rank 1 Shunkov group with solvable finite subgroups, in every Shunkov group acting freely on an abelian group, and in the groups of affine transformations of neardomains with finite elements.     

有限交换群的Burnside环的增广理想的连续商群的结构

本文对有限交换群的Burnside环的增广理想的n次幂与n 1次幂之商进行了讨论,完整地给出了这类连续商群的结构.     

Characterization of Locally Finite Simple Groups of Type G2 over Fields of Odd Characteristics in the Class of Periodic Groups

Mathematical Notes - We prove that a periodic group is locally finite, given that each of its finite subgroups lies in a subgroup isomorphic to a finite simple group G2 of Lietypeovera field of odd...     

Factorizations of Self-similar Groups Associated to the First Grigorchuk Group

In this paper, we construct two self-similar groups factorizable by their locally finite 2-subgroups, generators of which are constructed from the generators of the first Grigorchuk group. Both groups contain the first Grigorchuk group, the second also contains the binary adding machine. The first group is a 2-group but not locally finite and the second group is neither locally finite nor a 2-group. Therefore each of them is a new example providing a negative answer to well-known solved problems. Surprisingly, definitions of both groups involve the Gray code.     

Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable

Thompson’s theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson’s theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.     

On the socle of an endomorphism algebra

The socle of an endomorphism algebra of a finite dimensional module of a finite dimensional algebra is described. The results are applied to the modular Hecke algebra of a finite group with a cyclic Sylow subgroup.     

Recognizing the Automorphism Groups of Mathieu Groups Through Their Orders and Large Degrees of Their Irreducible Characters

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders and at most two irreducible character degrees of their character tables.     

关于有限Abel p-群的自同构群

从有限Abel p-群P的型不变量出发,给出了其自同构群AutP的阶的计算公式,并利用|AutP|的计算公式得到了下面3个结果:1.由有限Abel p-群的型不变量的两种变换得到了其自同构群的阶的变化规律;2.用群的阶、秩、幂指数三个量界定了有限Abel p-群的自同构的阶;3.对部分Frattini子群为p阶群的有限p-群,确定了其自同构群的阶何时达到最小值和最大值.