Negative K-Theory of Generalized Quaternion Groups and Binary Polyhedral Groups

Classical group representation theory is used to determine which finite groups have finite negative K-theory. There follows a computation of the K _?1 of integral group rings ZG for all finite non-abelian subgroups of the group SU(2) of unit quaternions. In principle, the method applies to any finite group.     

Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups

It is known that a number of algebraic properties of the braidgroups extend to arbitrary finite Coxeter-type Artin groups.Here we show how to extend the results to more general groupsthat we call Garside groups. Define a Gaussian monoid to be a finitely generated cancellativemonoid where the expressions of a given element have boundedlengths, and where left and right lowest common multiples exist.A Garside monoid is a Gaussian monoid in which the left andright lowest common multiples satisfy an additional symmetrycondition. A Gaussian group is the group of fractions of a Gaussianmonoid, and a Garside group is the group of fractions of a Garsidemonoid. Braid groups and, more generally, finite Coxeter-typeArtin groups are Garside groups. We determine algorithmic criteriain terms of presentations for recognizing Gaussian and Garsidemonoids and groups, and exhibit infinite families of such groups.We describe simple algorithms that solve the word problem ina Gaussian group, show that these algorithms have a quadraticcomplexity if the group is a Garside group, and prove that Garsidegroups have quadratic isoperimetric inequalities. We constructnormal forms for Gaussian groups, and prove that, in the caseof a Garside group, the language of normal forms is regular,symmetric, and geodesic, has the 5-fellow traveller property,and has the uniqueness property. This shows in particular thatGarside groups are geodesically fully biautomatic. Finally,we consider an automorphism of a finite Coxeter-type Artin groupderived from an automorphism of its defining Coxeter graph,and prove that the subgroup of elements fixed by this automorphismis also a finite Coxeter-type Artin group that can be explicitlydetermined. 1991 Mathematics Subject Classification: primary20F05, 20F36; secondary 20B40, 20M05.     

Smooth Groups

A group is called smooth if it has a finite maximal chain of subgroups in which any two intervals of the same length are isomorphic (as lattices). We show that every finite smooth group G is a semidirect product of a p-group by a cyclic group; in particular, G is soluble. We determine the exact structure of G if G is not a p-group.     

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.     

Groups with every subgroup ascendant-by-finite

A subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.     

Hopficity and Co-Hopficity in Soluble Groups

We show that a soluble group satisfying the minimal condition for its normal subgroups is co-hopfian and that a torsion-free finitely generated soluble group of finite rank is hopfian. The latter property is a consequence of a stronger result: in a minimax soluble group, the kernel of an endomorphism is finite if and only if its image is of finite index in the group.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 10, pp. 1335 – 1341, October, 2004.     

Periodic Endomorphisms of Polycyclic Groups

Let G be a polycyclic group. As a consequence of known results, any periodic group of automorphisms of G is finite and there is an upper bound (depending only on G) for its order. On the other hand, a periodic semigroup of endomorphisms of G need not be finite but we prove that it is locally finite. Also we show that the order of periodic endomorphisms of G is bounded.     

Groups of binary relations

It was shown in [3] that every finite group is the maximal subgroup of a semigroupB _x of all binary relations on some finite set X. This result is extended here to arbitrary groups.     

Finite Homomorphic Images of Groups of Finite Rank

Let π be a finite set of primes. We prove that each soluble group of finite rank contains a finite index subgroup whose every finite homomorphic π-image is nilpotent. A similar assertion is proved for a finitely generated group of finite rank. These statements are obtained as a consequence of the following result of the article: Each soluble pro-π-group of finite rank has an open normal pronilpotent subgroup.

     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

关于几乎可解群

本文推广了关于局部有限群的Asar定理及p.Hall—Kulatilaka,Kargapolov定理.     

一类不能作为自同构群的奇阶群

本文考虑如下问题：怎样的有限群可以作为另一个有限群的全自同构群？我们首先证明，若有限群Ｋ有一个正规Ｓｙｌｏｗｐ－子群使得｜Ｋ：Ｚ（Ｋ）｜ｐ＝ｐ２，那么Ｋ有２阶自同构．利用这个结果，我们证明了，若奇阶群Ｇ具有阶Ｐｓｍ（１≤ｓ≤３），ｐ为｜Ｇ｜的最小素因子，ｐｍ，ｍ无立方因子，则Ｇ不可能作为全自同构群．     

Semicharacters of Groups

We define the notion of a semicharacter of a group G: A function from the group to ?*, whose restriction to any abelian subgroup is a homomorphism. We conjecture that for any finite group, the order of the group of semicharacters is divisible by the order of the group. We prove that the conjecture holds for some important families of groups, including the Symmetric groups and the groups GL(2, q).     

Quasi-decompositions for Self-Small Abelian Groups

Abstract

We will prove a Krull-Schmidt type theorem for the quasi decompositions of a self-small abelian group of finite torsion free rank, which extends the classical result proved for finite rank torsion free abelian groups.     

Closed Subgroups of Profinite Groups

Theorem 1 asserts that in a finitely generated prosoluble group,every subgroup of finite index is open. This generalises anold result of Serre about pro-p groups. It follows by a standardargument from Theorem 2: in a d-generator finite soluble group,every element of the derived group is equal to a product of72d² + 46d commutators. This result about finite soluble groupsis proved by induction on the order of the group, and is elementarythough rather intricate. The essence of the proof lies in reducingthe problem to one about the number of solutions of quadraticequations over a finite field. Corollaries include the following.Let be a finitely generated prosoluble group. Then each termof the lower central series of and each power subgroup ⁿ isclosed. 1991 Mathematics Subject Classification: 20E18, 20D10.     

Engel-4 Groups of Exponent 5

We show that if G is a group of exponent 5, and if G satisfiesthe Engel-4 identity [x,y,y,y,y]=1, then G is locally finite.By a result of Traustason, this implies that Engel-4 5-groupsare locally finite. We also show that a group of exponent 5is locally finite if and only if it satsifies the identity This result implies that a group of exponent 5 is locally finiteif its three generator subgroups are finite. 1991 MathematicsSubject Classification: 20D15, 20F45, 20F50.     

Sequences of Endomorphism Groups of Abelian Groups

In the paper, Problem 18.3 of the book “Abelian groups” (2015) by L. Fuchs is solved in the case of Abelian groups with finite p-ranks. For an Abelian group A, a sequence of groups (A_n) is considered, where A₀ = A and A_n+1 = End A_n. It is shown that, if all p-ranks of the group A are finite, then this sequence can stabilize either after A₀ or after A₁.

     

On the Bergman Property for the Automorphism Groups of Relatively Free Groups

A group G is said to have the Bergman property (the propertyof uniformity of finite width) if given any generating X withX = X^–1 of G, we have that G = X^k for some natural k,that is, every element of G is a product of at most k elementsof X. We prove that the automorphism group Aut(N) of any infinitelygenerated free nilpotent group N has the Bergman property. Also,we obtain a partial answer to a question posed by Bergman byestablishing that the automorphism group of a free group ofcountably infinite rank is a group of uniformly finite width.     

Cycle-Closed Permutation Groups

A finite permutation group is cycle-closed if it contains all the cycles of all of its elements. It is shown by elementary means that the cycle-closed groups are precisely the direct products of symmetric groups and cyclic groups of prime order. Moreover, from any group, a cycle-closed group is reached in at most three steps, a step consisting of adding all cycles of all group elements. For infinite groups, there are several possible generalisations. Some analogues of the finite result are proved.     

Unfused Involutions in Finite Groups

ABSTRACT

Let x be a p-element of a finite group G. We say that x is unfused in G if, for some Sylow p-subgroup S of G containing x, all G-conjugates of x in S are S-conjugates. It is shown (using the classification of finite simple groups) that a finite group that contains an unfused involution has a chief factor of order 2.