The Universal Functorial Equivariant Lefschetz Invariant

We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K₀ of the category of “ ϕ -endomorphisms of finitely generated free RΠ(G, X)-modules”. We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds. Received: February 2006     

Finite Group Actions with Fixed Points on Homology 3-Spheres

 If a finite group acts freely on a homology 3-sphere, then it has periodic cohomology. To say that a finite group F has periodic cohomology is equivalent to say that any Sylow subgroup of F of odd order is cyclic and a Sylow 2-subgroup of F is either cyclic or a quaternion group. In this paper we consider more generally smooth actions of finite groups G on homology 3-spheres which may have fixed points. We prove that any Sylow subgroup of G of odd order is either cyclic or the direct sum of two cyclic groups. Moreover, we show that if G has odd order, then it splits as a semidirect product of a subgroup A and a normal subgroup B such that B acts freely and there exist some simple closed curves in the homology 3-sphere which are fixed pointwise by some non-trivial element of A. We discuss the relation between these algebraic results and some classical constructions of the theory of 3-manifolds. Received 25 September 1997; in revised form 2 June 1998     

On a Commuting Graph on Conjugacy Classes of Groups

We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.     

Maximal subrings and <Emphasis Type="Italic">E</Emphasis>-groups

For a finite group G, let E(G) denote the near-ring of functions generated by the semigroup, End(G), of endomorphisms of G. We characterize when E(G) is maximal as a subnear-ring of M ₀(G). A group G is an E-group if E(G) is a ring. We give a new characterization of finite E-groups and investigate the problem of determining, for finite E-groups, when E(G) is maximal as a ring in M₀(G). Received: 26 June 2006     

Groups with given properties of finite subgroups

Suppose that in every finite even order subgroup F of a periodic group G, the equality [u, x]² = 1 holds for any involution u of F and for an arbitrary element x of F. Then the subgroup I generated by all involutions in G is locally finite and is a 2-group. In addition, the normal closure of every subgroup of order 2 in G is commutative.     

Characterization of Finite Groups With Some S-quasinormal Subgroups

A subgroup of a finite group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we give a characterization of a finite group G under the assumption that every subgroup of the generalized Fitting subgroup of prime order is S-quasinormal in G.     

On Solvable R* Groups of Finite Rank

Abstract

For an element a of a group G,let S(a) denote the semigroup generated by all conjugates of a in G. We prove that if G is solvable of finite rank and 1 ? S(a) for all 1 ≠ a ∈ G,then ?a ^G ?/?b ^G ? is a periodic group for every b ∈ S(a). Conversely if every two generator subgroup of a finitely generated torsion-free solvable group G has this property then G has finite rank,and if every finitely generated subgroup has this property then every partial order on G can be extended to a total order.     

Characterization of Finite Groups With Some <Emphasis Type="Italic">S</Emphasis>-quasinormal Subgroups

A subgroup of a finite group G is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G. In this paper we give a characterization of a finite group G under the assumption that every subgroup of the generalized Fitting subgroup of prime order is S-quasinormal in G.     

Criterion for Invertibility of Endomorphisms and Test Rank of Metabelian Products of Abelian Groups

Let G be a finitely generated torsion-free group, which is a metabelian product of Abelian groups. We establish a criterion for endomorphisms satisfying some extra restriction to be invertible in G. Test sets for IA-endomorphisms of G are described and its test rank computed.     

On Infinite Camina Groups

A group G is called a Camina group if G′ ≠ G and each element x ∈ G?G′ satisfies the equation x ^G  = xG′, where x ^G denotes the conjugacy class of x in G. Finite Camina groups were introduced by Alan Camina in 1978, and they had been studied since then by many authors. In this article, we start the study of infinite Camina groups. In particular, we characterize infinite Camina groups with a finite G′ (see Theorem 3.1) and we show that infinite non-abelian finitely generated Camina groups must be nonsolvable (see Theorem 4.3). We also describe locally finite Camina groups, residually finite Camina groups (see Section 3) and some periodic solvable Camina groups (see Section 5).     

Abelian-by-G Groups,for G Finite,from the Model Theoretic Point of View

Let G be a finite group. We prove that the theory af abelian-by-G groups is decidable if and only if the theory of modules over the group ring ?[G] is decidable. Then we study some model theoretic questions about abelian-by-G groups, in particular we show that their class is elementary when the order of G is squarefree. Mathematics Subject Classification: 03C60, 03B25.     

ZS-metacyclic groups and their endomorphism near-rings

ItG is a group written additively, the inner automorphisms and the endomorphisms additively generate near-ringsI(G) andE(G) respectively. IfI(G)=E(G), i.e., if every endomorphism is a sum of inner automorphisms, we callG anI-E group. In this paper we describe a class ofI-E groups which includes two of the four known classes ofI-E groups and which contains infinitely many other examples. The order ofI(G) is obtained and its radical determined.Supported in part by the Natural Sciences and Engineering Research Council of Canada.     

On the number of maximal subgroups of a finite solvable group

For a finite solvable group G and prime number p, we use elementary methods to obtain an upper bound for \mathfrak m_p(G){\mathfrak {m}_{p}(G)} , defined as the number of maximal subgroups of G whose index in G is a power of p. From this we derive an upper bound on the total number of maximal subgroups of a finite solvable group in terms of its order. This bound improves existing bounds, and we identify conditions on the order of a finite solvable group under which this bound is best possible.     

Influence of certain permutable subgroups on finite smooth groups

A subgroup H of a finite group G is said to be permutable in G if it permutes with every subgroup of G. In this paper, we determine the finite groups which have a permutable subgroup of prime order and whose maximal subgroups are totally (generalized) smooth groups.     

Relative Group Cohomology and The Orbit Category

Let G be a finite group and ? be a family of subgroups of G closed under conjugation and taking subgroups. We consider the question whether there exists a periodic relative ?-projective resolution for ? when ? is the family of all subgroups H ≤ G with rk H ≤ rkG ? 1. We answer this question negatively by calculating the relative group cohomology ?H*(G, 𝔽₂) where G = ?/2 × ?/2 and ? is the family of cyclic subgroups of G. To do this calculation we first observe that the relative group cohomology ?H*(G, M) can be calculated using the ext-groups over the orbit category of G restricted to the family ?. In second part of the paper, we discuss the construction of a spectral sequence that converges to the cohomology of a group G and whose horizontal line at E ₂ page is isomorphic to the relative group cohomology of G.     

On irreducible algebras of conformal endomorphisms over a linear algebraic group

We study the algebra of conformal endomorphisms Cend ^G,G _n of a finitely generated free module M _n over the coordinate Hopf algebra H of a linear algebraic group G. It is shown that a conformal subalgebra of Cend _n acting irreducibly on M _n generates an essential left ideal of Cend ^G,G _n if enriched with operators of multiplication on elements of H. In particular, we describe such subalgebras for the case where G is finite. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 60, Algebra, 2008.     

Abelian groups with normal endomorphism rings

A ring is said to be normal if all of its idempotents are central. It is proved that a mixed group A with a normal endomorphism ring contains a pure fully invariant subgroup G ⊕ B, the endomorphism ring of a group G is commutative, and a subgroup B is not always distinguished by a direct summand in A. We describe separable, coperiodic, and other groups with normal endomorphism rings. Also we consider Abelian groups in which the square of the Lie bracket of any two endomorphisms is the zero endomorphism. It is proved that every central invariant subgroup of a group is fully invariant iff the endomorphism ring of the group is commutative.     

Notes on NE-subgroups of finite groups

In this paper, we first analyze the structure of a finite nonsolvable group in which every cyclic subgroup of order 2 and 4 of every second maximal subgroup is an NE-subgroup. Next, we prove that a finite group G is solvable if every nonnilpotent subgroup of G is a PE-group.     

Almost fixed-point-free automorphisms of prime order

Let ϕ be an automorphism of prime order p of the group G with C _G (ϕ) finite of order n. We prove the following. If G is soluble of finite rank, then G has a nilpotent characteristic subgroup of finite index and class bounded in terms of p only. If G is a group with finite Hirsch number h, then G has a soluble characteristic subgroup of finite index in G with derived length bounded in terms of p and n only and a soluble characteristic subgroup of finite index in G whose index and derived length are bounded in terms of p, n and h only. Here a group has finite Hirsch number if it is poly (cyclic or locally finite). This is a stronger notion than that used in [Wehrfritz B.A.F., Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo (in press)], where the case p = 2 is discussed.