Volume 7, Issue 4

It is a known fact that the subgroup Ω 2 ( G ) generated by all elements of order at most 4 in a finite 2-group G has a strong influence on the structure of the whole group G . For example, if Ω 2 ( G ) is metacyclic, then G is also metacyclic (N. Blackburn). Here we consider the case Ω 2 ( G ) = C 2 x D , where C 2 is cyclic of order 2 and D is any 2-group of maximal class and we show that ❘ G : Ω 2 ( G )❘ ≤ 2 and the structure of G is uniquely determined. We determine also the structure of a finite 2-group G whose elements of order 4 generate the subgroup Ω * 2 ( G ) ≅ C 2 × Q 2 n , where Q 2 n is generalized quaternion of order 2 n . Finally, we show that a finite p -group G all of whose non-cyclic subgroups are generated by elements of order p is cyclic or of exponent p or a dihedral 2-group.

Certain orthogonal subgroups of finite unitary groups belonging to the fifth Aschbacher class C 5 are studied and their maximality is proved using the geometry of permutable polarities.

We classify imprimitive groups acting highly transitively on blocks and satisfying conditions common in geometry. They can be realized as subgroups of twisted wreath products.

We address the question of the subnormality of a subgroup H in a group G when H is subnormal in two subgroups U and V such that G = 〈 U, V 〉. In particular, we consider the case in which G is nilpotent-by-abelian and the case in which G = UV is factorized by U and V .

A group is discriminating if and only if it discriminates its direct square, and square-like if and only if it is universally equivalent to its direct square. It is known that any discriminating group is square-like. These notions were introduced and studied in a series of papers by Baumslag, Myasnikov and Remeslennikov and by Fine, Gaglione, Myasnikov and Spellman. We prove that any square-like group is elementarily equivalent to a countable discriminating group. This answers a question of the second group of authors. We provide an explicit universal–existential axiom system for the class of square-like groups. We show that the theory of the class of discriminating groups is computably enumerable but undecidable. We give a criterion for determining whether a group is discriminating. We propose a construction method for discriminating groups and use it to construct in various group varieties many discriminating non-abelian groups that do not embed their squares. We construct square-like, non-discriminating nilpotent p -groups of arbitrary nilpotency class; all previously known square-like, non-discriminating groups were abelian.

It is shown that a pro- p group which is both relatively free and p -adic analytic must be nilpotent-by-finite, confirming a conjecture of Aner Shalev.

We explicitly construct finitely many generators for a subgroup of finite index in the unit group of an arbitrary integral semigroup ring ℤ S (with identity) of a finite semigroup S , subject to some restrictions on the simple epimorphic images of degree 1 and 2 of the rational semigroup algebra ℚ S . For a Mal'tsev nilpotent semigroup S even more precise information is obtained for the generators coming from the radical, and furthermore we can overcome most of the restrictions imposed on the simple images mentioned above. This extends previous work on group rings, matrices over group rings and some other orders.

Let K be a principal ideal domain, G a finite group, and M a KG -module which is a free K -module of finite rank on which G acts faithfully. A generalized crystallographic group is a non-split extension ℭ of M by G such that conjugation in ℭ induces the G -module structure on M . (When K = ℤ, these are just the classical crystallographic groups.) The dimension of ℭ is the K -rank of M , the holonomy group of ℭ is G , and ℭ is indecomposable if M is an indecomposable KG -module. We study indecomposable torsion-free generalized crystallographic groups with holonomy group G when K is ℤ, or its localization ℤ (p) at the prime p , or the ring ℤ p of p -adic integers. We prove that the dimensions of such groups with G non-cyclic of order p 2 are unbounded. For K = ℤ, we show that there are infinitely many non-isomorphic such groups with G the alternating group of degree 4 and we study the dimensions of such groups with G cyclic of certain orders.