Volume 9, Issue 1

We set up a Grothendieck spectral sequence which generalizes the Lyndon–Hochschild–Serre spectral sequence for a group extension K ↣ G ↠ Q by allowing the normal subgroup K to be replaced by a subgroup, or family of subgroups which satisfy a weaker condition than normality. This is applied to establish a decomposition theorem for certain groups as fundamental groups of graphs of Poincaré duality groups. We further illustrate the method by proving a cohomological vanishing theorem which applies for example to Thompson's group F .

Clifford Theory concerns the representations over a field F of a finite group G and one of the normal subgroups N of G . It is often viewed as a series of reduction theorems. Clifford classes are certain equivalence classes of G/N -algebras over F . They have been used to describe the Schur indices of the irreducible characters of certain families of classical groups. In the present paper, we show how Clifford classes can also be used to explain and to reflect the reduction theorems of classical Clifford theory. We show that certain products of characters correspond to certain products of Clifford classes. We show that induction, restriction, inflation, and extension of the base field can all be defined naturally for Clifford classes. We prove that the properties of these elementary operations on Clifford classes imply all of the standard Clifford-theoretic reductions in the case when the field is ℂ. They also provide important information in the case when the field F is an arbitrary field of characteristic zero.

A subgroup H of a finite group G is called an NE -subgroup if it satisfies N G ( H ) ∩ H G = H , where H G denotes the normal closure of H in G . A finite group G is called a PE -group if every minimal subgroup of G is an NE -subgroup of G . A group G is called a 𝒯-group if every subnormal subgroup of G is normal in G . In this paper, first we give two new characterizations of finite solvable 𝒯-groups in terms of the requirement that certain subgroups are NE -subgroups, and then we obtain new necessary conditions for supersolvability and nilpotency. Finally we classify the finite simple groups all of whose second maximal subgroups are PE -subgroups.

A construction is given of an infinite primitive Jordan permutation group which preserves a ‘limit’ of betweenness relations. There is a previous construction due to Adeleke of a Jordan group of this kind. The main difference is that in this paper the group arises as the automorphism group of an relational structure. It is 2-transitive, 3-homogeneous, but not 3-transitive.

In [], A. Nies introduces the notion of a quasi-finitely axiomatizable (QFA) group. He proves that the free nilpotent group of class 2 with 2 generators is QFA, and that it is a prime model of its theory. Our results, stated below in full generality, provide, in particular, a complete characterization of QFA nilpotent groups (in particular, any finitely generated nonabelian free nilpotent group is QFA). They imply that each finitely generated nilpotent group is QFA if and only if it is a prime model of its theory.

In [], F. Oger and G. Sabbagh prove that a finitely generated nilpotent group is quasi-finitely axiomatizable if and only if it is a prime model of its theory. Here we investigate the relations between these two properties for larger classes of groups.

For any integer n > 1, the variety of n -Bell groups is defined by the law w ( x 1 , x 2 ) = [ x 1 n , x 2 ][ x 1 , x 2 n ] −1 . Bell groups were studied by R. Brandl in [2], and by R. Brandl and L.-C. Kappe in [3]. In this paper we determine the structure of these groups. We prove that if G is an n -Bell group then G / Z 2 ( G ) has finite exponent depending only on n . Moreover, either G / Z 2 ( G ) is locally finite or G has a finitely generated subgroup H such that H / Z ( H ) is an infinite group of finite exponent. Finally, if G is finitely generated, then the subgroup H may be chosen to be the finite residual of G .

Let F be a free group and let w ∈ F . For a group G , let G w denote the set of all w -values in G and w ( G ) the verbal subgroup of G corresponding to w . A word w is called boundedly concise if, for each group G such that | G w | ≤ m , we have | w ( G )| ≤ c for some integer c = c ( m ) depending only on m . The main theorem of the paper says that if w is a boundedly concise word and G is a group such that | x G w | ≤ m for all x ∈ G then | x w ( G ) | ≤ d for all x ∈ G and some integer d = d ( m , w ) depending only on m and w .