GROUPS WITH MANY ABELIAN SUBGROUPS Groups in which

GROUPS WITH MANY ABELIAN SUBGROUPS

Groups in which every non-abelian subgroup has finite index (joint work with Francesco de Giovanni, Carmela Musella and Yaroslav P. Sysak)

We shall say that a group G is an X-group if it is an infinite group in which every non-abelian subgroup has finite index. G non-abelian X-group G is finitely generated if G is soluble, G is non-periodic the largest periodic normal subgroup of G is abelian

1. Let G be a non-abelian cyclic-by-finite group and let T be the largest periodic normal subgroup of G. Then G is an X-group if and only if either G/T is infinite cyclic or G/T is isomorphic to D and T=Z(G).

2. Let G be an X-group and let H be the hypercentre of G. Then either H has finite index in G or H=Z(G).

3. Let G be a group. The hypercentre of G has finite index in G if and only if G is finite-by-hypercentral.

4. Let G be a non-abelian X-group in which the hypercentre has finite index, and let T be the set of all periodic elements of G. Then T is a finite abelian subgroup of G, and one of the following conditions holds: (i) G=<a> T, where [T, a] {1} (ii) G=<a> (Tx), where T Z(G) and 1 [a, b] T (iii) G=<a> (Tx<c>x), where c 1, Tx<c> Z(G) and [a, b]=tcn for some t T and n>0.

5. Let G be a soluble-by-finite non-abelian X-group, and let T be the largest periodic normal subgroup of G. Then T is a finite abelian subgroup of G, and one of the following conditions holds: (i) G is nilpotent-by-finite. (ii) G= (Ax. T), where is infinite cyclic, A is torsion -free abelian, T Z(G), C(A)={1}, and each non-trivial subgroup of acts on AT/T rationally irreducibly.

THEOREM Let G be a group, and let T be the largest periodic normal subgroup of G. Then G is a nonabelian X-group if and only if G is finitely generated and one of the following conditions holds: (1) G/Z(G) is a non-(abelian-by-finite) just-infinite group in which any two distinct maximal abelian subgroups have trivial intersection. (2) G is soluble with derived length at most 3, T is a finite abelian subgroup, and G satisfies one of the following: (i) G=<a> T, where [T, a] {1}. (ii) G=<a> (Tx), where T Z(G) and 1 [a, b] T. (iii) G=<a> (Tx<c>x), where c 1, Tx<c> Z(G) and [a, b]=tcn for some t T and n>0. (iv) G= (Ax. T), where is infinite cyclic, A is torsion-free abelian, T Z(G), C(A)={1}, and each non-trivial subgroup of acts on AT/T rationally irreducibly. (v) G=( A) x. T, where is infinite cyclic, A is a torsion-free abelian normal subgroup, C(A)=<bn> for some n>1, and for each proper divisor m of n <bm> acts on A rationally irreducibly. (vi) G=K A, where A is a torsion-free abelian normal subgroup, K is finite, CK(A)=T, K/T is cyclic and each element of KT acts on A rationally irreducibly. (vii) G=<d>(<a> (Tx<c>x)), where c 1, T x<c> Z(G), [a, b]=tcn for some t T and n>0, d 3 Tx<c>, [d, a]=a 2 bc-1, [d, b]=a-1 b.