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<b>), where T Z(G) and 1 [a,b]T

(iii) G=<a> (Tx<c>x<b>), where c 1, Tx<c> Z(G) and [a,b]=tcn for some tT 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=<b> (AxT), where <b> is infinite cyclic, A is torsion-free abelian, TZ(G), C<b>(A)={1}, and each non-trivial subgroup of <b> 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 non-abelian 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<b>), where T Z(G) and 1 [a,b]T.(iii) G=<a> (Tx<c>x<b>), where c 1, Tx<c> Z(G) and [a,b]=tcn for some tT and n>0.(iv) G=<b> (AxT), where <b> is infinite cyclic, A is torsion-free abelian, TZ(G),

C<b>(A)={1}, and each non-trivial subgroup of <b> acts on AT/T rationally irreducibly. (v) G=(<b> A) xT, where <b> is infinite cyclic, A is a torsion-free abelian normal

subgroup, C<b>(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 K\T acts on A rationally irreducibly.

(vii) G=<d>(<a> (Tx<c>x<b>)), where c 1, T x<c> Z(G), [a,b]=tcn for some tT and n>0, d3 Tx<c>, [d,a]=a2bc-1, [d,b]=a-1b.