GROUPS WITH MANY ABELIAN SUBGROUPS

1. GROUPS WITH MANY ABELIAN SUBGROUPS

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

3. We shall say that a group Gis anX-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

4. 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).

5. 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).

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

7. 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: • G=<a> T, where [T,a]{1} • G=<a> (Tx<b>), where T  Z(G) and 1 [a,b]T • G=<a> (Tx<c>x<b>), where c  1, Tx<c>  Z(G) and [a,b]=tcn for some tT and n>0.

8. 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: • G is nilpotent-by-finite. • G=<b> (AxT), where <b> is infinite cyclic, A is torsion-free abelian, TZ(G), C<b>(A)={1}, and each non-trivial subgroup of <b> acts on AT/T rationally irreducibly.

9. 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: • G=<a> T, where [T,a]{1}. • G=<a> (Tx<b>), where T  Z(G) and 1 [a,b]T. • G=<a> (Tx<c>x<b>), where c  1,Tx<c>  Z(G) and [a,b]=tcn for some tT and n>0. • G=<b> (AxT), where <b> is infinite cyclic, A is torsion-free abelian, TZ(G), C<b>(A)={1}, and each non-trivial subgroup of <b> acts on AT/T rationally irreducibly. • 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. • 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. • G=<d>(<a> (Tx<c>x<b>)), where c  1,T x<c> Z(G), [a,b]=tcn for some tT and n>0, d3 Tx<c>, [d,a]=a2bc-1, [d,b]=a-1b.