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 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

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: (i)G= T, where [T,a]  {1} (ii)G= (Tx ), where T  Z(G) and 1  [a,b]  T (iii)G= (Tx x ), where c  1, Tx  Z(G) and [a,b]=tc n 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: (i)G is nilpotent-by-finite. (ii)G= (AxT), 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.

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: (i)G= T, where [T,a]  {1}. (ii)G= (Tx ), where T  Z(G) and 1  [a,b]  T. (iii)G= (Tx x ), where c  1, Tx  Z(G) and [a,b]=tc n for some t  T and n>0. (iv)G= (AxT), 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) xT, where is infinite cyclic, A is a torsion-free abelian normal subgroup, C (A)= for some n>1, and for each proper divisor m of n acts on A rationally irreducibly. (vi)G=K A, where A is a torsion-free abelian normal subgroup, K is finite, C K (A)=T, K/T is cyclic and each element of K\T acts on A rationally irreducibly. (vii)G= ( (Tx x )), where c  1, T x  Z(G), [a,b]=tc n for some t  T and n>0, d 3  Tx, [d,a]=a 2 bc -1, [d,b]=a -1 b.