Finiteness conditions on characteristic closures and cores of subgroups

We characterise groups in which every abelian subgroup has finite index in its characteristic closure. In a group with this property every subgroup H has finite index in its characteristic closure and there is an upper bound for this index over all subgroups H of G. For every prime p we construct groups G with this property that are infinite nilpotent p-groups of class 2 and exponent p in which G = Z(G) is finite and Aut G acts trivially on G/G. We also characterise abelian groups with the dual property that every subgroup has finite index over its characteristic core. In 1955, in his paper [5], B.H. Neumann began a systematic study of finiteness conditions in group theory defined by restrictions on the conjugacy classes of subgroups. Among other results he proved that every subgroup H of a group G has finite index in its normal closure H if and only if G′ is finite; as Tomkinson pointed out in [10] the same conclusion may be drawn if the finiteness of |H : H | is required only for abelian subgroups. Several results of a similar nature, and many generalizations, have appeared in the literature. A dual condition was considered in [1]: a group G is called core-finite (or CF ) if H/HG is finite for all H ≤ G; as usual HG denotes the normal core of H in G. The class of CF -groups is rather more elusive than the class considered by B.H. Neumann. In fact the existence of many infinite groups all of whose proper subgroups are finite, like Tarski groups, makes clear that a simple description of CF -groups is out of range. Nonetheless, it was proved that CF -groups are abelian-by-finite under rather general hypotheses: for instance if they have no periodic quotient which is not locally finite (see [8]). Here we shall consider another variation along the same lines. Instead of considering normal closures and cores of subgroups we deal with characteristic closures and cores. In other words, we deal with properties related to orbits under the action of the full automorphism group AutG of a group G on subgroups, rather than (G-)conjugacy classes. This can be compared, for instance, to [4, 7], where groups G with finite (AutG)-orbits of subgroups are studied. For every subgroup H of the group G let H and H" be the characteristic closure and the characteristic core of H in G, respectively, that is: H is the smallest characteristic subgroup of G containing H and H" is the largest characteristic subgroup of G contained in H . The absence of a reference to G in this notation will never cause ambiguity in what follows. We say that G has the property P (respectively P) if |H : H | (respectively |H : H"|) is finite for every subgroup H of G. We define the property P a by weakening the condition and requiring only that |H : H | is finite for all abelian subgroups H of G. We will also say that G satisfies P (respectively P) boundedly if there is a (finite) upper bound for |H : H | (respectively |H : H"|), where H ranges over all subgroups of G. It is clear that P a -groups have the property considered by B.H. Neumann and Tomkinson, hence they are finite-byabelian, while P-groups are core-finite, hence they are abelian-by-finite under suitable hypotheses. However, even in these classes it is not immediately obvious how to set apart groups which are P or P from the others. For instance, the well-known examples of complicated abelian torsion-free groups in which the inversion map is the only nontrivial automorphism and so all subgroups are characteristic suggest that there is no hope of obtaining a completely explicit description of the groups in these classes. We shall classify abelian groups in P or in P up to the description of the just-mentioned torsion-free groups (see Theorems 2.2 and 2.6). It turns out that the two properties are equivalent for periodic abelian groups, but in the nonperiodic case P is a stronger property than P. Another consequence of these results is that every abelian group satisfying either of the two properties satisfies it boundedly. It is worth remarking that, unlike what happens in the abelian case, nonabelian P-groups need not satisfy P. Indeed, if G = U ! 〈x〉, where |AutU | = 2 and U is infinite, x has order 2 and u = u−1 for all u ∈ U (for instance, G might be the infinite dihedral group) then G ∈ P but G′ = U is infinite, hence G / ∈ P. Apart from this observation, we shall not here carry on the study of P-groups to the nonabelian case, but we extend the results on abelian P-groups to arbitrary groups. We shall prove the following: 2000 Mathematics Subject Classification. 20E07, 20F24, 20E36. This research was initiated during visits of the first author to Cardiff University and Bucknell University. He thanks both institutions for generous and warm hospitality. He also gratefully acknowledges financial support from UPIMDS of Università Federico II, Napoli and INdAM/GNSAGA (Italy).