On groups covered by relatively subnormal Černikov local systems

Abstract

Let \(\mathcal L_{\mathfrak F}\) be the class of groups having a local system \(\{X_i : i\in I\}\) of finite subgroups such that \(X_i\) is subnormal in \(X_j\) whenever \(X_i\leq X_j\). It has been shown by Rae in [19] that the class of soluble \(\mathcal L_{\mathfrak F}\)-groups is closer to the class of soluble periodic FC-groups than might be expected. The aim of this paper is to prove that, under some additional finite-rank assumptions, one can extend Rae's results to local systems of Černikov subgroups, showing for example that the locally nilpotent residual is always covered by normal Černikov subgroups of the group, and that the factor group by the Hirsch–Plotkin radical has Černikov conjugacy classes of elements (see Theorem 5.9).

In [2], Reinhold Baer introduced a characteristic subgroup of a group which coincides with the hypercentre in the finite case (we call this subgroup the Baer centre of the group); actually, as shown in [4], this subgroup coincides with the hypercentre even in periodic FC-groups. Extending these results, we prove that this equivalence holds in many relevant universes of locally finite groups (see Theorem 6.2) and in particular in certain classes of locally finite groups having local systems of the above-mentioned type (see Theorem 6.9).

Finally, in order to better understand the behaviour of the Baer centre in our context, we introduce and study a new class of groups that is strictly contained between the classes of periodic FC-groups and periodic BFC-groups, and that could be very useful from a computational point of view (see Section 7).