Groups with the maximal condition on non FC-subgroups (Q1409602)

Let \(G\) be a group. The group \(G\) satisfies the maximal condition for non-FC-subgroups or shortly Max-(non-FC) (respectively, for non-BFC-subgroups, for short Max-(non-BFC)) if the ordered by inclusion set of all non-FC-subgroups (respectively non-BFC-subgroups) satisfies the maximal condition. The groups satisfying Max-(non-BFC) have been studied by the same authors in the papers [Algebra Colloq. 10, No. 2, 177-193 (2003; Zbl 1012.20031); Proc. Edinb. Math. Soc., II. Ser. 45, No. 3, 513-522 (2002; Zbl 1012.20032)]. It is not difficult to see that a group satisfying Max-(non-FC) either is a locally FC-group or is finitely generated. In the paper under review the authors consider the first case, that is locally FC-groups satisfying Max-(non-FC). The main results of this paper are following. Theorem 3.7. Let \(G\) be a locally FC-group satisfying Max-(non-FC) and let \(T\) be the periodic part of \(G\). If \(G/T\) is not finitely generated, then either \(G\) is an FC-group or \(G\) satisfies Max-(non-BFC). Theorem 3.13. Let \(G\) be a locally FC-group satisfying Max-(non-FC). If \(G\) is soluble, then either \(G\) is an FC-group or \(G\) satisfies Max-(non-BFC). Theorem 3.14. Let \(G\) be a locally nilpotent group satisfying Max-(non-FC). Then \(G\) is a group of one of the following types: (1) \(G\) is an FC-group; (2) \(G\) is finitely generated; (3) \(G\) satisfies Max-(non-BFC); (4) \(G\) is hyperabelian and the hyperfinite residual \(R\) satisfies the following properties: (4A) \(R\) is a perfect, \(F\)-perfect \(p\)-subgroup of \(G\); (4B) \(R\) is a non-FC-subgroup of \(G\); (4C) every proper subgroup of \(R\) is an FC-group, and hence is hypercentral; (4D) \(G/R\) is a finitely generated FC-group.