## Solvable groups with many BFC-subgroups.(English)Zbl 0978.20016

The author considers groups with the maximal condition for non-BFC-subgroups. Denote by $$L_{\text{non-BFC}}(G)$$ the set of all non-BFC-subgroups of a group $$G$$. A group $$G$$ satisfies the maximal condition for non-BFC-subgroups (the condition max-(non-BFC)) if (ordered by inclusion) the set $$L_{\text{non-BFC}}(G)$$ satisfies the maximal condition. The purpose of the paper is to obtain a description of soluble groups satisfying max-(non-BFC).
However, some proofs contain gaps, therefore some results are not correct. For example, let $$P=\langle c_n\mid(c_1)^p=1$$, $$(c_{n+1})^p=c_n\rangle$$ be a Prüfer $$p$$-group, $$A=\langle a\rangle$$ an infinite cyclic group, $$B=\langle b_n\mid(b_{n+1})^p=b_n\rangle\cong\mathbb{Q}_p$$. Consider the semidirect product of $$P\times A$$ on $$B$$, defined by the rules: $$b^{-1}cb=1$$ for all elements $$c\in P$$, $$b\in B$$, $$(b_n)^{-1}a_nb_n=a_nc_n$$ for each positive integer $$n$$. This group $$G$$ satisfies max-(non-BFC), but $$[G,G]=P$$ is infinite and $$G/[G,G]$$ is not finitely generated. This example shows that Proposition 1.4 is not correct. The wreath product of two infinite cyclic groups gives a counterexample to Lemma 1.5. Therefore the main result of this paper is still in doubt.

### MSC:

 20F16 Solvable groups, supersolvable groups 20F24 FC-groups and their generalizations 20F22 Other classes of groups defined by subgroup chains 20E34 General structure theorems for groups 20E15 Chains and lattices of subgroups, subnormal subgroups

### Keywords:

non-BFC-subgroups; maximal condition; solvable groups
Full Text: