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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
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