Influence of permutizers of subgroups on the structure of finite groups.
The `permutiser' of a subgroup \(H\) in a group \(G\) is the subgroup generated by all cyclic subgroups of \(G\) that permute with \(H\). A group \(G\) satisfies the `permutiser condition' if a proper subgroup of \(G\) is a proper subgroup of its permutiser, and satisfies the `maximal permutiser condition' if the permutiser of every maximal subgroup of \(G\) coincides with \(G\). In this paper the authors study conditions under which a finite group satisfying the (maximal) permutiser condition is supersoluble. For instance, they characterise the supersoluble groups as the groups in which every normal subgroup with index at most~\(2\) satisfies the permutiser condition (Theorem~3.1.1). They also characterise the supersoluble groups as the groups \(G\) satisfying the maximal permutizer condition and contained in a quotient-closed class of groups containing \(G\) but not the symmetric group \(S_4\) (Theorem~3.1.3). This theorem is used to prove the equivalence of several group properties for groups satisfying the maximal permutizer condition. In the last part of the paper, the authors study some conditions under which a group \(G\) with a normal subgroup \(H\) such that \(G/H\) is supersoluble turns out to be supersoluble. This happens when \(G\) satisfies the maximal permutiser condition and \(H\) is \(V_4\)-free, where \(V_4\) is the Klein four-group (Theorem~3.2.1). When \(H\cap M\) is either \(H\) or a maximal subgroup of \(H\) for every maximal subgroup \(M\) of \(G\) and \(H\) satisfies the maximal permutiser condition, then \(G\) is \(p\)-supersoluble for all odd primes \(p\), \(G\) is supersoluble if \(H\) has odd order, and \(G\) is supersoluble if \(H\) is \(V_4\)-free (Theorem~3.2.2). Moreover, if every almost maximal subgroup \(L\) of \(H\) is supplemented in \(H\) by a cyclic subgroup, then \(G\) is supersoluble (Theorem~3.2.3). As a consequence, if \(H\) has all Sylow subgroups cyclic, then \(G\) is supersoluble (Corollary~3.2.4), and the classical result about the supersolubility of groups with all Sylow subgroups cyclic is obtained (Corollary~3.2.5).
scientific article; zbMATH DE number 5299446