A contribution to the theory of finite supersoluble groups
Criteria for \(p\)-supersolubility of \(p\)-soluble groups and for supersolubility of groups are provided in terms of subgroups possessing the cover-avoidance property (CAP). A subgroup \(M\) of a group \(G\) is said to be a CAP-subgroup of \(G\), if each chief factor of \(G\) is either covered or avoided by \(M\). Let \(G\) be a \(p\)-soluble group. Then the following statements are equivalent: (i) \(G\) is \(p\)-supersoluble; (ii) all \(p\)-subgroups of \(G\) are CAP-subgroups of \(G\), (iii) all maximal subgroups of the Sylow \(p\)-subgroups of \(G\) are CAP-subgroups of \(G\); (iv) all maximal subgroups of \(O_{p'p}(G)\) containing \(O_{p'}(G)\) are CAP-subgroups of \(G\); (v) there exists a normal subgroup \(H\) of \(G\) such that \(G/H\) is \(p\)-supersoluble and all maximal subgroups of any Sylow \(p\)-subgroup of \(H\) are CAP-subgroups of \(G\); (vi) there exists a normal subgroup \(H\) of \(G\) such that \(G/H\) is \(p\)-supersoluble and all maximal subgroups of \(O_{p'p} (H)\) containing \(O_{p'}(H)\) are CAP-subgroups of \(G\) (Corollary 1). The criterion for supersolubility is of similar nature. Let \(G\) be a group and let \(p\) denote the largest prime dividing the order of \(G\). Assume that for all primes \(q\) different from \(p\) every maximal subgroup of the Sylow \(q\)-subgroups of \(G\) is a CAP-subgroup of \(G\). Then (i) \(G\) possesses a Sylow tower and (ii) \(G/O_ p (G)\) is supersoluble.
\(p\)-supersolubility of \(p\)-soluble groups
scientific article; zbMATH DE number 492571