On \(c\)-supplemented subgroups of finite groups.
A subgroup \(H\) of \(G\) is \(c\)-supplemented in \(G\) if there is a subgroup \(K\) of \(G\) such that \(G=HK\) and \(H\cap K\subseteq H_G\), the largest normal subgroup of \(G\) contained in \(H\). The generalized Fitting subgroup of \(G\) is denoted by \(F^*(G)\), the maximal supersolvably embedded subgroup by \(Z_U(G)\). One of the main results of this article (Theorem 3.7) states the following: If for every prime \(p\) dividing \(|F^*(G)|\) there is an integer \(k\geq 1\) such that \(p^k\mid |F^*(G)|\) and all the subgroups of \(F^*(G)\) of the orders \(p^k\) and \(p^{k-1}\) are \(c\)-supplemented in \(G\); in addition all cyclic subgroups of order \(4\) are \(c\)-supplemented in \(G\) if \(k=1\) for \(p=2\), then \(G\) is supersolvable. From this it follows that a normal subgroup \(N\) of \(G\) is contained in \(Z_U(G)\) if the subgroups of \(N\) of the form mentioned before are \(c\)-supplemented in \(G\); further \(G\) belongs to the saturated formation \(\mathcal I\) containing all supersolvable groups if \(G/N\) does (Theorem 1.6). -- Many special cases which have been treated earlier are cited in the last chapter.
scientific article; zbMATH DE number 6138242
scientific article