Finite groups with a seminormal Hall subgroup.
The paper studies finite groups with a seminormal Hall subgroup. A subgroup \(H\) of a group \(G\) is said to be seminormal in \(G\) if there is a subgroup \(K\) in \(G\) such that \(HK=G\) and the product \(HK_1\) is a proper subgroup of \(G\) for any subgroup \(K_1\) of \(K\) which differs from \(K\). In this case, we refer to the subgroup \(K\) as a supercomplement to \(H\) in \(G\). The main results proved in the paper are the following: (1) If \(G\) is a finite group having a seminormal Hall \(\pi\)-subgroup \(H\), then the group \(G\) is \(\pi\)-solvable in each of the following cases: (i) \(H\) is 2-nilpotent (i.e., \(H\) has a normal Hall \(2'\)-subgroup); (ii) \(H\) is solvable and \(3\not\in\pi\). (2) If every noncyclic Sylow subgroup of a group \(G\) is seminormal, then the group \(G\) is solvable and its third commutator subgroup is nilpotent.
scientific article; zbMATH DE number 5155560