Finite groups with many seminormal subgroups (Q1327569)

A subgroup \(H\) of a group \(G\) is said to be seminormal if there exists a subgroup \(K\) of \(G\) such that \(G = HK\) and \(H\) commutes with every subgroup of \(K\). \textit{X. Su} [J. Math., Wuhan Univ. 8, 5-10 (1988; Zbl 0687.20024)] and \textit{P. Wang} [J. Algebra 148, 289-295 (1992; Zbl 0778.20011)] have studied finite groups with ``many'' seminormal subgroups, and sufficient conditions for such groups to be supersoluble or nilpotent have been given. In this paper the author studies finite groups whose Sylow subgroups are seminormal and proves the following result: Let \(G\) be a finite group which cannot be decomposed as a direct product of two non-trivial subgroups with coprime orders and which is not a \(p\)-group (\(p\) prime); then every Sylow subgroup of \(G\) is seminormal if and only if \(G = PN\), where \(N\) is an abelian normal Hall subgroup of \(G\) and \(P\) is a Sylow subgroup inducing on each Sylow subgroup of \(N\) a non-trivial group of power automorphisms. Moreover, the author proves that, if \(G\) is a \(p\)-group (\(p\) odd prime) whose cyclic subgroups are seminormal, then \(G\) is quasihamiltonian (i.e. any two subgroups of \(G\) commute). If \(p = 2\), the result is no longer true since the dihedral group of order 8 is an obvious counterexample.