On the quasi-normality and subnormality of subgroups of finite groups
A subgroup \(H\) of the group \(G\) is quasi-normal if \(HK=KH\) for all subgroups \(K\) of \(G\). A well-known result of Ore says that every quasi-normal subgroup of a finite group is subnormal. It is shown that if the index of the quasi-normal subgroup \(H\) of the finite group \(G\) is a prime or of the form \(2^rm\) where \(r=1,2\) and \(m\) is an odd square free number, then \(H\) is normal in \(G\). On the other hand, for an odd prime \(p\) and \(n\geq 3\) or for \(p=2\) and \(n\geq 4\) there exists a finite group \(G\) of order \(p^n\) with a quasi-normal subgroup \(H\) with index \(p^{n-1}\) in \(G\) such that \(H\) is not normal in \(G\).
scientific article; zbMATH DE number 1417647