On seminormal subgroups of finite groups
Let \(G\) be a finite group. \(G\) is said to be an MS-group if every maximal subgroup of any Sylow subgroup of \(G\) is S-semipermutable in \(G\). MS-groups have been studied recently for instance in [the first author et al., Bull. Aust. Math. Soc. 90, No. 2, 220--226 (2014; Zbl 1310.20018)]. A subgroup \(H\) of \(G\) is said to be seminormal (S-seminormal) in \(G\) if it is normalized by every subgroup (Sylow subgroup) \(K\) of \(G\) such that \(\gcd(|H|,|K|) = 1\), these notions were introduced in [\textit{J. C. Beidleman} and \textit{M. F. Ragland}, Cent. Eur. J. Math. 9, No. 4, 915--921 (2011; Zbl 1245.20016)]. The authors study a subclass of MS-groups: \(G\) is said to be an MSN-group if every maximal subgroup of any Sylow subgroup of \(G\) is seminormal in \(G\). Their main results are as follows: An MSN-group is a split extension of a nilpotent Hall subgroup by a cyclic group. Let \(\pi\) denote the set of all primes \(p\) dividing \(|G^{\mathfrak{N}}|\) and \(\theta_N\) (\(\theta_C\)) the set of all primes \(p\notin\pi\) such that a Sylow \(p\)-subgroup is noncyclic (cyclic). \(G\) is an MSN-group if and only if (i) \(G\) is an MS-group; (ii) if \(p\in\pi\) and \(q\in\theta_N\) then \(p\) and \(q\)-Sylow subgroups commute; (iii) if \(p\in\pi\) and \(q\in\theta_C\) then \(p\) -Sylow subgroups and maximal subgroups of \(q\)-Sylow subgroups commute. Moreover, soluble PST-groups which are at the same time MSN-groups are examined. Note that in [\textit{X. Su}, J. Math., Wuhan Univ. 8, No. 1, 5--10 (1988; Zbl 0687.20024)] the expression `seminormal subgroup' is used in a different sense.
scientific article; zbMATH DE number 6715754