Ramadan, M.; Mohamed, M. Ezzat.; Heliel, A. A. On \(c\)-normality of certain subgroups of prime power order of finite groups. (English) Zbl 1082.20008 Arch. Math. 85, No. 3, 203-210 (2005). Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is called \(c\)-normal in \(G\) if \(G\) has a normal subgroup \(N\) such that \(G=NH\) and \(N\cap H\leq\text{Core}(H)\), where \(\text{Core}(H)\) is the intersection of all the conjugates of \(H\) in \(G\). The authors prove the following main results: Let \(\mathcal F\) be a saturated formation containing the formation of supersoluble groups. Then \(G\in\mathcal F\) if and only if there is a normal subgroup \(H\) of \(G\) such that \(G/H\in\mathcal F\) and all maximal subgroups of the Sylow subgroups of \(H\) are \(c\)-normal in \(G\) (Theorem 3.3). The same result holds if we replace the maximal subgroups of the Sylow subgroups of \(H\) by the subgroups of prime order or order \(4\) of \(H\) (Theorem 3.9). The proofs of the above results rely on the following \(p\)-supersolubility criterion: Let \(G\) be a finite \(p\)-soluble group for a prime \(p\). Then \(G\) is \(p\)-supersoluble if there is a normal subgroup \(H\) of \(G\) such that \(G/H\) is \(p\)-supersoluble and either all maximal subgroups of the Sylow subgroups of \(H\) are \(c\)-normal in \(G\) or the subgroups of prime order or order \(4\) of \(H\) are \(c\)-normal in \(G\) (Theorems 3.1 and 3.7). Reviewer: Adolfo Ballester-Bolinches (Burjasot) Cited in 1 ReviewCited in 28 Documents MSC: 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks 20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure 20D40 Products of subgroups of abstract finite groups 20E28 Maximal subgroups Keywords:finite groups; \(c\)-normality; supersoluble groups; saturated formations; normally complemented subgroups PDF BibTeX XML Cite \textit{M. Ramadan} et al., Arch. Math. 85, No. 3, 203--210 (2005; Zbl 1082.20008) Full Text: DOI OpenURL