Abstract
Let G be a finite group. We say that \({\mathfrak{Z}}\) is a complete set of Sylow subgroups of G if for each prime p dividing the order of \({G, \mathfrak{Z}}\) contains exactly one Sylow p-subgroup of G, G p say. A subgroup of G is said to be \({\mathfrak{Z}}\)-permutable in G if it permutes with every member of \({\mathfrak{Z}}\). A subgroup H of G is said to be weakly \({\mathfrak{Z}}\)-permutable in G if there exists a subnormal subgroup K of G such that G = HK and \({H \cap K \leq H_\mathfrak{Z}}\), where \({H_{\mathfrak{Z}}}\) is the subgroup of H generated by all those subgroups of H which are \({\mathfrak{Z}}\)-permutable in G . In this paper, we prove that G is supersolvable if the maximal subgroups of \({G_{p} \cap F ^{\ast}(G)}\) are weakly \({\mathfrak{Z}}\)-permutable in G, for every \({G_{p} \in \mathfrak{Z}}\), where \({F^{\ast} (G)}\) is the generalized Fitting subgroup of G. Also, we prove that if \({\mathfrak{F}}\) is a saturated formation containing the class of all supersolvable groups, then \({G \in \mathfrak{F}}\) if and only if there is a normal subgroup H in G such that \({G/H \in \mathfrak{F}}\) and the maximal subgroups of \({G_{p} \cap F^{\ast}(H)}\) are weakly \({\mathfrak{Z}}\)-permutable in G, for every \({G_{p} \in \mathfrak{Z}}\).
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References
Asaad M., Heliel A.A.: On permutable subgroups of finite groups. Arch. Math. 80, 113–118 (2003)
Ballester-Bolinches, A.; Esteban-Romero, R.; Asaad, M.: Products of Finite Groups. In: Expositions in Mathematics, vol. 53. Walter de Gruyter, Berlin (2010)
Doerk, K.; Hawkes, T.: Finite Soluble Groups. In: Expositions in Mathematics, vol. 4. Walter de Gruyter, Berlin (1992)
Heliel, A.A.; Al-Shomrani, M.M.; Al-Gafri, T.M.: On weakly \({\mathfrak{Z}}\)-permutable subgroups of finite groups. J. Algebra Appl. (2015). doi:10.1142/S0219498815500620
Huppert, B.; Blackburn, N.: Finite Groups III. Springer, Berlin (1982)
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1979)
Kegel O.H.: Sylow–Gruppen und Subnormalteiler endlicher Gruppen. Math. Z. 78, 205–221 (1962)
Li Y., Heliel A.A.: On permutable subgroups of finite groups II. Commun. Algebra 33, 3353–3358 (2005)
Li, Y.; Wang, Y.; Wei, H.: The influence of \({\pi}\)-quasinormality of some subgroups of a finite group. Arch. Math. 81, 245–252 (2003)
Wei, H.; Wang, Y.; Li, Y.: On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups II. Commun. Algebra 31, 4807–4816 (2003)
Wang, Y.: C-normality of groups and its properties. J. Algebra 180, 954–965 (1996)
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Heliel, A.A., Al-Shomrani, M.M. & Al-Gafri, T.M. On weakly \({\mathfrak{Z}}\)-permutable subgroups of finite groups II. Arab. J. Math. 5, 63–68 (2016). https://doi.org/10.1007/s40065-015-0129-6
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DOI: https://doi.org/10.1007/s40065-015-0129-6