Publication Date:
2015-06-13
Description:
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 \({G_{p} \in \mathfrak{Z}}\) .
Print ISSN:
2193-5343
Electronic ISSN:
2193-5351
Topics:
Mathematics
Permalink