On π-quasinormally embedded subgroups of finite group

A subgroup of a group G is said to be π-quasinormal in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be π-quasinormally embedded in G if for each prime dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some π-quasinormal subgroups of G. In this paper we investigate the influence of π-quasinormally embedded of maximal or minimal subgroup of Sylow subgroups of the generalized Fitting subgroup of a finite group. The main theorems are as follows: Theorem 1.1 Let F be a saturated formation containing U. Suppose that G is a group. Then G F if and only if there is a normal subgroup H of G such that G/H F, and all maximal subgroups of any Sylow subgroup of F*(H) are π-quasinormally embedded in G. Theorem 1.2 Let F be a saturated formation containing U and let G be a group. Then G F if and only if there is a normal subgroup H in G such that G/H F and the subgroups of prime order or order 4 of the generalized Fitting subgroup F*(H) are π-quasinormally embedded in G. Theorem 1.3 Let F be a saturated formation such that N F. Let G be a group such that every element of order 4 of the generalized Fitting subgroup F*(GF) is π-quasinormally embedded in G. Then G belongs to F if and only if x lies in the F-hypercenter ZF(G) of G for every element x of prime order of F*(GF). These results extended recent results of Asaad et al. [On S-quasinormally embedded subgroups of finite groups, J. Pure Appl. Algebra 165 (2001) 129–135], Li et al. [The influence of minimal subgroups on the structure of finite groups, Proc. Amer. Math. Soc., in press; The influence of π-quasinormality of maximal subgroups of Sylow subgroups of a finite group, Arch. Math., in press].