Narrow Results

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let H_sG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ H_sG.

Our main result here is the following theorem. Let be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that . Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then .

Let $G$ be a finite group and $H$ a subgroup of $G$. Then $H$ is said to be $s$-permutably embedded in $G$ if a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-permutable subgroup of $G$ for every prime dividing the order of $H$. We say that $H$ is nearly$S\mathrm{\Phi }$-embedded in $G$ if $G$ has a normal subgroup $N$ such that $HN$ is $s$-permutable in $G$ and $H\cap N\le \mathrm{\Phi }(H)H_{seG}$, where $H_{seG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-permutably embedded in $G$. In this paper, we study the properties of the nearly $S\mathrm{\Phi }$-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.