Let $mathfrak{F}$ be a formation and $G$ a finite group. A subgroup $H$ of $G$ is said to be weakly $mathfrak{F}_{s}$-quasinormal in $G$ if $G$ has an $S$-quasinormal subgroup $T$ such that $HT$ is $S$-quasinormal in $G$ and $(Hcap T)H_{G}/H_{G}leq Z_{mathfrak{F}}(G/H_{G})$, where $Z_{mathfrak{F}}(G/H_{G})$ denotes the $mathfrak{F}$-hypercenter of $G/H_{G}$. In this paper, we study the structur...

A subgroup H of a group G is called ss-quasinormal in G if there is a subgroup B of G such that G = HB and H permutes with every Sylow subgroup of B; H is called weakly s-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We fix in every non-cyclic Sylow subgr...

Let $D$ be a division ring with center $F$, and $G$ subnormal or quasinormal subgroup of $D^*$. We show that if is locally solvable, then contained in $F$.

The intersection graph of quasinormal subgroups a group $G$, denoted by $\Gamma_{\mathrm{q}}(G)$, is defined as follows: the vertex set consists all nontrivial, proper and two distinct vertices $H$ $K$ are adjacent if $H\cap K$ nontrivial. In this paper, we show that when $G$ an arbitrary nonsimple group, diameter $\Gamma_{\mathrm{q}}(G)$ in $\{0,1,2,\infty\}$. Besides, general skew linear grou...