Narrow Results

According to Mao et al. (2016), let $H$ be a subgroup of a finite group $G$; then $H$ is said to be generalized $S\mathrm{\Phi }$-supplemented in $G$ if there exists a subnormal subgroup $T$ of $G$ such that $G=HT$ and $(H\cap T)H_{sG}/H_{sG}\le \mathrm{\Phi }(H/H_{sG})$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $S$-permutable in $G$. In this paper, we extend the concept of generalized $S\mathrm{\Phi }$-supplemented subgroups and generalize the main result of Mao et al. (2016).

Let $G$ be a finite group, $p$ a prime, $P$ a Sylow $p$-subgroup of $G$ and $d$ a power of $p$ such that $1<d<\left|P\right|$. Let $G_p^{\ast }$ denote the unique smallest normal subgroup of $G$ for which the corresponding factor group is abelian of exponent dividing $p-1$. Let ${𝔉}_1$, ${𝔉}_2$, ${𝔉}_3$ be classes of all $p$-groups, $p$-nilpotent groups and $p$-supersolvable groups, respectively, $G^{𝔉}$ be the $𝔉$-residual of $G$. Let $X\in \{{(G_p^{\ast })}^{{𝔉}_1},G^{{𝔉}_2},G^{{𝔉}_3}\}$. A subgroup $H$ of a finite group $G$ is said to have $\mathrm{\Pi }$-property in $G$, if for any $G$-chief factor $L/K$, $|G/K:N_{G/K}((H\cap L)K/K)|$ is a $\pi ((H\cap L)K/K)$-number. A normal subgroup $E$ of $G$ is said to be $p$-hypercyclically embedded in $G$ if every $p$-$G$-chief factor of $E$ is cyclic, where $p$ is a fixed prime. In this paper, we prove that $E$ is $p$-hypercyclically embedded in $G$ if and only if for some $p$-subgroups $H$ of $E$, $H\cap X$ have $\mathrm{\Pi }$-property in $G$.

A subgroup $H$ of a group $G$ is called an $\mathcal{\mathscr{H}}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g\cap N_T(H)\le H$ for all $g\in G$. In this paper, we obtain new criteria for a normal subgroup to be contained in the $p𝔉$-hypercenter of a finite group by assuming that some of its subgroups are $\mathcal{\mathscr{H}}C$-subgroups. Our results generalize and uniform many known results.

A subgroup H of G is called ℳ-supplemented in G if there exists a subgroup B of G such that G = HB and H_i B < G for every maximal subgroup H_i of H. In this paper, we use ℳ-supplemented subgroups to study the structure of finite groups and obtain some new characterization about solvability and p-supersolvability for a fixed prime p. Some results in the literature are corollaries of our theorems.