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Search: MSC category 20D10 ( Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17] )

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1. CMB Online first

Xu, Yong; Zhang, Xinjian
$m$-embedded Subgroups and $p$-nilpotency of Finite Groups
Let $A$ be a subgroup of a finite group $G$ and $\Sigma : G_0\leq G_1\leq\cdots \leq G_n$ some subgroup series of $G$. Suppose that for each pair $(K,H)$ such that $K$ is a maximal subgroup of $H$ and $G_{i-1}\leq K \lt H\leq G_i$, for some $i$, either $A\cap H = A\cap K$ or $AH = AK$. Then $A$ is said to be $\Sigma$-embedded in $G$; $A$ is said to be $m$-embedded in $G$ if $G$ has a subnormal subgroup $T$ and a $\{1\leq G\}$-embedded subgroup $C$ in $G$ such that $G = AT$ and $T\cap A\leq C\leq A$. In this article, some sufficient conditions for a finite group $G$ to be $p$-nilpotent are given whenever all subgroups with order $p^{k}$ of a Sylow $p$-subgroup of $G$ are $m$-embedded for a given positive integer $k$.

Keywords:finite group, $p$-nilpotent group, $m$-embedded subgroup
Categories:20D10, 20D15

2. CMB 2014 (vol 57 pp. 648)

Tang, Juping; Miao, Long
On the ${\mathcal F}{\Phi}$-Hypercentre of Finite Groups
Let $G$ be a finite group, $\mathcal F$ a class of groups. Then $Z_{{\mathcal F}{\Phi}}(G)$ is the ${\mathcal F}{\Phi}$-hypercentre of $G$ which is the product of all normal subgroups of $G$ whose non-Frattini $G$-chief factors are $\mathcal F$-central in $G$. A subgroup $H$ is called $\mathcal M$-supplemented in a finite group $G$, if there exists a subgroup $B$ of $G$ such that $G=HB$ and $H_1B$ is a proper subgroup of $G$ for any maximal subgroup $H_1$ of $H$. The main purpose of this paper is to prove: Let $E$ be a normal subgroup of a group $G$. Suppose that every noncyclic Sylow subgroup $P$ of $F^{*}(E)$ has a subgroup $D$ such that $1\lt |D|\lt |P|$ and every subgroup $H$ of $P$ with order $|H|=|D|$ is $\mathcal M$-supplemented in $G$, then $E\leq Z_{{\mathcal U}{\Phi}}(G)$.

Keywords:${\mathcal F}{\Phi}$-hypercentre, Sylow subgroups, $\mathcal M$-supplemented subgroups, formation
Categories:20D10, 20D20

3. CMB 2014 (vol 57 pp. 277)

Elkholy, A. M.; El-Latif, M. H. Abd
On Mutually $m$-permutable Product of Smooth Groups
Let $G$ be a finite group and $H$, $K$ two subgroups of G. A group $G$ is said to be a mutually m-permutable product of $H$ and $K$ if $G=HK$ and every maximal subgroup of $H$ permutes with $K$ and every maximal subgroup of $K$ permutes with $H$. In this paper, we investigate the structure of a finite group which is a mutually m-permutable product of two subgroups under the assumption that its maximal subgroups are totally smooth.

Keywords:permutable subgroups, $m$-permutable, smooth groups, subgroup lattices
Categories:20D10, 20D20, 20E15, 20F16

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