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Let $sigma ={sigma_{i} | iin I}$ be some partition of the set of all primes $Bbb{P}$ and let $G$ be a finite group. Then $G$ is said to be $sigma $-full if $G$ has a Hall $sigma _{i}$-subgroup for all $i$. A subgroup $A$ of $G$ is said to be $sigma$-permutable in $G$ provided $G$ is $sigma $-full and $A$ permutes with all Hall $sigma _{i}$-subgroups $H$ of $G$ (that is, $AH=HA$) for all $i$. We obtain a characterization of finite groups $G$ in which $sigma$-permutability is a transitive relation in $G$, that is, if $K$ is a ${sigma}$-permutable subgroup of $H$ and $H$ is a ${sigma}$-permutable subgroup of $G$, then $K$ is a ${sigma}$-permutable subgroup of $G$.
Let $sigma ={sigma_i |iin I}$ is some partition of all primes $mathbb{P}$ and $G$ a finite group. A subgroup $H$ of $G$ is said to be $sigma$-subnormal in $G$ if there exists a subgroup chain $H=H_0leq H_1leq cdots leq H_n=G$ such that either $H_{i-1
Let $mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${cal L}_{mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}in mathfrak{F}$. A chief factor $H/K$ of $G$ is $mathfrak{F}$-central in $G$ if $(H/K)rtimes
We construct several series of explicit presentations of infinite hyperbolic groups enjoying Kazhdans property (T). Some of them are significantly shorter than the previously known shortest examples. Moreover, we show that some of those hyperbolic Ka
In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.
Let G be a group and DS(G) = { H| H is any subgroup of G}. G is said to be a DC-group if DS(G) is a chain. In this paper, we prove that a finite DC-group is a semidirect product of a Sylow p-subgroup and an abelian p-subgroup. For the case of G being