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Register a new userA Robinson characterization of finite $Psigma T$-groups
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Authors
Alexander N. Skiba
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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 $G$.
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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 $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is a finite $sigma_j$-group for some $j in I$ for $i = 1, ldots, n$. We call a finite group $G$ a $T_{sigma}$-group if every $sigma$-subnormal subgroup is normal in $G$. In this paper, we analyse the structure of the $T_{sigma}$-groups and give some characterisations of the $T_{sigma}$-groups.
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