Sufficient conditions for the $T(T_{0})$-solvability of finite groups

Abstract: Let $G$ be a finite group. We say that $G$ is a $T_{0}$-group if its Frattini quotient group $G/\Phi{(G)}$ is a $T$-group, where by a $T$-group we mean a group in which every subnormal subgroup is normal. In this paper, we investigate the structure of the group $G$ if $G$ is the product of two solvable $T$-groups ($T_{0}$-groups) $H$ and $K$ such that $H$ permutes with every subgroup of $K$ and $K$ permutes with every subgroup of $H$ (that is, $H$ and $K$ are mutually permutable) and that $(|G:H|, |G:K|)=1$. Some structure theorems are also discussed.

Keywords: $T$-groups, $T_ {0}$-groups, $PST$-groups, permutable subgroups, solvable groups, supersolvable groups, nilpotent groups

Classification (MSC2000): 20D10; 20D15, 20D20

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