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Keywords:
finite group; supersolublilty; c-permutable subgroup; maximal subgroup series
finite group; supersolublilty; c-permutable subgroup; maximal subgroup series
Summary:
Let $H$, $K$ be subgroups of $G$. Then $H$ and $K$ are called c-permutable in $G$ if there exists an element $x\in G$ such that $HK^x = K^xH$. By imposing c-permutability conditions on the members of a maximal subgroup series of $G$, we give some new criteria for the supersolubility of a finite group. We prove that if $G$ is a soluble group with a maximal subgroup series $1=M_0< M_1<\cdots < M_{s-1}< M_s=G$, where $M_i$ is c-permutable in $G$ for every $i=0,1,\cdots ,s$, then $G$ is supersoluble. On this basis, if $G$ is not soluble, but we have the condition that $M_i$ is c-permutable in $M_{i+1}$ $(0\leq i\leq s-1)$, then $G$ is also supersoluble. These results provide new characterizations of supersoluble groups.
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