# On solvability of finite groups with some *ss*-supplemented subgroups

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## Abstract

A subgroup *H* of a finite group *G* is said to be *ss*-supplemented in *G* if there exists a subgroup *K* of *G* such that *G* = *HK* and *H* ∩ *K* is *s*-permutable in *K*. In this paper, we first give an example to show that the conjecture in A.A. Heliel’s paper (2014) has negative solutions. Next, we prove that a finite group *G* is solvable if every subgroup of odd prime order of *G* is *ss*-supplemented in *G*, and that *G* is solvable if and only if every Sylow subgroup of odd order of *G* is *ss*-supplemented in *G*. These results improve and extend recent and classical results in the literature.

## Keywords

*ss*-supplemented subgroup solvable group supersolvable group

## MSC 2010

20D10 20D20## Preview

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© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2015