Geodesic
Some criteria for the nonsimplicity of finite groups
Problemy fiziki, matematiki i tehniki, no. 2 (2018), pp. 60-68.

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Let $|G|=\prod_{i=1}^n p_i^{\alpha_i}$, where $p_i$ are prime numbers, $p_i\ne p_j$ for $i\ne j$. Let $\pi(G)=\{p_1,\dots,p_n\}$, $s\in\pi(G)$ and let $\mathfrak{T}$ is the set of some Sylow subgroups of the group $G$, that are taken one at a time for every $p_i\in\pi(G)\setminus\{s\}$, $i=\overline{1,n-1}$. It is proved that if every subgroup from the set $\mathfrak{T}$ normalises some non-identity $s$-subgroup from $G$, $s>3$, then $G$ has solvable normal subgroup $R$ and $s$ divide $|R|$.
Keywords: finite group, Sylow subgroup
Mots-clés : $s$-solvable group. 
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E. M. Palchik; S. Yu. Bashun. Some criteria for the nonsimplicity of finite groups. Problemy fiziki, matematiki i tehniki, no. 2 (2018), pp. 60-68. http://geodesic.mathdoc.fr/item/PFMT_2018_2_a8/

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