Geodesic
Chains in finite groups
Problemy fiziki, matematiki i tehniki, no. 4 (2019), pp. 70-73.

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Let $\mathbb{N}$ and $\mathbb{P}$ be the set of all positive integers and all primes, respectively. A subgroup $H$ of $G$ is called $\mathbb{P}^\infty$-subnormal in $G$ ($H$ $\mathbb{P}^\infty$-$sn$ $G$) if there is a chain $H=H_0\subset H_1\subset\dots\subset H_{n-1}\subset H_n=G$ such that $|H_i:H_{i-1}|\in\mathbb{P}^\infty$ for every $i=1,\dots,n$, where $\mathbb{P}^\infty=\{p^k\mid p\in\mathbb{P}, k\in\{0\}\subset\mathbb{N}\}$. We obtained finite simple non-abelian groups $G$ with $1$ $\mathbb{P}^\infty$-$sn$ $G$.
Keywords: finite group, $\mathbb{P}^\infty$-subnormal subgroup.
Mots-clés : simple non-abelian group 
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V. N. Tyutyanov; A. A. Trofimuk. Chains in finite groups. Problemy fiziki, matematiki i tehniki, no. 4 (2019), pp. 70-73. http://geodesic.mathdoc.fr/item/PFMT_2019_4_a13/

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