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Problemy Fiziki, Matematiki i Tekhniki (Problems of Physics, Mathematics and Technics), 2013, Issue 4(17), Pages 47–54
(Mi pfmt270)
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This article is cited in 1 scientific paper (total in 1 paper)
MATHEMATICS
On some generalizations of permutability and $S$-permutability
Xiaolan Yia, A. N. Skibab a Zhejiang Sci-Tech University, Hangzhou, China
b F. Scorina Gomel State University, Gomel, Belarus
Abstract:
Let $H$ and $X$ be subgroups of a finite group $G$. Then we say that $H$ is: $X$-quasipermutable (respectively, $X_S$-quasipermutable) in $G$ provided $G$ has a subgroup $B$ such that $G=N_G(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (respectively, with all Sylow subgroups) $V$ of $B$ such that $(|H|,|V|)=1$; $X$-propermutable (respectively, $X_S$-propermutable) in $G$ provided $G$ has a subgroup $B$ such that $G=N_G(H)B$ and $H$ $X$-permutes with $B$ and with all subgroups (respectively, with all Sylow subgroups) of $B$.
In this paper we analyze the influence of $X$-quasipermutable, $X_S$-quasipermutable, $X$-propermutable and $X_S$-propermutable subgroups on the structure of $G$.
Keywords:
finite group, $X$-quasipermutable subgroup, $X_S$-quasipermutable subgroup, $X$-propermutable subgroup, $X_S$-propermutable subgroup, Sylow subgroup, Hall subgroup, $p$-soluble group, $p$-supersoluble group, maximal subgroup, saturated formation, $PST$-group, $PT$-group.
Received: 20.05.2013
Citation:
Xiaolan Yi, A. N. Skiba, “On some generalizations of permutability and $S$-permutability”, PFMT, 2013, no. 4(17), 47–54
Linking options:
https://www.mathnet.ru/eng/pfmt270 https://www.mathnet.ru/eng/pfmt/y2013/i4/p47
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