W. Guo, A. N. Skiba, K. P. Sham, “<nobr>$X$</nobr>-quasinormal subgroups”, Sibirsk. Mat. Zh., 2007, Volume 48, Number 4,Pages <nobr>742

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$X$-quasinormal subgroups

W. Guo^a, A. N. Skiba^b, K. P. Sham^c

^a Department of Mathematics, Xuzhou Normal University, Xuzhou, P. R. China
^b Francisk Skaryna Gomel State University, Faculty of Mathematics, Gomel, Belarus
^c Depatrment of Mathematics, The Chinese University of Hong Kong, Hong Kong, P. R. China (SAR)

Abstract: Considering two subgroups $A$ and $B$ of a group $G$ and $\varnothing\ne X\subseteq G$, we say that $A$ is $X$-permutable with $B$ if $AB^x=B^xA$ for some element $x\in X$. We use this concept to give new characterizations of the classes of solvable, supersolvable, and nilpotent finite groups.

Keywords: Sylow subgroup, supplement to a subgroup, maximal subgroup, nilpotent group, supersolvable group, solvable group, $X$-quasinormal subgroup.

Received: 26.01.2006