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$$X$$-quasinormal subgroups. (Russian, English) Zbl 1153.20304
Sib. Mat. Zh. 48, No. 4, 742-759 (2007); translation in Sib. Math. J. 48, No. 4, 593-605 (2007).
Summary: Considering two subgroups $$A$$ and $$B$$ of a group $$G$$ and $$\emptyset\neq X\subseteq G$$, 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.

MSC:
 20D40 Products of subgroups of abstract finite groups
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