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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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