On conditional permutability and factorized groups.
Two subgroups \(A,B\) of a finite group \(G\) are called tcc-permutable if for all pairs of subgroups \(X\subseteq A\) and \(Y\subseteq B\) there is an element \(g\in\langle X,Y\rangle\) such that \(X\) permutes with \(Y^g\). This is a generalization of total permutability. The authors show that certain results following from total permutability are still true for tcc-permutability. For instance, if \(G=G_1\cdots G_k\) is the product of pairwise tcc-permutable subgroups \(G_j\) and \(F\) is a saturated formation containing all supersoluble groups, then \(G_j\in F\) for all \(j\) and \(G\in F\) are equivalent (Theorem 5).
completely conditionally permutable subgroups
scientific article; zbMATH DE number 6342896
scientific article