Conditional permutability of subgroups and certain classes of groups (Q511867)

Let \(G\) be a finite group. Two subgroups \(A\) and \(B\) of \(G\) are said to be \textit{c-permutable} if \(AB^g=B^gA\) for some \(g\in G\), and they are called \textit{cc-permutable} if the element \(g\) can be chosen in \(\langle A,B\rangle\). Moreover, \(A\) and \(B\) are said to be \textit{tcc-permutable} if every subgroup of \(A\) is cc-permutable with all subgroups of \(B\). This paper deals with the structure of a finite group \(G=AB\) factorized by two tcc-permutable subgroups \(A\) and \(B\). Among other results, the authors prove for instance that all chief factors of \(G\) are simple if and only if \(A\) and \(B\) have the same property. Moreover, \(G\) has the property that each subnormal subgroup permutes with all maximal subgroups if and only if the same holds in \(A\) and in \(B\).