Mutually permutable subgroups and Fitting classes.
All groups considered are finite. Recall that two subgroups \(A\) and \(B\) of a group \(G\) are said to permute if \(AB\) is a subgroup of \(G\). Subgroups \(A\) and \(B\) are called mutually permutable if every subgroup of \(A\) permutes with \(B\) and every subgroup of \(B\) permutes with \(A\). A Fitting class is a class of groups which is both \(S_n\)-closed and \(N_0\)-closed, that is, a class of groups \(\mathfrak F\) is a Fitting class if \(\mathfrak F\) has the following two properties: 1. If \(G\in\mathfrak F\) and \(N\) is a subnormal subgroup of \(G\), then \(N\in\mathfrak F\), and 2. If \(N_1\) and \(N_2\) are subnormal subgroups of a group \(G\) and \(G=\langle N_1,N_2\rangle\), then \(G\in\mathfrak F\). \textit{J. C. Beidleman} and \textit{H. Heineken} [Arch. Math. 85, No. 1, 18-30 (2005; Zbl 1103.20015)] have shown that in a mutually permutable product \(G=AB\) the commutator subgroups \(A'\) and \(B'\) of \(A\) and \(B\), respectively, are subnormal in \(G\). In this paper the dual statement is proved. That is, if \(G=AB\) is a mutually permutable product of \(A\) and \(B\), then \(A\in\mathfrak F\) and \(B\in\mathfrak F\) implies \(G'\in\mathfrak F\) where \(\mathfrak F\) is a Fitting class.
scientific article; zbMATH DE number 5162043
scientific article