The normal Fitting class generated by a finite soluble group (Q1210157)

For a given group \(G\), we denote by \(\text{Fit}(G)\) [resp. NormFit\((G)\)] the intersection of all the [normal] Fitting classes to which \(G\) belongs: this is called the [normal] Fitting class generated by \(G\). Transfer Fitting pairs were introduced by \textit{H. Laue}, \textit{H. Lausch} and \textit{G. R. Pain} [Math. Z. 154, 257-260 (1977; Zbl 0337.20013)] while \textit{T. R. Berger} [Proc. Lond. Math. Soc., III. Ser. 42, 59-86 (1981; Zbl 0456.20006)] proved that such Fitting pairs in fact suffice to describe the minimal normal Fitting class. The paper shows that for any group \(G\), NormFit\((G)\) can also be described in terms of transfer Fitting pairs. In particular, a characterization of NormFit\((S_ 3)\), where \(S_ 3\) is the symmetric group on 3 objects is given.