Solvable finite groups with a particular configuration of Fitting sets
In this paper groups are finite. \textit{W. Anderson} [PhD thesis, Michigan State Univ. (1973)] defined a Fitting set of a group \(G\) as a collection \(\mathcal F\) of subgroups of \(G\) such that: i) every normal subgroup of a group in \(\mathcal F\) belongs to \(\mathcal F\); ii) if \(S\) and \(T\) are in \(\mathcal F\) and normalise each other, then \(ST\) belongs to \(\mathcal F\); iii) if \(S\) belongs to \(\mathcal F\), so does each of its conjugates in \(G\). In view of a result by \textit{K. Doerk} and \textit{T. Hawkes} [Finite Soluble Groups (1992; Zbl 0753.20001), VIII, 3.3], the author calls a Fitting set \(\mathcal F\) of \(G\) elementary if there exists a subgroup \(H\) of \(G\) such that \({\mathcal F}=s_n H^G\). If \({\mathcal F}\) is a Fitting class and \(G\) is a group, then the trace of \(\mathcal F\) in \(G\), \(\text{tr}_{\mathcal F} (G)\) is defined to be the set of all subgroups of \(G\) that belong to \(\mathcal F\); clearly, \(\text{tr}_{\mathcal F} (G)\) is a Fitting set of \(G\). Let \({\mathcal N}^2\) denote the class of groups having nilpotency length at most 2. Theorems 2.3 and 2.5: Let \(G\) be a soluble but not nilpotent group; then the following statements are equivalent: (i) \(\text{tr}_{{\mathcal N}^2} (G)\) is a disjoint union of elementary Fitting sets of \(G\); (ii) every Fitting set of \(G\) is a disjoint union of elementary Fitting sets; (iii) \(G \simeq \left\{\left(\begin{smallmatrix} x \\ \alpha x+\beta\end{smallmatrix}\right):\;\alpha \in H \leq\text{GF}(p^n)^\times,\;\beta \in \text{GF} (p^n) \right\}\), where \(n\) is the order of \(p\) modulo \(q\), for each \(q\) dividing the order of \(H\); (iv) every Fitting set of \(G\) is a disjoint union of at most two elementary Fitting sets.
scientific article; zbMATH DE number 817258