On subgroups of \(ZJ\) type of an \({\mathfrak F}\)-injector for Fitting classes \({\mathfrak F}\) between \({\mathfrak E}_{p^*p}\) and \({\mathfrak E}_{p^*}{\mathfrak S}_ p\) (Q1896477)

All groups considered here are finite. Let \(\mathbf F\) be a Fitting class such that \({\mathbf E}_{p^*p} \subseteq {\mathbf F} \subseteq {\mathbf E}_p * {\mathbf S}_p\). \textit{M. J. Iranzo} and \textit{M. Torres} [Rend. Semin. Mat. Univ. Padova 82, 233-237 (1989; Zbl 0707.20011)] showed that all finite groups have a conjugacy class of \(\mathbf F\) injectors which are \(\{(O_{p^*} (G)P)_{\mathbf F} \mid P\in \text{Syl}_p(G)\}\). In this paper for a \(p\)-stable group \(G\) (\(p\) an odd prime) such that \(O_p (G) \neq 1\) it is proved that \(1 \neq Z(J(K))\) with \(K\) an \(\mathbf F\)-injector of \(G\) and \(J(K)\) the Thompson subgroup of \(K\). When \(G\) is \(p\)-constrained and \(O_{p'} (G)=1\), then \(K=P \in \text{Syl}_p (G)\), so the classical Glauberman \(ZJ\)-theorem is obtained. For any group \(K\), the subgroup \(ZJ^*(K)\) is introduced inductively, and it is proved for a group \(G\) such that \(O_{p'} (F(G)) \leq Z(J(K))\) the equivalence: i) \(G\) is nilpotent constrained, ii) \(C_G (ZJ^* (K)) \leq ZJ^* (K)\).