An analogue for some Fitting classes to a theorem of Stellmacher (Q1909728)

A group \(H\) is weakly \(p\)-stable if for every \(h\in H\), \([V, h, h]=1\) implies \(h C_H(V)\in O_p (H/C_H (V))\) for every abelian normal \(p\)-subgroup \(V\) of \(H\) and for \(V=O_p(H)\). The following theorem of Stellmacher is well-known: Let \(p\) be an odd prime and \(S\) a \(p\)-group. Then there exists a characteristic subgroup \(W(S)\) of \(S\) so that \(Z(S)\subseteq W(S)\) and \(W(S)\) is normal in every weakly \(p\)-stable group \(H\) such that \(C_H (O_p (H))\subseteq O_p(H)\) with \(S\in\text{Syl}_p (H)\). In this paper is proved an analogue for some Fitting classes of the theorem of Stellmacher. Let \(C^*_G(H)\) be the largest normal subgroup of \(N_G (H)\) acting nilpotently on \(H\). In what follows \(\mathcal F\) will denote a Fitting class with \(\text{char} ({\mathcal F})=\pi\). Let \(H\) be a group such that each subnormal subgroup of \(H\) has a unique conjugacy class of \(\mathcal F\)-injectors. A group \(H\) verifies (*) if \(C^*_H (H_{\mathcal F})\subseteq F(H)\) and verifies (**) if \(C_H(H_{\mathcal F})\subseteq H_{\mathcal F}\). If \(S\) is a group, an embedding of \(S\) is a pair \((\tau, H)\) where \(H\) is a group and \(\tau\) is a monomorphism from \(S\) to \(H\). Two embeddings \((\tau_1, H_1)\) and \((\tau_2, H_2)\) of \(S\) are equivalent if there exists an isomorphism \(\varphi\) from \(H_1\) onto \(H_2\) such that \(\tau_1\varphi=\tau_2\). Let \(\pi\) be a set of primes. A group \(H\) is weakly \(\mathfrak N\)-stable if for every \(x\in H\), \([V, x, x] =1\) implies \(x C_H(V)\in F_\pi (H/C_H(V))\), for every abelian normal \(\pi\)-subgroup \(V\) of \(H\) and \(V=F_\pi (H)\). Theorem. Let \(S\) be an \(\mathcal F\)-group and \(\mathcal H\) a class of embeddings of \(S\) which are weakly \({\mathfrak N}_\pi\)-stable groups verifying (*) and such that \(S\) is an \(\mathcal F\)-injector of \(H\), for every \(H\in {\mathcal H}\). Then \(Z=Z(S)\subseteq\text{Core}_{\mathcal H} (S)\) where \(\text{Core}_{\mathcal H}(S)\) is the largest subgroup of \(S\) which is normal in every \(H\in\mathcal H\). Corollary. Let \(S\) be an \(\mathcal F\)-group. Then there exists a characteristic subgroup \(W(S)\) such that \(Z(S)\subseteq W(S)\) and \(W(S)\) is normal in every weakly \({\mathfrak N}_\pi\)-stable group \(H\) verifying (**) such that \(S\in\text{Inj}_{\mathcal F}(H)\). We recover Stellmacher's theorem quoted at the beginning by taking \({\mathcal F}={\mathfrak N}_p\) in the above Corollary.