The local \(C(G,T)\) theorem. (Q2498867)

A finite group \(G\) has characteristic \(p\), \(p\) a prime, if \(F^*(G)=O_p(G)\). For a subgroup \(S\) of a Sylow \(p\)-subgroup \(T\) of \(G\) one defines \(C(G,T)=\langle N_C(G)\mid 1\neq C\text{\,char\,}G\rangle\). In this paper finite groups of characteristic \(p\) are determined for which \(G\neq C(G,T)\) holds. For \(p=2\) this result -- called ``local \(C(G,T)\)-theorem'' -- was proven by \textit{M. Aschbacher} [Proc. Lond. Math. Soc., III. Ser. 43, 450-477 (1981; Zbl 0486.20012)]. It plays an important role in the classification of finite simple groups. \textit{D. Gorenstein} and \textit{R. Lyons} gave under the \(\mathcal K\)-group assumption an alternative proof [Isr. J. Math. 82, No. 1-3, 227-279 (1983; Zbl 0790.20029)]. In the paper under review the local \(C(G,T)\)-theorem is proved for every prime \(p\). The proof is completely independent of the work of Aschbacher and Gorenstein-Lyons. The authors also prove two variations of the local \(C(G,T)\)-theorem: again for \(T\in\text{Syl}_p(G)\) they define \(C^*(G,T)=\langle C_G(\Omega_1(Z(T))),C(G,B(T))\rangle\) and \(C^{**}(G,T)=\langle C_G(\Omega_1(Z(T))),C_G(J(T))\rangle\). Here \(J(T)\) is the Thompson subgroup of \(T\) and \(B(T)=C_T(\Omega_1(Z(J(T))))\) is the Baumann subgroup of \(T\). The authors prove the local \(C^*(G,T)\)-theorem and the local \(C^{**}(G,T)\)-theorem, i.e., they determine the finite groups of characteristic \(p\) with \(G\neq C^*(G,T)\) and \(G\neq C^{**}(G,T)\), respectively. This paper is a contribution to the project of the investigation of groups of local characteristic \(p\) [see \textit{U. Meierfrankenfeld}, \textit{B. Stellmacher} and \textit{G. Stroth}, in Groups, combinatorics and geometry, Durham 2001, World Scientific, 155-192 (2003; Zbl 1031.20008)].