Using the classification of the finite simple groups, the authors prove the following generalization of Glauberman’s \(Z^*\)-theorem. Let \(\pi\) be a set of primes and let \(x\) be a \(\pi\)-element of the finite group \(G\). Then the following are equivalent: (i) \(G=O_{\pi'}(G)C_G(x)\). (ii) For each \(g\in G\setminus C_G(x)\), the subgroup \(\langle x,x^g\rangle\) has a non-trivial normal \(\pi\)-complement, and furthermore \([O_{\pi'}(\langle x,x^g\rangle),x]\cap C_G(x)\leq O_{\pi'}(C_G(x))\). (iii) For each \(g\in G\setminus C_G(x)\), the subgroup \(\langle x,x^g\rangle\) has a non-trivial normal \(\pi\)-complement, and furthermore \(\langle[O_{\pi'}(\langle x,x^g\rangle),x]\cap C_G(x)\mid g\in G\rangle\) is a \(\pi'\)-group. When \(\pi=\{2\}\) and \(x\) is an involution, this theorem becomes the \(Z^*\)-theorem.
Authors’ comment: The reviewer has been misleading that the proof of theorem 1 and 2 make use of the classification of finite simple groups. They are purely character-theoretic (with some standard group theory), and do not make use of the classification of finite simple groups.