On orbits of automorphism groups. II.
Let \(A\) be a group of automorphisms of a finite group \(G\), let \(C_G(A)\) be the fixed point subgroup, and \(k_A(G)\) the number of \(A\)-orbits in \(G\). It is proved that if \(|C_G(A)|\) is coprime to \(k_A(G)\), then \(C_G(A)\subseteq [G,A]\) and the core of \(C_G(A)\) in \(G\) is contained in the centre of \([G,A]\). An automorphism \(\alpha\) is said to be class-avoiding, if \(x^\alpha\) belongs to the conjugacy class of \(x\) in \(G\) only if \(x^\alpha=x\). Any fixed-point free automorphism is class-avoiding by a result of Glauberman. Extending (and using) Thompson's celebrated theorem, the authors prove that if a class avoiding automorphism \(\alpha\in\Aut\,G\) of prime order coprime to \(|G|\) is such that \(C_G(\alpha)\leq[G,\alpha]\), then \(G\) is nilpotent. Similarly, based on the classification, solubility of \(G\) is derived for any class-avoiding automorphism \(\alpha\) if \((|\alpha|,|G|)=1\) and \(C_G(\alpha)\leq[G,\alpha]\). For part I cf. Sib. Mat. Zh. 46, No. 3, 533-537 (2005); translation in Sib. Math. J. 46, No. 3, 413-416 (2005; Zbl 1096.20025).
scientific article; zbMATH DE number 5565247
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