Finite groups with an automorphism of prime order whose centralizer has small rank.
Let \(\varphi\) be an automorphism of prime order \(p\) of a finite group \(G\), and \(C_G(\varphi)\) its fixed point subgroup. By the classical theorem of \textit{J. Thompson}, [Proc. Natl. Acad. Sci. USA 45, 578-581 (1959; Zbl 0086.25101)], if \(C_G(\varphi)=1\) (that is, \(\varphi\) is regular), then \(G\) is nilpotent. It is natural to expect that if \(\varphi\) is ``almost regular'', then \(G\) must be ``almost nilpotent''. The ``almost regular'' result of \textit{P. Fong}, [Osaka J. Math. 13, 483-489 (1976; Zbl 0372.20010)], and Hartley-Meixner-Pettet, [see \textit{B. Hartley, T. Meixner}, Arch. Math. 36, 211-213 (1981; Zbl 0447.20014) and \textit{M. R. Pettet}, J. Algebra 72, 404-412 (1981; Zbl 0473.20018)], gives the existence of a nilpotent subgroup of index bounded in terms of \(p\) and \(|C_G(\varphi)|\). In this paper, the authors prove an analogue of the Hartley-Meixner-Pettet theorem for soluble groups and an analogue of Fong's theorem in the case where the orders of \(G\) and \(\varphi\) are coprime (the examples in this paper show that there can be no rank analogue of Fong's theorem if the orders of \(G\) and \(\varphi\) are not coprime). Three rather complicated theorems are proved in the present paper.
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