Abstract
We consider a group G with an automorphism of finite, usually prime, order. If G has finite Hirsch number, and also if G satisfies various stronger rank restrictions, we study the consequences and equivalent hypotheses of having only finitely many fixed-points. In particular we prove that if a group G with finite Hirsch number \({\mathfrak{h}}\) admits an automorphism \({\varphi}\) of prime order p such that \({\vert C_{G}(\varphi) \vert = n < \infty,}\) then G has a subgroup of finite index bounded in terms of p, n and \({\mathfrak{h}}\) that is nilpotent of p-bounded class.
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Bettio, E., Jabara, E. & Wehrfritz, B.A.F. Groups Admitting an Automorphism of Prime Order with Finite Centralizer. Mediterr. J. Math. 11, 1–12 (2014). https://doi.org/10.1007/s00009-013-0317-6
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DOI: https://doi.org/10.1007/s00009-013-0317-6