Finite Groups of Bounded Rank with an Almost Regular Automorphism of Prime Order

Author: Khukhro E.I.  

Publisher: Springer Publishing Company

ISSN: 0037-4466

Source: Siberian Mathematical Journal, Vol.43, Iss.5, 2002-09, pp. : 955-962

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Abstract

We prove that if a finite group G of rank r admits an automorphism varphi of prime order having exactly m fixed points, then G has a varphi-invariant subgroup of (r,m)-bounded index which is nilpotent of r-bounded class (Theorem 1). Thus, for automorphisms of prime order the previous results of Shalev, Khukhro, and Jaikin-Zapirain are strengthened. The proof rests, in particular, on a result about regular automorphisms of Lie rings (Theorem 3). The general case reduces modulo available results to the case of finite p-groups. For reduction to Lie rings powerful p-groups are also used. For them a useful fact is proved which allows us to “glue together” nilpotency classes of factors of certain normal series (Theorem 2).