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• W2036938863 abstract "Let φ be an automorphism of prime order p of a finite group G, and let r be the (Prüfer) rank of the fixed-point subgroup CG(φ). It is proved that if G is nilpotent, then there exists a characteristic subgroup C of nilpotency class bounded in terms of p such that the rank of G/C is bounded in terms of p and r. For infinite (locally) nilpotent groups a similar result holds if the group is torsion-free (due to Makarenko), or periodic, or finitely generated; but examples show that these additional conditions cannot be dropped, even for nilpotent groups. As a corollary, when G is an arbitrary finite group, the combination with the recent theorems of the author and Mazurov gives characteristic subgroups R ⩽ slant N ⩽ slant G such that N/R is nilpotent of class bounded in terms of p while the ranks of R and G/N are bounded in terms of p and r (under the additional unavoidable assumption that p∤ |G| if G is insoluble); in general it is impossible to get rid of the subgroup R. The inverse limit argument yields corresponding consequences for locally finite groups." @default.
• W2036938863 created "2016-06-24" @default.
• W2036938863 creator A5076019015 @default.
• W2036938863 date "2007-12-11" @default.
• W2036938863 modified "2023-10-18" @default.
• W2036938863 title "Groups with an automorphism of prime order that is almost regular in the sense of rank" @default.
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• W2036938863 doi "https://doi.org/10.1112/jlms/jdm097" @default.
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