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Journal of the London Mathematical Society Advance Access originally published online on December 11, 2007
Journal of the London Mathematical Society 2008 77(1):130-148; doi:10.1112/jlms/jdm097
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© 2007 London Mathematical Society

Groups with an automorphism of prime order that is almost regular in the sense of rank

E. I. Khukhro

School of Mathematics
Cardiff University
Cardiff
CF23 9ED
United Kingdom

Let {varphi} 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({varphi}). 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{nmid} |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.

2000 Mathematics Subject Classification 20D45 (primary), 20C05, 20D15, 20F18, 20F40, 20F50 (secondary).

Received May 16, 2007; published online December 11, 2007.


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