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On the Number of p-Regular Elements in Finite Simple Groups

Published online by Cambridge University Press:  01 February 2010

László Babai ,
Péter P. Pálfy  and
Jan Saxl
László Babai
Affiliation:
Department of Computer Science, University of Chicago, 1100 East 58th Street, Chicago, IL 60637, USA, laci@cs.tod.uchicago.tod.edu
Péter P. Pálfy
Affiliation:
Rényi Institute and Eötvös University, 13–15 Reáltanoda u., 1053 Budapest, Hungary, ppp@renyi.tod.hu
Jan Saxl
Affiliation:
Gonville and Caius College, Trinity Street, Cambridge CB2 1TA, United Kingdom, J.Saxl@dpmms.tod.cam.tod.ac.tod.uk

Abstract

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A p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n – 1 is the dimension of the projective space on which S acts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1/15. For the alternating group An, this proportion is at least 26/(27√n), and for sporadic simple groups, at least 2/29.

We also show that for an arbitrary field F, if the simple group S is a quotient of a finite subgroup of GLn(F) then for any prime p, the proportion of p-regular elements in S is at least min{1/31, 1/(2n)}.

Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998).Our result shows that in finite simple groups, p-regular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomial-time Monte Carlo algorithms for matrix groups.

Finally we complement our lower bound results with the following upper bound: for all n ≥ 2 there exist infinitely many prime powers q such that the proportion of elements of odd order in PSL(n,q) is less than 3/√n.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 12 , 2009 , pp. 82 - 119
Copyright
Copyright © London Mathematical Society 2009

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