On the Number ofp-Regular Elements in Finite Simple Groups

Abstract

AbstractAp-regular element in a finite group is an element of order not divisible by the prime numberp. We show that for every primepand every finite simple groupS, a fair proportion of elements ofSisp-regular. In particular, we show that the proportion ofp-regular elements in a finite classical simple group (not necessarily of characteristicp) is greater than 1/(2n), wheren– 1 is the dimension of the projective space on whichSacts 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 fieldF, if the simple groupSis a quotient of a finite subgroup ofGLn(F) then for any primep, the proportion ofp-regular elements inSis 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 alln≥ 2 there exist infinitely many prime powersqsuch that the proportion of elements of odd order inPSL(n,q) is less than 3/√n.