Abstract: 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 sqrt{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 at least 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/sqrt{n}.
