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 A n , 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 GL n (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 P SL(n, q) is less than 3/ √ n.

No abstract

No abstract

We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name of G, assuming in the case of

Given a black-box group G isomorphic to some finite simple group of Lie type and the characteristic of G, we compute the standard name of G by a Monte Carlo algorithm. The running time is polynomial in the input length and in the time requirement for the group operations in G.The algorithm chooses a relatively small number of (nearly) uniformly distributed random elements of G, and examines the divisibility of the orders of these elements by certain primitive prime divisors. We show that the divisibility statistics determine G, except that we cannot distinguish the groups PΩ(2m + 1, q) and PSp(2m, q) in this manner when q is odd and m ≥ 3. These two groups can, however, be distinguished by using an algorithm of Altseimer and Borovik.