On some non-isomorphic simple groups with equalities on their number of elements orders

Abstract

For all prime powers q such that , we prove that and have the same number of involutions and there exist divisors of such that these groups have the same number of elements of order r. If , a similar result is proved for the groups and . These provide an infinite sequence of pairs of non-isomorphic simple groups that have the same number of involutions and for various odd integers r, the same number of elements of order r. Our results address two recent questions.