The number of elements of prime order

Abstract

It is expected (but not proved yet) that a simple group is determined by the set of element orders with multiplicities. There are some variations of this problem: it has been shown that some simple groups are determined simply by the set of element orders or simply by the set of multiplicities of element orders and the order of the group. A common feature of all of these variations is that the hypothesis is an unboundedly large set of integers. In this note, we propose that just two of these integers should suffice to essentially determine most simple groups: the number of elements of order p, where p is the largest prime divisor of the order of the simple group.