Finite simple groups with the same numbers of elements of orders p and q Let $G$ be a nonabelian finite simple group, and let $p$ and $q$ be
distinct prime divisors of the order of $G$. Is it true that the
number of elements of $G$ of order $p$ never equals the number of elements
of $G$ of order $q$?
Remark: My husband ran a GAP computation checking all nonabelian finite simple groups
of order less than 400000000, and did not find a counterexample.
 A: Here is a partial (though not complete) answer which shows that the number of such elements is often different for pairs $\{p,q\}.$  As long as the prime $p$ divides $|G|$, the number of elements of order $p$ in $G$ is not divisible by $p$ (by virtue of Frobenius's theorem that the number of solutions of $x^{p} = 1$ in $G$ is a multiple of $p$, and the identity is present). Also, the number of elements of order $p$ in $G$ is a multiple of $p-1$, since distinct subgroups of order $p$ have only the identity in common, and each such subgroup contains $p-1$ elements of order $p$.
Hence if $p$ and $q$ are different primes which divide $|G|$ , and either $q|p-1$ or $p|q-1$, then the number of elements of order $p$ in $G$ is different from the number of elements of order $q$ in $G$ (consider the case $q|p-1$. Then the number of elements of order $p$ in $G$ is divisible by $p-1$, so by $q$, but the number of elements of order $q$ in $G$ is not divisible by $q$). Note that this eliminates the case $q = 2$ from all consideration in this problem.
Another (easy) reduction is when a Sylow $p$-subgroup $P$ of $G$ centralizes no element of order $q$ in $G$, but the prime $q$ divides $|G|$. Then the number of elements of order $q$ in $G$ is divisible by $p$, but the number of elements of order $p$ in $G$ is not. Hence we only need to consider cases where $q$ divides $|O_{p^{\prime}}(C_{G}(P))|$ and $p$ divides $|O_{q^{\prime}}(C_{G}(Q))|,$ where $Q$ is a Sylow $q$-subgroup of $G$.
Such considerations eliminate quite a number of pairs of primes for any given simple group $G$ (and similar arguments hold, to show for example that that if $P$ and $Q$ are both Abelian and commute, then $PQ$ can't be contained in a TI Hall subgroup).
But I imagine that a definitive answer to the question requires both use of the classification of finite simple groups, and very detailed examination of properties of simple groups of Lie type.
Later edit: If $G$ is a finite simple group of Lie type in characteristic $p$, then $C_{G}(U) = Z(U)$ for $U$ a Sylow $p$-subgroup of $G$, so that for any prime $r \neq p$, the number of elements of order $r$ in $G$ is divisible by $p$,  whereas the number of elements of order $p$ in $G$ is not divisible by $p$ ( the latter as remarked above in general). Hence for such a group $G$ we only need to consider pairs of primes $r,s$ such that the elements of order $r$ and the elements of order $s$ are semisimple.
