Finite simple groups have many classes of prime order elements
Jessica Anzanello, Pablo Spiga
TL;DR
This work strengthens prior bounds that restrict the order of a finite non-abelian simple group $T$ by the number of Aut$(T)$-conjugacy classes of elements of prime-power order. It replaces prime-power with prime-order counts via $\text{mpr}_p(T)$ and the maximal $\text{mpr}(T)$, and introduces the refined invariant $\text{mpr}^*(T)$ for Lie-type groups, proving $|T|\le f(\text{mpr}(T))$ (and $|T|\le f(\text{mpr}^*(T))$ in Lie types) for some increasing $f$. The method combines controlling normalizers of cyclic subgroups with cyclotomic-prime arguments (Stewart) to locate primitive divisors, translating Aut$(T)$-class data into explicit size bounds; the results are established separately for alternating, classical, and exceptional groups and connect to number-theoretic conjectures and derangement-graph phenomena. These bounds enhance our understanding of how the distribution of prime-order elements constrains the global size of finite simple groups.
Abstract
Let $T$ be a finite non-abelian simple group. Giudici, Morgan and Praeger have shown that the order of $T$ is bounded above by a function depending on the maximum number of $\mathrm{Aut}(T)$-classes of elements of $T$ of prime-power order. In this note, we strengthen this result by showing, in particular, that prime-power can be replaced by prime.