Finite simple groups have many classes of $p$-elements
Michael Giudici, Luke Morgan, Cheryl E. Praeger
TL;DR
The paper proves that the order of any finite nonabelian simple group $T$ is bounded above by an increasing function of $m(T)$, where $m(T)=\max_p m_p(T)$ and $m_p(T)$ counts Aut$(T)$-classes of $p$-power order elements. It extends Pyber's bounds by replacing total conjugacy-class data with $p$-element class data and develops a framework (pp-boundedness) to relate Aut$(G)$-classes to normalisers of cyclic subgroups, then applies it to alternating, classical, and exceptional groups via detailed analysis of maximal tori and subgroup structure. The paper derives explicit bounds for each family (alternating, classical, exceptional, sporadic) and combines them to obtain an overall bound $|T|\le f(m(T))$ for some increasing function $f$, with comments on the potential for sharper functions. These results have implications for relative Brauer groups of finite field extensions and connect to broader themes in the distribution of conjugacy classes and $p$-elements in finite groups. The methods blend group-theoretic structure (tori, normalisers, unipotent classes) with combinatorial counting of Aut$(T)$-classes to achieve global size bounds from local element-order data.
Abstract
For an element $x$ of a finite group $T$, the $\mathrm{Aut}(T)$-class of $x$ is the set $\{ x^σ\mid σ\in \mathrm{Aut}(T)\}$. We prove that the order $|T|$ of a finite nonabelian simple group $T$ is bounded above by a function of the parameter $m(T)$, where $m(T)$ is the maximum, over all primes $p$, of the number of $\mathrm{Aut}(T)$-classes of elements of $T$ of $p$-power order. This bound is a substantial generalisation of results of Pyber, and of Héthelyi and Külshammer, and it has implications for relative Brauer groups of finite extensions of global fields.