Finite simple groups have many classes of \(p\)-elements

For an element \(x\) of a finite group \(T\), the \(\mathrm{Aut}(T)\)-class of \(x\) is the set \(\{ x^\sigma\mid \sigma\in \mathrm{Aut}(T)\}\). 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 K\"ulshammer, and it has implications for relative Brauer groups of finite extensions of global fields.