FINITE SIMPLE GROUPS HAVE MANY CLASSES OF p-ELEMENTS

For an element x of a finite group T, the Aut(T)-class of x is {x sigma | sigma is an element of 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 Aut(T)-classes of elements of T of p-power order. This bound is a substantial generalisation of the results of Pyber (1992) and of H & eacute;thelyi and K & uuml;lshammer (2005), and it has implications for relative Brauer groups of finite extensions of global fields.