Abstract: The generation of finite simple groups has been a thriving area of research for many years. Since it was established that each is generated by a pair of elements, many interesting refinements have followed: for instance, determining the existence of generating pairs of prescribed orders.
More recently the notion of width has provided an additional perspective on generation, measuring how efficiently a chosen subset generates a group. For example we may ask, can every element be written as a product of at most 2, or perhaps 3, elements from a fixed conjugacy class? Answering such questions relies on a range of tools involving subgroup structure and character theory.
In this talk we will examine the width of finite simple groups with respect to elements of a fixed prime order. We will report on sharp bounds for particular families, and answer questions concerning Lie-type groups of unbounded rank.