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Involution centralizers in matrix group algorithms

Introduction In this talk all groups are finite (by definition), and all simple groups are non-abelian. Let us first define our terms: an involution in a group G is an element t of order 2, i.e. t = 1 and t 6= 1. Its centralizer CG(t) is the subgroup of elements which commute with it, so CG(t) = {g ∈ G | tg = gt}. It has long been accepted in abstract group theory that the way to study simple groups is via their involution centralizers. But this orthodoxy has been slow to filter through into computational group theory. Part of the purpose of this talk is to advertise the usefulness of involution centralizer methods in this field.