It is well-known that the number of conjugacy classes of a finite group G tends to infinity as the size of G tends to infinity. There is no such result for a general infinite group. In this talk I will discuss the situation when G is a profinite group and show that the number of conjugacy of G is then uncountable unless G is finite. The proof depends on many classical results on finite groups and in particular the classification of the finite simple groups.
By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I'll talk about ongoing work to classify the totally deranged elements of almost simple groups, and I'll mention how this solves a question of Garzoni about invariable generating sets for simple groups.
I will discuss joint work with Bob Guralnick and Russ Woodroofe. We investigate invariable generation of finite simple groups by two elements of prime or prime power order. We apply our results to a problem raised by Ken Brown on the topology of the order complex of the poset of all cosets of all proper subgroups of an arbitrary finite group.
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