Shelly Garion

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Let G be a finite simple group. We show that the commutator map α : G × G → G is almost equidistributed as |G| → ∞. This somewhat surprising result has many applications. It shows that a for a subset X ⊆ G we have α −1 (X)/|G| 2 = |X|/|G| + o(1), namely α is almost measure preserving. From this we deduce that almost all elements g ∈ G can be expressed as(More)
The Product Replacement Algorithm is a practical algorithm for generating random elements of a finite group. The algorithm can be described as a random walk on a graph whose vertices are the generating k-tuples of the group (for a fixed k). We show that there is a function c(r) such that for any finite simple group of Lie type, with Lie rank r, the Product(More)