Aner Shalev

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We consider the asymptotic complexity of manipulating matrix groups over finite fields. The question is, given a matrix group G by a list of generators, what can we say in polynomial time about the structure of G? While considerable progress has been made recently in identifying the nonabelian composition factors of a matrix group, the fundamental question(More)
We prove that random d-regular Cayley graphs of the symmetric group asymp-totically almost surely have girth at least (log d−1 |G|) 1/2 /2 and that random d-regular Cay-ley graphs of simple algebraic groups over F q asymptotically almost surely have girth at least log d−1 |G|/ dim(G). For the symmetric p-groups the girth is between log log |G| and (log |G|)(More)
Let w be a non-trivial word in two variables. We prove that the probability that two randomly chosen elements x, y of a nonabelian finite simple group S satisfy w(x, y) = 1 tends to 0 as |S| → ∞. As a consequence, we obtain a new short proof of a well-known conjecture of Magnus concerning free groups, as well as some applications to profinite groups.
Let G be a finite simple group and let S be a normal subset of G. We determine the diameter of the Cayley graph Γ(G, S) associated with G and S, up to a multiplicative constant. Many applications follow. For example, we deduce that there is a constant c such that every element of G is a product of c involutions (and we generalize this to elements of(More)
Waring's classical problem deals with expressing every natural number as a sum of g(k) kth powers. Recently there has been considerable interest in similar questions for nonabelian groups, and simple groups in particular. Here the kth power word is replaced by an arbitrary group word w = 1, and the goal is to express group elements as short products of(More)