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- László Babai, Aner Shalev
- 2008

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 survey recent progress, made using probabilistic methods , on several conjectures concerning nite groups. 1 Random generation In recent years probabilistic methods have proved useful in the solution of several diicult problems concerning nite groups; these involve conjectures on nite simple groups and on nite permutation groups. In some cases the… (More)

- Alexander Gamburd, Shlomo Hoory, Mehrdad Shahshahani, Aner Shalev, Bálint Virág
- Random Struct. Algorithms
- 2009

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.

We prove that if a finitely generated profinite group G is not generated with positive probability by finitely many random elements, then every finite group F is obtained as a quotient of an open subgroup of G. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley… (More)

- László Pyber, Aner Shalev
- Combinatorica
- 1996

- Julia Kempe, Aner Shalev
- SODA
- 2005

We employ concepts and tools from the theory of finite permutation groups in order to analyse the Hidden Subgroup Problem via Quantum Fourier Sampling (QFS) for the symmetric group. We show that under very general conditions both the weak and the random-strong form (strong form with random choices of basis) of QFS fail to provide any advantage over… (More)

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)

- Shelly Garion, Aner Shalev
- 2007

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)