108 research outputs found
Commutator maps, measure preservation, and T-systems
Let G be a finite simple group. We show that the commutator map is almost equidistributed as the order of G goes to infinity. This
somewhat surprising result has many applications. It shows that for a subset X
of G we have , namely is almost measure
preserving. From this we deduce that almost all elements can be
expressed as commutators where x,y generate G. This enables us to
solve some open problems regarding T-systems and the Product Replacement
Algorithm (PRA) graph. We show that the number of T-systems in G with two
generators tends to infinity as the order of G goes to infinity. This settles a
conjecture of Guralnick and Pak. A similar result follows for the number of
connected components of the PRA graph of G with two generators. Some of our
results apply for more general finite groups, and more general word maps. Our
methods are based on representation theory, combining classical character
theory with recent results on character degrees and values in finite simple
groups. In particular the so called Witten zeta function plays a key role in
the proofs.Comment: 28 pages. This article was submitted to the Transactions of the
American Mathematical Society on 21 February 2007 and accepted on 24 June
200
Mixing and generation in simple groups
AbstractLet G be a finite simple group. We show that a random walk on G with respect to the conjugacy class xG of a random element xβG has mixing time 2. In particular it follows that (xG)2 covers almost all of G, which could be regarded as a probabilistic version of a longstanding conjecture of Thompson. We also show that if w is a non-trivial word, then almost every pair of values of w in G generates G
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