From: Douglas Zare Subject: Re: Commuting permutations in S_n Date: Wed, 18 Jul 2001 17:33:19 -0400 Newsgroups: sci.math Summary: Probability that two given elements generate the symmetric group or A_n Robert Israel wrote: > In article , > Sharon wrote: > >If we choose at random uniformly two permutations p1,p2 in the > >symmetric group S_n what is the probability that p1,p2 commute with > >each other ?[...] > [...] > So the probability of commuting is P(n)/n!, where P(n) is the number > of partitions of n. > > I don't recall the asymptotics of P(n), but surely the limiting > probability must be 0. Partitions grow subexponentially (exp(c sqrt(n))) and n! grows superexponentially. For the question asked elsewhere on this thread, about the limiting probability that two elements generate a group, for the alternating and symmetric groups this was settled by Dixon: 40 #4985 60.10 (20.00) Dixon, John D. The probability of generating the symmetric group. Math. Z. 110 1969 199--205. and there has been a lot of more recent work generalizing this result (with much different techniques) to infinite families of finite simple groups. For two elements not to generate the whole group, they must lie in the same maximal subgroup, so the study of the maximal subgroups of the finite simple groups was relevant. Probabilistic arguments have also been used to show that almost all finite simple groups are quotients of the modular group: Shalev and Liebeck showed that the probability that an element of order 2 and an element of order 3 generate the whole group goes to 1 for certain families of finite simple groups. Douglas Zare