# Finite Groups and Probabilistic Combinatorics

## ABSTRACT

A number of combinatorial problems with the symmetric group can be resolved using probabilistic methods. In this talk we discuss some of these of problems, including the following two:
1) Given a permutation s \in S_n, and a random conjugate t of it, what is the probability that s and t commute?
2) Given a word w(X,Y), what is the probability that two random elements x,y \in S_n satisfy w(x,y)=1?

Applications include recognition algorithms for finite groups, and a connection to Magnus's conjecture about a residual property of free groups. The talk assumes no group theoretic background and should be accessible to a general audience.

Speaker's Contact Info: akos(at-sign)math.ohio-state.edu