First Hitting Times for Some Random Walks on Finite Groups

We consider a random walk on a finite group G based on a generating set that is a union of conjugacy classes. Let the nonnegative integer valued random variable T denote the first time the walk arrives at the identity element of G, if the starting point of the walk is uniformly distributed on G. Under suitable hypotheses, we show that the distribution function F of T is almost exponential, and we give an error term.