An upper bound on random walks on dihedral groups

In this paper we look at an upper bound for the rate of convergence to stationarity of two specific random walks on the dihedral group. The first theorem discusses a random walk generated with equal probabilities by one rotation and one flip. We show that roughly p2 steps are sufficient for the walk to become close to uniformly distributed on all of D2p where p≥3 is an integer. Next we take a random walk on the dihedral group generated by a random k-subset of the dihedral group. The later theorem shows that it is sufficient to take roughly p2/(k−1) steps in the typical random walk to become close to uniformly distributed on all of D2p. We note that there are at least one rotation and one flip in the k-subset or the random walk generated by this subset has periodicity problems or will not generate all of D2p.