In 1988 Persi Diaconis considered the following random walk on Zn . Pick k values from Zn and then repeatedly choose one of these values at random. This choice determines a random walk starting at 0 and the natural question that arises is, how long does it take the walk to get close to uniformly distributed on Zn for "most" choices of the k values. In 1989 A. Greenhalgh, in his thesis, gave a lower bound for a random walk on Zn . A paper written by J. Dai and M. Hildebrand in 1997 considered random walks on Zn without restrictions on n and in the case where k is a constant. It is important to note that the set of k values were chosen at random but were subject to certain divisibility restrictions. With these restrictions, it will take slightly over n2/(k-1) steps for the position of the typical random walk to be close to uniformly distributed on Zn . Using the same divisibility restriction this thesis will find a similar bound for a random walk on ZpxZ pn where p is a constant.; In 1992 C. Dou considered some cases of the random walk on an abelian group of order n where n is not prime, but instead n = p1··· pt where t is no more than a constant. The p1,···,p t are all prime, with p1 ≥ ··· ≥ pt and p1 ≤ Ap t for some constant A. We will show that if you relax some of the conditions that Dou had on t, m, and A then we can extend his work to include more examples that Dou's restrictions could not.; In 1997 Farid Bassiri in his thesis proved some bounds on a random walk on the dihedral group. This thesis will give a comprehensive study of the random walks on the dihedral group generated by a k-subset.
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