Random walks on a finite group

In the early twentieth century, Markov, Poincare and Borel discussed the special instance of the convergence of random walks on finite groups associated with card shuffling, the underlying group in this discussion is the symmetric group. Consider, for example, the shuffling method used by good card players called riffle shuffling, a subject to which Persi Diaconis gives a marvelous introduction in [1]. The motivation of this thesis is to address the particular question, ``In a deck of cards, how many times of top-in shuffle should be performed before the top card goes back to the original position?" At first, we introduce several types of popular shuffling methods and describe the process of each with representations of the symmetric group, Markov chains and switching systems. Furthermore, with the support of C++ software, we simulate top-in shuffling for 6 cards and extend the simulation to 12 cards, obtaining conjectures about the relations that the data satisfies. Finally we shuffle the entire deck of cards to acquire the inherent statistics characteristic of top-in shuffle.