Find a generating set for Sym(p)

Let p be a prime. Show that S_p = \langle \sigma, \tau \rangle where \sigma is any transposition and \tau any p-cycle.


Let \sigma = (a_1\ a_2) and \tau = (a_1\ b_2\ \ldots\ b_p). (We have a_2 = b_i for some i.) By a previous exercise, \tau^k(a_1) = a_2 for some k. Because \tau has prime order, \langle \sigma, \tau \rangle = \langle \sigma, \tau^k \rangle. Relabeling the elements \{ 1, \ldots, n \}, by the previous exercise we have S_p = \langle \sigma, \tau^k \rangle = \langle \sigma, \tau \rangle.

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