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I was wondering how to find a minimal set of generators for the symmetric groups. Would it be difficult to fill-in the following table?

##\begin{array}{cl}

S_3&=\big<(1\;2),(2\;3)\big> \\

S_4&=\big<(1\;2\;3\;4),(1\;2\;4\;3)\big>\\

\vdots\\

S_{500}

\end{array}

##

Is there a procedure to find them other than by trial an error? How about just ##S_{10}## for that matter?

I know how to find generator orders for the integer mod groups simply by finding their isomorphic additive groups. Can I do something similar to find for examples the orders of the generators needed to generate the symmetric groups?

Thanks,

Jack