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Thread: S_n generated by two elements

  1. #1
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    S_n generated by two elements

    Hello all , i have tried to show it with no progress .

    Can someone help me with that one.

    Thanks a lot .
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  2. #2
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    The transposition $\displaystyle \sigma = (1\;2)$ and the cycle $\displaystyle \tau = (1\;2\;3\;\ldots\; n)$ generate $\displaystyle S_n$. In fact $\displaystyle \tau^{k-1}\sigma\tau^{-k+1} = (k\;k{+}1)$, so all transpositions of consecutive numbers are in the subgroup generated by $\displaystyle \sigma$ and $\displaystyle \tau$. Then any transposition can be expressed as a product of those, for example $\displaystyle (1\;3) = (1\;2)(2\;3)(1\;2)$. Finally, it is well known that the transpositions generate the whole of $\displaystyle S_n$.
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