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Unsolvable series

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Find the derived series for S_4

Show that S'_n=A_n if n is not equal to 2 and conclude that S_n is not solvable if n is greater than or equal to 5.

I need a detailed proof of this to study please.

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Solution Summary

This provides an example of concluding a series is not solvable.

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We have

S_4 > A_4 > (A_4)' = { double transpositions } > {( )}

If n = 2, [S_n, S_n] = { ( ) } = A_n.

Consider n > 2, and let a = (1 2) and b = (1 2 3); then

aba^{-1} b^{-1} = (1 ...

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