Ok, I don't know how to go about this.

Prove that there is no permutation α such that α(12) α^-1 = (123)

Prove that there is no permutation α such that α(12) α^-1 = (124)(567)

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- Oct 30th 2007, 08:48 PMchris27Permutations
Ok, I don't know how to go about this.

Prove that there is no permutation α such that α(12) α^-1 = (123)

Prove that there is no permutation α such that α(12) α^-1 = (124)(567) - Oct 30th 2007, 10:37 PMSoltras
Well (1 2 3) = (1 2)(2 3), which is an even permutation.

So $\displaystyle \alpha (1 2) \alpha^{-1}$ must be even, but this is impossible because if $\displaystyle \alpha$ is an even permutation, then $\displaystyle \alpha (1 2)$ is odd, so $\displaystyle \alpha (1 2)\alpha^{-1}$ is odd.

But if $\displaystyle \alpha$ is odd, then $\displaystyle \alpha (1 2)$ is even, so $\displaystyle \alpha (1 2) \alpha^{-1}$ is odd. So the left side is never an even permutation.

The same reasoning applies to the second problem. - Oct 31st 2007, 05:42 AMThePerfectHacker
Two cycles cannot be conjugate if they have different lengths for that will imply thier orders are not the same.