1. ## Permutations

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)

2. Originally Posted by chris27

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)
Well (1 2 3) = (1 2)(2 3), which is an even permutation.
So $\alpha (1 2) \alpha^{-1}$ must be even, but this is impossible because if $\alpha$ is an even permutation, then $\alpha (1 2)$ is odd, so $\alpha (1 2)\alpha^{-1}$ is odd.
But if $\alpha$ is odd, then $\alpha (1 2)$ is even, so $\alpha (1 2) \alpha^{-1}$ is odd. So the left side is never an even permutation.

The same reasoning applies to the second problem.

3. Two cycles cannot be conjugate if they have different lengths for that will imply thier orders are not the same.