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)
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.