# Permutations

• Oct 30th 2007, 08:48 PM
chris27
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)
• Oct 30th 2007, 10:37 PM
Soltras
Quote:

Originally Posted by chris27
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.