# Prove both even or both odd

• Oct 13th 2008, 12:21 PM
mandy123
Prove both even or both odd
Let alpha and beta belong to S_n. Prove that (beta)(alpha)(beta^-1) and alpha are both even or both odd.
• Oct 13th 2008, 12:43 PM
ThePerfectHacker
Quote:

Originally Posted by mandy123
Let alpha and beta belong to S_n. Prove that (beta)(alpha)(beta^-1) and alpha are both even or both odd.

Notice that $\beta (a_1,a_2) \beta^{-1} = (\beta(a_1), \beta(a_2))$.
Therefore if $\alpha = \tau_1 \cdot ... \cdot \tau_k$ where $\tau_i$ are transpositions then $\beta \alpha \beta^{-1} = (\beta \tau_1 \beta^{-1})(\beta \tau_2 \beta^{-1}) ... (\beta \tau_k \beta^{-1})$. Thus, we see that $\alpha$ and $\beta \alpha \beta^{-1}$ have the same parity.