Thread: Prove both even or both odd

1. 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.

2. 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.

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Let  and  belong to Sn. Prove that –1 and  are both even or both odd

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