Originally Posted by
Drexel28 Here is a rough outline of the proof. Fill in the blanks.
1. Suppose that $\displaystyle \sigma,\tau\in A_n$ then $\displaystyle \sigma,\tau$ are both the product of an even number of transpositions. So argue that $\displaystyle \sigma\tau$ is also the product of an even number of transpositions.
2. Let $\displaystyle \sigma\in S_n$ be a transposition. What is $\displaystyle \sigma^2$? What does that imply?
3. Let $\displaystyle \tau\in S_n$ such that $\displaystyle \tau=\prod_{i=1}^n t_i$ where $\displaystyle t_i$ are transpositions. Using 2. what is $\displaystyle \tau^{-1}$ in terms of $\displaystyle \prod_{i=1}^n t_i$? (if this doesn't make sense...try doing some small example and see the pattern)