Hw7 Problem 13

### Problem 13

Prove if $\sigma, \tau \in S_n$, then $\sigma \circ \tau$ is an even permutation if both $\sigma, \tau$ are even or both are odd.

(i.e. Prove if $\sigma, \tau$ are both even, then $\sigma \circ \tau$ is even.)

(i.e. Prove if $\sigma, \tau$ are both odd, then $\sigma \circ \tau$ is even.)

Solution

Prove: If $\sigma, \tau$ are both even, then $\sigma \circ \tau$ is even.

Since $\sigma$ and $\tau$ are both even permutations, each of their number of transpositions are even. So, their product of transpositions is even since the sum of two even numbers is even. Hence, $\sigma \circ \tau$ can be written as an even number of transpositions.

Prove: If $\sigma, \tau$ are both odd, then $\sigma \circ \tau$ is even.

Since $\sigma$ and $\tau$ are both odd permutations, each of their number of transpositions are odd. So, product of their of transpositions is even since the sum of two odd numbers is even. Hence, $\sigma \circ \tau$ can be written as an even number of transpositions.