Return to Homework 7, Glossary, Theorems
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.