# Definition The *symmetric group on $n$ letters*, denoted $S_n$, is the [[group]] of all [[Types of Maps|bijective]] [[Map|maps]] from the set $\{1,2, \ldots, n\}$ to itself. The group operation is the [[Composite Map|composition]] of maps. These maps are known as permutations. $S_n$ has $n!$ [[group element|elements]]. # $\text{sgn}$ homomorphism Consider the following map: $ \text{sgn}: S_n \rightarrow \mathbb{Z}_2 $ Given a permutation $\sigma \in S_n$, then $ \text{sgn}(\sigma) = \left\{\begin{array} 0+1\text{ if $\sigma$ consist of an even number of transpositions}\\-1\text{ if $\sigma$ consist of an odd number of transpositions}\end{array}\right. $ where [[Transposition|transpositions]] are permutations that change only two letters and leave the rest unchanged. If $\text{sgn}(\sigma) = +1$ then we say $\sigma$ is even, and otherwise, $\sigma$ is odd. It is straightforward to prove that $\text{sgn}$ is a homomorphism.