Alternating Group
Definition
The alternating group $A_n$ is the subset $A_n \subseteq S_n$ of the symmetric group consisting of only even elements.
Properties
The alternating group $A_n$ accounts for exactly half of the elements of $S_n$ (and thus the other half are odd). That is, $\lvert A_n \rvert = \frac 1 2 \lvert S_n \rvert$ pf
Fix $n$. Let $A_n$ and $B_n$ respectively denote the even and odd elements of $S_n$. We will show that $A_n \cong B_n$; since together they form a partition of $S_n$, this implies that each contains exactly half of the elements. That $A_n \cong B_n$ is witnessed by the following two injections: \begin{align*} f : A_n \to B_n &= (\sigma \mapsto (1\ 2)\sigma) \\ \tilde f : B_n \to A_n &= (\sigma \mapsto (1\ 2)\sigma) \end{align*} Both of these are injective because \begin{align*} (1\ 2)\sigma &= (1\ 2)\rho \\ (1\ 2)(1\ 2)\sigma &= (1\ 2)(1\ 2)\rho \\ \sigma &= \rho \end{align*} Since there is an injection in both directions $A_n \leftrightarrow B_n$, we have that $A_n \cong B_n$.

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