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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