What is the order of the alternating group $A_4$? When I write out all the elements of $S_4$, I count only 11 transpositions. But in my text, the order of $A_4$ is $12$. What am I missing?
$A_4=\{(12)(34),(13)(24),(14)(23),(123),(124),(132),(134),(142),(143),(234),(243)$
$|A_4|=11$

Consider the sign homomorphism $\varphi:S_n\rightarrow \{\pm 1\}$, which maps even permutations to 1 and odd permutations to -1. Notice that $\ker \varphi = A_n$, and since $\varphi$ is surjective, 
\begin{align*}
\lvert \ker \varphi \lvert = \frac{\lvert S_n \lvert}{\lvert \{\pm 1\}\lvert}=\frac{n!}{2}.
\end{align*}
