Definition 10

In the notation of Theorem 2, the number $\varepsilon(\sigma)$ (also denoted $\varepsilon_\sigma$) is called the signature of the permutation $\sigma$. The kernel of the homomorphism $\varepsilon$ is called the alternating group of E.

$\sigma$ is called even (resp. odd) if $\varepsilon(\sigma) = 1$ (resp. $\varepsilon(\sigma) = -1$). The alternating group of E is denoted by $\mathfrak{A}_E$. It is a normal subgroup of $\mathfrak{S}_E$. When $E = \{1, n\}$, it is simply denoted by $\mathfrak{A}_n$. When the cardinal $n$ of E is $\geq 2$, it is a subgroup of index 2 and hence of order $n! / 2$. It can be shown that, for $n = 3$ or $n \geq 5$, the group $\mathfrak{A}_n$ is a simple group (cf. Exercise 16).