Alternating group

Definition

Let $n$ be a positive integer. The alternating group $A_n$ is the set of the even permutations in $S_n$, with composition as the operation.

Remarks

For $n \geq 2$, $A_n$ is the kernel of the sign homomorphism, so it is a normal subgroup of $S_n$. For $n \geq 2$, $A_n$ is also the commutator subgroup of $S_n$. The group $A_3$ is cyclic of order $3$, and $A_4$ has order $12$. The group $A_n$ is abelian if and only if $n \leq 3$.

Explanations

L1 human

The alternating group keeps only the even permutations. The product of two even permutations is even, so these form a group. For $n \geq 5$ it is one of the most important examples of a simple group.