The alternating group $A_n$ is defined as a certain subgroup of the symmetric group $S_n$: namely, the collection of all elements which can be made by multiplying together an even number of transpositions. This is a well-defined notion (proof).

Examples

• A cycle of even length is an odd permutation in the sense that it can only be made by multiplying an odd number of transpositions. For example, $\left(132\right)$ is equal to $\left(13\right)\left(23\right)$.

• A cycle of odd length is an even permutation, in that it can only be made by multiplying an even number of transpositions. For example, $\left(1354\right)$ is equal to $\left(54\right)\left(34\right)\left(14\right)$.

• The alternating group $A_4$ consists precisely of twelve elements: the identity, $\left(12\right)\left(34\right)$, $\left(13\right)\left(24\right)$, $\left(14\right)\left(23\right)$, $\left(123\right)$, $\left(124\right)$, $\left(134\right)$, $\left(234\right)$, $\left(132\right)$, $\left(143\right)$, $\left(142\right)$, $\left(243\right)$.

Properties

• $A_n$ is generated by its $3$-cycles. (Proof.)
• $A_n$ is simple. (Proof.)
• The conjugacy classes of $A_n$ are easily characterised.