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).

%%%knows-requisite(Normal subgroup): $A_n$ is a Normal subgroup of $S_n$; it is the quotient of $S_n$ by the sign homomorphism. %%%

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, $(132)$ is equal to $(13)(23)$.

• 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, $(1354)$ is equal to $(54)(34)(14)$.

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

Properties

%%%knows-requisite(Normal subgroup): The alternating group $A_n$ is of [index_of_a_subgroup index] $2$ in $S_n$. Therefore $A_n$ is normal in $S_n$ (proof). Alternatively we may give the homomorphism explicitly of which $A_n$ is the kernel: it is the sign homomorphism. %%%

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