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.