The alternating group is defined as a certain subgroup of the symmetric group : 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).
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, is equal to .
The alternating group consists precisely of twelve elements: the identity, , , , , , , , , , , .