A cycle that contains finitely many elements is a finite cycle
and the cycle length is the number of elements it contains. Let
be a permutation written as a finite product of disjoint cycles
of finite length. The order of is the least common multiple of
the lengths of the cycles. A transposition is a cycle of length
two. Clearly a transposition has order two, but there are
permutations of order two that are not transpositions. One
important application of transpositions is that every
permutation may be written as a product of transpositions
(although not necessarily disjoint and not uniquely). A
permutation is an even permutation if it is a product of an even
number of transpositions and an odd permutation if it is a
product of an odd number of transpositions. As the definition
suggests, a permutation cannot be both odd and even. However,
the decomposition into a product of transpositions is not unique
nor is the number of transpositions unique. For example, the
permutation may be written as , or . Of course,