If G is infinite, then G is isomorphic to the group Z of integers. Permutation groups 3.1.4. Every permutation in S n can be written as a product of disjoint cycles. The cycles that appear in the product are unique. If a permutation in S n is written as a product of disjoint cycles, then its order is the least common multiple of the lengths of its cycles.
Every group is isomorphic to a permutation group.
The group of rigid motions of a regular n -gon is called the n th dihedral group , denoted by D n . If a permutation is written as a product of transpositions in two ways, then the number of transpositions is either even in both cases or odd in both cases. A permutation is called even if it can be written as a product of an even number of transpositions, and odd if it can be written as a product of an odd number of transpositions. The set of all even permutations of S n is a subgroup of S n . The set of all even permutations of S n is called the alternating group on n elements, and will be denoted by A n .