 A cyclic group is a group \(G\) with an element \(g\) such that every element of \(G\) is of the form \(g^i\) for some positive integer \(i\). Cyclic groups are abelian because \(g^{i}g^j = g^{i + j} = g^j g^{i}\). The group of integers under addition, and the groups of integers mod \(n\) for any positive integer \(n\), all form cyclic groups, with 1 as a generator in every case. All cyclic groups are isomorphic to one of these. There are always other generators when the group is of order 3 or more, for example \(-1\), and for groups of prime order every nonzero element is a generator.