Commutative groups, those groups in which operand order does not change an equation's result, form Abelian groups that commute: "7 × 3 = 3 × 7". When this condition is not satisfied, we say the expression is non-commutative. Cyclic groups are a special case of commutative Abelian groups—sets that are monogenous—generated by a single element—and invertible with a single operation. Consider a set that, if we iterated over every other element with a particular operation, we'd be able to derive all of the elements of the set. For a finite cyclic group, let G be the group, n be the size of the set, and e be the identity element, such that g i = g j whenever i ≡ j ( mod n ); like so. G = {e, g, g 2 , ... , g n−1 } The commutative property also holds over the additive group of ā¤, or the integers, which are isomorphic to any infinite cyclic group. Similarly, the additive group of ā¤/nā¤, integers modulo n, is isomorphic to the finite cyclic group of order...