## Quick Reference

Let *a* be an element of a multiplicative group *G*. The elements *a*^{r}, where *r* is an integer (positive, zero or negative), form a subgroup of *G*, called the subgroup generated by *a*. A group *G* is cyclic if there is an element *a* in *G* such that the subgroup generated by *a* is the whole of *G*. If *G* is a finite cyclic group with identity element *e*, the set of elements of *G* may be written {*e*, *a*, *a*^{2},…, *a*^{n−1}}, where *a*^{n}=*e* and *n* is the smallest such positive integer. If *G* is an infinite cyclic group, the set of elements may be written {…, *a*^{−2}, *a*^{−1}, *e*, *a*, *a*^{2},…}.

By making appropriate changes, a cyclic additive group (or group with any other operation) can be defined. For example, the set {0, 1, 2,…*n*−1} with addition modulo *n* is a cyclic group, and the set of all integers with addition is an infinite cyclic group. Any two cyclic groups of the same order are isomorphic.

*Subjects:*
Mathematics.

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