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# cyclic group

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Let a be an element of a multiplicative group G. The elements ar, 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, a2,…, an−1}, where an=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, a2,…}.

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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