Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a "generator" of the group) such that, when written multiplicatively, every element of the group is a power of a. Equivalently, an element a of a group G generates G precisely if the only subgroup of G that contains a is G itself.

Related Topics:
Group theory - Group - Generated - Subgroup

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For any positive integer n, there is a cyclic group Cn of order n.

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Unlike the name suggests, the elements of a cyclic group do not have to form a cycle: the infinite cyclic group is the additive group of integers Z.

Related Topics:
Cycle - Integer

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The cyclic groups are the simplest groups and they are completely classified: every cyclic group is isomorphic to one of those mentioned.

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