# cyclic group

## English

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### Noun

cyclic group (plural cyclic groups)

1. A group generated by a single element.
• 1986, N. S. Gopalakrishnan, University Algebra, New Age International, 2nd Edition, page 22,
Proposition 1.5.6. Any subgroup of an infinite cyclic group is also an infinite cyclic group.
• 2002, Serge Lang, Algebra, 3rd edition, Springer, page 24:
If ${\displaystyle u:G_{1}\rightarrow \mathbb {Z} /m\mathbb {Z} }$ and ${\displaystyle v:G_{2}\rightarrow \mathbb {Z} /m\mathbb {Z} }$ are isomorphisms of two cyclic groups with ${\displaystyle \mathbb {Z} /m\mathbb {Z} }$, then ${\displaystyle v^{-1}\circ u:G_{1}\rightarrow G_{2}}$ is an isomorphism.
• 2003, Alexander Retakh (translator), Ėrnest Borisovich Vinberg, A Course in Algebra, [2001, Э. Б. Винберг, Курс алгебры, Factorial Press] American Mathematical Society, page 152,
Cyclic groups are the simplest groups imaginable.

#### Usage notes

More precisely, there exists at least one element g such that every other element of the group may be obtained by repeatedly applying the group operation (or its inverse) to g. The group operation is required to be invertible and associative. The element g is called a generator of G.

Every infinite cyclic group is isomorphic to the additive group of ${\displaystyle \mathbb {Z} }$, the integers. Any finite cyclic group of order n is isomorphic to the additive quotient group ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$: the integers modulo n.