# Definition

A cyclic group is a group $$(G, +)$$ (hereafter abbreviated as simply $$G$$) with a single generator, in the sense that there is some $$g \in G$$ such that for every $$h \in G$$, there is $$n \in \mathbb{Z}$$ such that $$h = g^n$$, where we have written $$g^n$$ for $$g + g + \dots + g$$ (with $$n$$ terms in the summand). That is, “there is some element such that the group has nothing in it except powers of that element”.

We may write $$G = \langle g \rangle$$ if $$g$$ is a generator of $$G$$.

# Examples

• $$(\mathbb{Z}, +) = \langle 1 \rangle = \langle -1 \rangle$$

• The group with two elements (say $$\{ e, g \}$$ with identity $$e$$ with the only possible group operation $$g^2 = e$$) is cyclic: it is generated by the non-identity element. Note that there is no requirement that the powers of $$g$$ be distinct: in this case, $$g^2 = g^0 = e$$.

• The integers modulo $$n$$ form a cyclic group under addition, for any $$n$$: it is generated by $$1$$ (or, indeed, by $$n-1$$).

• The symmetric groups $$S_n$$ for $$n > 2$$ are not cyclic. This can be deduced from the fact that they are not abelian (see below).

# Properties

## Cyclic groups are abelian

Suppose $$a, b \in G$$, and let $$g$$ be a generator of $$G$$. Suppose $$a = g^i, b = g^j$$. Then $$ab = g^i g^j = g^{i+j} = g^{j+i} = g^j g^i = ba$$.

## Cyclic groups are countable

The elements of a cyclic group are nothing more nor less than $$\{ g^0, g^1, g^{-1}, g^2, g^{-2}, \dots \}$$, which is an enumeration of the group (possibly with repeats).

Children:

Parents:

• Group

The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.