# 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.