Cyclic group

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.