nLab
cyclic group

Cyclic groups

Definition

A cyclic group is a quotient group of the free group on the singleton.

Generally, we consider a cyclic group as a group, that is without specifying which element comprises the generating singleton. But see Ring structure below.

Examples

There is (up to isomorphism) one cyclic group for every natural number nn, denoted

n/n. \mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z} \,.

For n>0n \gt 0, the order (cardinality) of n\mathbb{Z}_n is nn (so finite); for n=0n = 0, which is the group of integers

0 \mathbb{Z}_0 \coloneqq \mathbb{Z}

the order is countable but infinite.

If we identify the free group on the singleton with the additive group \mathbb{Z} of integers, then the infinite cyclic group is \mathbb{Z} itself, while the finite cyclic group of order nn is /n\mathbb{Z}/n, that is \mathbb{Z} modulo (the normal subgroup generated by) the integer nn. Of course, \mathbb{Z} itself is also /0\mathbb{Z}/0. One could also consider /n\mathbb{Z}/n for negative nn, but this is the same as /|n|\mathbb{Z}/{|n|}.

The cyclic group of order nn may also be identified with a subgroup of the multiplicative group of complex numbers (or algebraic numbers): the group of nnth roots of 11. For n=0n = 0, we may pick any non-zero complex number (or even something else) that is not a root of 11 (but there is no standard choice) and take the subgroup generated by it.

Notation

Besides ‘/n\mathbb{Z}/n’, the cyclic group of order nn may also be denoted in other ways: some more complicated variation of ‘/n\mathbb{Z}/n’ (to put ‘the [normal] subgroup generated by’ explicitly in the notation), or else the simplified form ‘ n\mathbb{Z}_n’ (which however conflicts with notation for the nn-adic integers). When written multiplicatively, it may be denoted ‘Z nZ_n’ (note the font change) or ‘C nC_n’; either letter here stands for ‘cyclic’ in one language or another. (It is a coincidence that the German words ‘Zahl’, which gives us ‘\mathbb{Z}’, and ‘zyklisch’, which gives us ‘ZZ’, begin with the same letter.)

Besides ‘\mathbb{Z}’, the infinite cyclic group may also be denoted in other ways: some variation of ‘(,+)(\mathbb{Z},+)’ to indicate that we are using addition of integers, or any of the above notations with either ‘00’ or ‘\infty’ in place of ‘nn’ (depending on whether we think of it as \mathbb{Z} modulo 00 or the cyclic group with order \infty).

When written additively, the notation for the elements of a cyclic group are usually just the notation for integers; for the finite cyclic group of order nn, we use the natural number less than nn. In the finite case, we may also use brackets or some other notation to indicate equivalence classes. When written multiplicatively, any letter (‘ee’, ‘xx’, ‘aa’, ‘ξ\xi’, etc) may be taken to stand for the generating element; then any other element is a power of this generator. When thought of as a multiplicative group of complex numbers, one generator is e 2iπ/n\mathrm{e}^{2\mathrm{i}\pi/n}, and the notation may reflect that.

Properties

Ring structure

Let AA be a cyclic group, and let xx be a generator of AA. Then there is a unique ring structure on AA (making the original group the additive group of the ring) such that xx is the multiplicative identity 11.

If we identify AA with the additive group /n\mathbb{Z}/n and pick (the equivalence class of) the integer 11 for xx, then the resulting ring is precisely the quotient ring? /n\mathbb{Z}/n.

In this way, a cyclic group equipped with the extra structure of a generator is the same thing (in the sense that their groupoids are equivalent) as a ring with the extra property that the underlying additive group is cyclic.

For n>0n \gt 0, the number of ring structures on the cyclic group /n\mathbb{Z}/n, which is the same as the number of generators, is ϕ(n)\phi(n), the Euler totient? of nn; the generators are those ii that are relatively prime to nn. While ϕ(1)=1\phi(1) = 1, otherwise ϕ(n)>1\phi(n) \gt 1 (another way to see that we have a structure and not just a property). For \mathbb{Z} itself, there are two ring structures, since 11 and 1-1 are the generators (and these are relatively prime to 00).

Group cohomology

For a discussion of the group cohomology of cyclic groups see at projective resolution in the section Cohomology of cyclic groups.

Relation to finite abelian groups

Proposition

Every finite abelian group is a direct sum of abelian groups over cyclic groups.

See at finite abelian group for details.

Revised on November 12, 2013 09:06:17 by Urs Schreiber (145.116.131.45)