Cyclic group:Z3

Verbal definition
The cyclic group of order 3 is defined as the unique group of order 3. Equivalently it can be described as a group with three elements $$e= x^0,x,x^2$$ where $$x^lx^m = x^{l+m}$$ with the exponent reduced mod 3. It can also be viewed as:


 * The quotient group of the group of integers by the subgroup of multiples of 3
 * The multiplicative group comprising the three cuberoots of unity (as a subgroup of the multiplicative group of nonzero complex numbers)
 * The member of family::alternating group on three elements
 * The group of orientation-preserving symmetries (rotational symmetries) of the equilateral triangle
 * The multiplicative group of the field of four elements. In particular, it is the $$GL(1,4)$$.

Endomorphisms
Any endomorphism of a cyclic group is determined by where it sends the generator. The cyclic group of order three has three endomorphisms:


 * The identity map is an endomorphism. This map sends every element to itself.
 * The square map is an endomorphism. This map sends $$x$$ to $$x^2$$, $$x^2$$ to $$x$$, and $$e$$ to itself.
 * The trivial map is an endomorphism. This map sends every element to $$e$$.

Automorphisms
Of the three endomorphisms, two are automorphisms: the identity map and the square map. These form a cyclic group of order two: the square map, applied twice, gives the identity map.

Subgroups
The group has no proper nontrivial subgroups: the only subgroups are the whole group and the trivial subgroup.

More generally, any nontrivial group with no proper nontrivial subgroup must be cyclic of prime order. Conversely, any cyclic group of prime order has no proper nontrivial subgroup.

Quotients
The cyclic group of order three has only two quotients: the whole group and the trivial quotient. This follows from the fact that this group is simple -- it has no proper nontrivial normal subgroup.

Other constructions
The holomorph of this group, i.e., the semidirect product of this group with its automorphism group, is isomorphic to the symmetric group on three letters. The cyclic group sits inside this as the alternating group and the automorphism group sits inside as a subgroup of order two.

Occurrence as a subgroup
The cyclic group of order 3 occurs as a subgroup in many groups. In general, a group contains a cyclic subgroup of order three if and only if its order is a multiple of three (this follows from Cauchy's theorem, a corollary of Sylow's theorem).

Occurrence as a normal subgroup
The cyclic group of order three occurs as a normal subgroup in some groups.

For instance, if a field contains non-identity cuberoots of unity, then the multiplicative group of the field contains a cyclic subgroup of order three. As a corollary, the general linear group contains a central subgroup of order three.

Normal subgroups of order three need not be central; for instance, in the symmetric group on three letters, the alternating group is a normal subgroup of order three, but is not central. However, for an odd-order group, any normal subgroup of order three is central. This follows from a more general fact: a normal subgroup whose order is the least prime divisor of the order of the group is central.

Other descriptions
The cyclic group of order three can be defined using the CyclicGroup function:

CyclicGroup(3)

It can also be defined as the alternating group of degree three, using the AlternatingGroup function:

AlternatingGroup(3)

Internal links

 * Linear representation theory of cyclic group:Z3
 * Galois extensions for cyclic group:Z3
 * Group cohomology of cyclic group:Z3