Cyclic group:Z2

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This particular group is a finite group of order: 2

Definition

The cyclic group of order 2 is defined as the unique group of order two. Explicitly it can be described as a group with two elements, say $e$ and $x$ such that $ex=xe=x$ and $e^{2}=x^{2}=e$ . It can also be viewed as:

The quotient group of the group of integers by the subgroup of even integers
The multiplicative group comprising $1$ and $-1$ (in this context it is also termed the sign group)
Bits under the XOR operation
The symmetric group on two elements
The general linear group $GL(1,3)$ (or equivalently, the multiplicative group of the field of order 3)
The multiplicative group of the ring $\mathbb {Z} /4\mathbb {Z}$

This group is denoted as $C_{2}$ , $\mathbb {Z} _{2}$ and sometimes as $\mathbb {Z} /2\mathbb {Z}$ .

Properties

Cyclicity

This particular group is cyclic

Abelianness

This particular group is Abelian

Nilpotence

This particular group is nilpotent

Solvability

This particular group is solvable

Simplicity

This particular group is simple: it has no proper nontrivial normal subgroup

Endomorphisms

Automorphisms

The cyclic group of order two has no nontrivial automorphisms. In fact, it is the only Abelian group with the property of having no nontrivial automorphisms.

Endomorphisms

The cyclic group of order two admits two endomorphisms: the identity map and the trivial map.

Subgroups

There are only two subgroups: the trivial subgroup and the whole group. Most of the nice subgroup properties are true for both.

Quotients

There are only two quotients: itself and the trivial quotient.

In larger groups

Occurrence as a subgroup

The cyclic group of order 2 occurs as a subgroup in many groups. In general, any group of even order contains a cyclic subgroup of order 2 (this follows essentially from Sylow's theorem and the fact that in a group whose order is a power of 2, there must be an element of order 2). Elements of order 2, which are generators for cyclic groups of order 2, are termed involutions.

Occurrence as a normal subgroup

The cyclic subgroup of order 2 may occur as a normal subgroup in some groups. Examples are the general linear group or special linear group over a field whose characteristic is not 2. This is the group comprising the identity and negative identity matrix.

Viewed another way, given a group $G$ , it often happens that we can find a group $H$ with a surjective homomorphism to $G$ whose kernel is a cyclic group of order 2. In some cases, the Schur multiplier of the group is the cyclic group of order $2$ . An example is the projective special linear group, whose Schur multiplier has order two and the universal central extension is the special linear group.

Occurrence as a quotient group

The cyclic group of order 2 occurs very often as a quotient. Put another way, given a group, we can often find a subgroup of index two. Any subgroup of index two is normal (more generally, any subgroup of least prime index is normal).

In these cases, the group of order two may or may not occur as a complement to the normal subgroup. Examples where it does occur as a complement are the alternating group in the symmetric group, or $GL^{+}(n,\mathbb {R} )$ in $GL(n,\mathbb {R} )$ .

Viewed another way, given a group $G$ , we may often be able to construct a group $H$ in which $G$ has index two.

Families

The cyclic group of order two lies in the family of cyclic groups, of general linear groups, and of symmetric groups.

Implementation in GAP

Group ID

The cyclic group is the only group of order 2, so its group ID is 1. The cyclic group can thus be described in GAP as:

SmallGroup(2,1)

Other descriptions

The cyclic group of order 2 can be constructed in many ways using GAP. The simplest is by the command

CyclicGroup(2)

Since it is the symmetric group on two elements, it can also be constructed as

SymmetricGroup(2)

It is also the general linear group $GL(1,3)$ and can hence be constructed as

GL(1,3)

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