Cyclic group:Z2

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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 $e x = x e = x$ 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

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

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.

Supergroups

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.

Occurrence

In arithmetic modulo two

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