Cyclic group:Z2
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Definition
Verbal 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 
and 
such that 
and 
. It can also be viewed as:
- The quotient group of the group of integers by the subgroup of even integers
- The multiplicative group comprising
and 
(in this context it is also termed the sign group)
- The additive group of the field of two elements.
- Bits under the XOR operation
- The symmetric group on two elements. In particular, it is a symmetric group on finite set and symmetric group of prime degree.
- The general linear group
(or equivalently, the multiplicative group of the field of order 3)
- The multiplicative group of the ring
- The group of units in
- The group
viz., the group of vectors in 
of unit length
This group is denoted as 
, 
and sometimes as 
.
Multiplication table
| Element | (identity element) |
(non-identity element)
|
|---|---|---|
| |
|
|
| |
|
|
Arithmetic functions
Basic arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 2#Arithmetic functions
Arithmetic functions of a counting nature
Group properties
Basic properties
| Property | Satisfied? | Explanation |
|---|---|---|
| group of prime order | Yes | |
| simple group | Yes | |
| cyclic group | Yes | |
| abelian group | Yes | Follows from cyclic implies abelian |
| nilpotent group | Yes | Follows from abelian implies nilpotent |
| solvable group | Yes | Follows from nilpotent implies solvable |
| supersolvable group | Yes | Follows from being a finitely generated abelian group. |
Other properties
| Property | Satisfied? | Explanation |
|---|---|---|
| rational-representation group | Yes | |
| rational group | Yes | |
| ambivalent group | Yes | |
| strongly ambivalent group | Yes | |
| T-group | Yes | |
| Schur-trivial group | Yes |
Endomorphisms
Automorphisms
The cyclic group of order two has no nontrivial automorphisms. In fact, it is the only nontrivial group with the property of having no nontrivial automorphisms. Further information: trivial automorphism group implies trivial or order two
Endomorphisms
The cyclic group of order two admits two endomorphisms: the identity map and the trivial map (the map sending both elements to the identity element).
Subgroup structure
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
Further information: supergroups of cyclic group:Z2
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 from Cauchy's theorem, which is a corollary of Sylow's theorem, though it can also be proved by a direct counting argument). Elements of order 2, which are generators for cyclic groups of order 2, are termed involutions.
Occurrence as a normal subgroup
The cyclic group 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.
It is also true that a normal subgroup of order two is central. More generally, it is true that a normal subgroup whose order is the least prime divisor of the order of the group is central. Thus, the existence of normal subgroups of order two indicates a nontrivial center.
Viewed another way, given a group 
, it often happens that we can find a group 
with a surjective homomorphism to whose kernel is a cyclic group of order 2. In some cases, the Schur multiplier of the group is the cyclic group of order 
. 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 
in 
.
Viewed another way, given a group , we may often be able to construct a group in which 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.
It also lies in the family of sphere groups: namely, it is the group of unit vectors in . The other two sphere groups are 
(the circle group) and 
(the unit quaternion group).
GAP imeplementation
Group ID
This finite group has order 2 and has ID 1 among the groups of order 2 in GAP's SmallGroup library. For context, there are groups of order 2. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(2,1)
For instance, we can use the following assignment in GAP to create the group and name it :
gap> G := SmallGroup(2,1);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [2,1]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Other descriptions
| Description | Functions used |
|---|---|
| CyclicGroup(2) | CyclicGroup |
| SymmetricGroup(2) | SymmetricGroup |
| GL(1,3) | GL |
Internal links
- Linear representation theory of cyclic group:Z2
- Permutation representation theory of cyclic group:Z2
- Properly discontinuous group actions of cyclic group:Z2
- Galois extensions for cyclic group:Z2
- Group cohomology of cyclic group:Z2
