Klein four-group
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Contents
Definition
Verbal definitions
The Klein four-group, usually denoted 
, is defined in the following equivalent ways:
- It is the direct product of the group
with itself
- It is the group comprising the elements
under coordinate-wise multiplication
- It is the unique non-cyclic group of order 4
- It is the subgroup of the symmetric group of degree four comprising the double transpositions, and the identity element.
- It is the Burnside group
: the free group on two generators with exponent two.
Multiplication table
The multiplication table with non-identity elements 
and identity element 
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The multiplication table can be described as follows (and this characterizes the group):
- The product of the identity element and any element is that element itself.
- The product of any non-identity element with itself is the identity element.
- The product of two distinct non-identity elements is the third non-identity element.
Elements
Further information: element structure of Klein four-group
Up to conjugation
There are four conjugacy classes, each containing one element (the conjugacy classes are singleton because the group is abelian.
Up to automorphism
There are two equivalence classes of elements upto automorphism: the identity element as a singleton, and all the non-identity elements. All the non-identity elements are equivalent under automorphism.
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 4#Arithmetic functions
Group properties
| Property | Satisfied? | Explanation | Comment |
|---|---|---|---|
| Abelian group | Yes | ||
| Nilpotent group | Yes | ||
| Elementary abelian group | Yes | ||
| Solvable group | Yes | ||
| Supersolvable group | Yes | ||
| Cyclic group | No | ||
| Rational-representation group | Yes | ||
| Rational group | Yes | ||
| Ambivalent group | Yes |
Endomorphisms
Automorphisms
The automorphism group is naturally identified with the group 
as follows. Each element of the automorphism group corresponds to a permutation of the three non-identity elements.
The holomorph, viz the direct product with the automorphism group, is the symmetric group on 4 elements.
Endomorphisms
The non-automorphism endomorphisms include:
- The trivial map
- Pick an arbitrary direct sum decomposition and an arbitrary two-element subgroup. Then the projection on the first direct factor for the decomposition, composed with the isomorphism to the other two-element subgroup, is an endomorphism.
Subgroups
Further information: subgroup structure of Klein four-group
Quick summary
| Item | Value |
|---|---|
| Number of subgroups | 5 As elementary abelian group of prime-square order for prime  : 
|
| Number of conjugacy classes of subgroups | 5 (same as number of subgroups, because the group is an abelian group |
| Number of automorphism classes of subgroups | 3 As elementary abelian group of order  : 
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| Isomorphism classes of subgroups | trivia group (1 time), cyclic group:Z2 (3 times, all in the same automorphism class), Klein four-group (1 time). |
Table classifying subgroups up to automorphism
Note that because abelian implies every subgroup is normal, all the subgroups are normal subgroups.
| Automorphism class of subgroups | List of subgroups | Isomorphism class | Order of subgroups | Index of subgroups | Number of conjugacy classes(=1 iff automorph-conjugate subgroup) | Size of each conjugacy class(=1 iff normal subgroup) | Total number of subgroups(=1 iff characteristic subgroup) | Isomorphism class of quotient (if exists) | Subnormal depth | Nilpotency class |
|---|---|---|---|---|---|---|---|---|---|---|
| trivial subgroup |  |
trivial group | 1 | 4 | 1 | 1 | 1 | Klein four-group | 1 | 0 |
| Z2 in V4 |  |
cyclic group:Z2 | 2 | 2 | 3 | 1 | 3 | cyclic group:Z2 | 1 | 1 |
| whole group |  |
Klein four-group | 4 | 1 | 1 | 1 | 1 | trivial group | 0 | 1 |
| Total (3 rows) | -- | -- | -- | -- | 5 | -- | 5 | -- | -- | -- |
Bigger groups
Groups containing it as a subgroup
- Alternating group:A4 which is the semidirect product of the Klein-four group by a cyclic group of order 3
- Symmetric group:S4 which is the holomorph of the Klein-four group, and in which the Klein-four group is a characteristic subgroup
- Dihedral group:D8 which is the dihedral group of order 8, acting on a set of four elements. It sits between the Klein-four group and the symmetric group on 4 elements
Note that the Klein-four group embeds in two ways inside the symmetric group, one, as double transpositions, the other, as the direct product of a pair of involutions. We usually refer to the former embedding, when nothing is explicitly stated.
Groups having it as a quotient
In general, whenever a group has a subgroup of index two that is not characteristic, then the intersection of that subgroup and any other automorph of it, is of index four, and the quotient obtained is the Klein-four group.
It may also occur as the intersection of index-two subgroups that are not automorphs of each other.
Some examples:
- The quaternion group, which has the Klein-four group as its inner automorphism group. The normal subgroups can be taken as those generated by the squareroots of
- The dihedral group of order eight, which has the Klein-four group as its inner automorphism group. Here, it is the quotient by the intersection of two subgroups of order four, one being a cyclic subgroup, the other being itself a Klein-four group.
Implementation in GAP
Group ID
This finite group has order 4 and has ID 2 among the groups of order 4 in GAP's SmallGroup library. For context, there are groups of order 4. It can thus be defined using GAP's SmallGroup function as:
SmallGroup(4,2)
For instance, we can use the following assignment in GAP to create the group and name it 
:
gap> G := SmallGroup(4,2);
Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:
IdGroup(G) = [4,2]
or just do:
IdGroup(G)
to have GAP output the group ID, that we can then compare to what we want.
Other descriptions
The group can also be defined using GAP's ElementaryAbelianGroup function as:
ElementaryAbelianGroup(4)
