# 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 :

Element/element | ||||
---|---|---|---|---|

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

### Upto conjugation

There are four conjugacy classes, each containing one element (the conjugacy classes are singleton because the group is Abelian.

### Upto 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`

### Normal subgroups

All subgroups are normal, since the group is abelian. There is a total of five subgroups: the whole group, the trivial subgroup, and two-element subgroups (viz., copies of the cyclic group of order 2).

### Characteristic subgroups

The Klein-four group is a characteristically simple group, since it is a direct power of a simple group. Hence, the only characteristic subgroups are the trivial subgroup and the whole group.

## 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)`