# Klein four-group

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*This particular group is the smallest (in terms of order):* non-cyclic group

*This particular group is a finite group of order:* 4

## 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 degre 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

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

Function | Value | Explanation |
---|---|---|

order | 4 | |

exponent | 2 | Cyclic subgroup of order two. |

nilpotency class | 1 | The group is abelian. |

derived length | 1 | The group is abelian. |

Frattini length | 1 | The group is elementary abelian. |

Fitting length | 1 | The group is abelian, hence nilpotent. |

minimum size of generating set | 2 | Elementary abelian of rank two. |

subgroup rank | 2 | |

max-length | 2 | |

rank as p-group | 2 | |

normal rank | 2 | |

characteristic rank of a p-group | 2 |

## 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 |

## 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

### 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

The Klein-four group is the second group of order 4 as per GAP's small-group enumeration, so it can be described in GAP as:

SmallGroup(4,2)