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

## Contents

## Definition

The Klein-four group 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 on 4 elements comprising the double transpositions, and the identity element.

## Group properties

## 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.