Klein four-group

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

Definition

The Klein-four group is defined in the following equivalent ways:

• It is the direct product of the group $\mathbb{Z}/2\mathbb{Z}$ with itself
• It is the group comprising the elements $(\pm 1, \pm 1)$ 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

Abelianness

This particular group is Abelian

Endomorphisms

Automorphisms

The automorphism group is naturally identified with the group $S_3$ 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.