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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
- 1 Definition
- 2 Elements
- 3 Group properties
- 4 Endomorphisms
- 5 Subgroups
- 6 Bigger groups
- 7 Implementation in GAP
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.
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.
This particular group is Abelian
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 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.
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).
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.
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.
- 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
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: