Klein 4-group

The Klein 4-group is the subgroupMathworldPlanetmathPlanetmath V (Vierergruppe) of S4 (see symmetric groupMathworldPlanetmathPlanetmath) consisting of the following 4 permutationsMathworldPlanetmath:


(see cycle notation). This is an abelian groupMathworldPlanetmath, isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to the productPlanetmathPlanetmathPlanetmath 22. The group is named after http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Klein.htmlFelix Klein, a pioneering figure in the field of geometric group theory.

1 Klein 4-group as a symmetry group

The group V is isomorphic to the automorphism groupMathworldPlanetmath of various planar graphsMathworldPlanetmath, including graphs of 4 vertices. Yet we have

Proposition 1.

V is not the automorphism group of a simple graphMathworldPlanetmath.


Suppose V is the automorphism group of a simple graph G. Because V contains the permutations (12)(34), (13)(24) and (14)(23) it follows the degree of every vertex is the same – we can map every vertex to every other. So G is a regular graphMathworldPlanetmath on 4 vertices. This makes G isomorphic to one of the following 4 graphs:


In order the automorphism groups of these graphs are S4, (12),(34), (12),(1234) and S4. None of these are V, though the second is isomorphic to V. ∎

Though V cannot be realized as an automorphism group of a planar graph it can be realized as the set of symmetriesMathworldPlanetmathPlanetmathPlanetmath of a polygonMathworldPlanetmathPlanetmath, in particular, a non-square rectangleMathworldPlanetmathPlanetmath.


We can rotate by 180 which corresponds to the permutation (13)(24). We can also flip the rectangle over the horizontal diagonal which gives the permutation (14)(23), and finally also over the vertical diagonal which gives the permutation (12)(34).


An important corollary to this realization is

Proposition 2.

Given a square with vertices labeled in any way by {1,2,3,4}, then the full symmetry group (the dihedral groupMathworldPlanetmath of order 8, D8) contains V.

2 Klein 4-group as a vector space

As V is isomorphic to 22 it is a 2-dimensional vector space over the Galois field 2. The projective geometry of V – equivalently, the lattice of subgroups – is given in the following Hasse diagam:


The automorphism group of a vector space is called the general linear groupMathworldPlanetmath and so in our context AutVGL(2,2). As we can interchange any basis of a vector space we can label the elements e1=(12)(34), e2=(13)(24) and e3=(14)(23) so that we have the permutations (e1,e2) and (e2,e3) and so we generate all permutations on {e1,e2,e3}. This proves:

Proposition 3.

AutVGL(2,2)S3. Furthermore, the affine linear group of V is AGL(2,2)=VS3.

3 Klein 4-group as a normal subgroup

Because V is a subgroup of S4 we can consider its conjugates. Because conjugation in S4 respects the cycle structureMathworldPlanetmath. From this we see that the conjugacy classMathworldPlanetmath in S4 of every element of V lies again in V. Thus V is normal. This now allows us to combine both of the previous sectionsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to outline the exceptional nature (amongst Sn families) of S4. We collect these into

Theorem 4.
  1. 1.

    V is a normal subgroupMathworldPlanetmath of S4.

  2. 2.

    V is contained in A4 and so it is a normal subgroup of A4.

  3. 3.

    V is the Sylow 2-subgroup of A4.

  4. 4.

    V is the intersectionMathworldPlanetmathPlanetmath of all Sylow 2-subgroups of S4, that is, the 2-core of S4.

  5. 5.


  6. 6.



We have already argued that V is normal in S4. Upon inspecting the elements of V we see V contains only even permutationsMathworldPlanetmath so VA4 and consequently V is normal in A4 as well. As |A4|=12 and |V|=4 we establish V is a Sylow 2-subgroup of A4. But V is normal so it the Sylow 2-subgroup of A4 (Sylow subgroups are conjugate.)

Now notice that the dihedral group D8 acts on a square and so it is represented as a permutation groupMathworldPlanetmath on 4 vertices, so D8 embeds in S4. As |D8|=8 and |S4|=24, D8 is a Sylow 2-subgroup of S4 and so all Sylow 2-subgroups of S4 are embeddingsPlanetmathPlanetmath of D8 (in particular various relabellings of the vertices of the square.) But by PropositionPlanetmathPlanetmath 2 we know that each embedding contains V. As there are 3 non-equal embeddings of D8 (think of the 3 non-equal labellings of a square) we know that the intersection of these D8 is a proper subgroupMathworldPlanetmath of D8. As V is a maximal subgroup of each D8 and contained in each, V is the intersection of all these embeddings.

Now the action of S4 by conjugation on the Sylow 2-subgroups D8 permutes all 3 (again Sylow subgroups are conjugate) so S4S3. Indeed, V is in the kernel of this action as V is in each D8. Indeed a three cycle (123) permutes the D8’s with no fixed pointPlanetmathPlanetmath (consider the relabellings) and (12) fixes only one. So S4 maps onto S3 and so the kernel is precisely V. Thus S4/V=S3.

Now we can embed S3 into S4 as (123),(12) so VS3=1, VS3=S4 so S4=VS3. Finally, AGL(2,2) acts transitively on the four points of the vector space V so AGL(2,2) embeds in S4. And by Proposition 3 we conclude S4AGL(2,2). ∎

We can make similarMathworldPlanetmathPlanetmath arguments about subgroups of symmetries for larger regular polygonsMathworldPlanetmath. Likewise for other 2-dimensional vector spaces we can establish similar structural properties. However it is only when we study we involve V that we find these two methods intersect in a this exceptionally parallelMathworldPlanetmathPlanetmath fashion. Thus we establish the exceptional structure of S4. For all other Sn’s, An is the only proper normal subgroup.

We can view the properties of our theorem in a geometric way as follows: S4 is the group of symmetries of a tetrahedronMathworldPlanetmathPlanetmath. There is an induced action of S4 on the six edges of the tetrahedron. Observing that this action preserves incidence relationsPlanetmathPlanetmath one gets an action of S4 on the three pairs of opposite edges.

4 Other properties

V is non-cyclic and of smallest possible order with this property.

V is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and regularPlanetmathPlanetmath. Indeed V is the (unique) regular representation of 22. The other 3 subgroups of S4 which are isomorphic to 22 are not transitive.

V is the symmetry group of the Riemannian curvature tensor.

Title Klein 4-group
Canonical name Klein4group
Date of creation 2013-03-22 12:49:02
Last modified on 2013-03-22 12:49:02
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 26
Author Algeboy (12884)
Entry type Topic
Classification msc 20K99
Synonym Klein four-group
Synonym Viergruppe
Related topic GroupsInField
Related topic Klein4Ring
Related topic PrimeResidueClass
Related topic AbelianGroup2