Klein 4-group (original) (raw)

1 Klein 4-group as a symmetry group

Proposition 1.

V is not the automorphism group of a simple graph.

Proof.

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 graph 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 symmetries of a polygon, in particular, a non-square rectangle.

We can rotate by 180∘ which corresponds to the permutation(13)⁢(24). We can also flip the rectangle over the horizontal diagonalwhich 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 group of order 8, D8) contains V.

2 Klein 4-group as a vector space

As V is isomorphic to ℤ2⊕ℤ2 it is a 2-dimensionalvector space over the Galois field ℤ2. The projective geometryof 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 group and so in our context Aut⁡V≅G⁢L⁢(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.

Aut⁡V≅G⁢L⁢(2,2)≅S3. Furthermore, the affine linear group of V is A⁢G⁢L⁢(2,2)=V⋊S3.

3 Klein 4-group as a normal subgroup

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

Theorem 4.

2. 1. V is contained in A4 and so it is a normal subgroup of A4.
3. 1. V is the Sylow 2-subgroup of A4.
4. 1. V is the intersection of all Sylow 2-subgroups of S4, that is, the 2-core of S4.
5. 1. S4/V≅S3.
6. 1. S4≅A⁢G⁢L⁢(2,2)≅V⋊S3.

Proof.

We have already argued that V is normal in S4. Upon inspecting the elements of V we see V contains only even permutations so V≤A4and consequently V is normal in A4 as well. As |A4|=12 and |V|=4we 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 group 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 embeddings of D8 (in particular various relabellings of the vertices of the square.) But by Proposition 2we 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 subgroup 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 D8permutes all 3 (again Sylow subgroups are conjugate) so S4↦S3. 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 point(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)⟩ soV∩S3=1, V⁢S3=S4 so S4=V⋉S3. Finally, A⁢G⁢L⁢(2,2)acts transitively on the four points of the vector space V soA⁢G⁢L⁢(2,2) embeds in S4. And by Proposition 3 we concludeS4≅A⁢G⁢L⁢(2,2). ∎

We can make similar arguments about subgroups of symmetries for larger regular polygons. 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 parallel 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 tetrahedron. There is an induced action of S4 on the six edges of the tetrahedron. Observing that this action preserves incidence relations 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 transitive and regular. Indeed V is the (unique) regular representation ofℤ2⊕ℤ2. The other 3 subgroups of S4 which are isomorphic toℤ2⊕ℤ2 are not transitive.

V is the symmetry group of the Riemannian curvature tensor.