Klein 4-group

The Klein 4-group is the subgroup $V$ (Vierergruppe) of $S_4$ (see symmetric group) consisting of the following 4 permutations:

$$(),(12)(34),(13)(24),(14)(23).$$

(see cycle notation). This is an abelian group, isomorphic to the product ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$. 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 group of various planar graphs, including graphs of 4 vertices. Yet we have

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 $S_4$, $⟨(12),(34)⟩$, $⟨(12),(1234)⟩$ and $S_4$. 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 ${180}^{\circ }$ 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 $\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{3}\mathrm{,}\mathrm{4}\mathrm{\}}$, then the full symmetry group (the dihedral group of order 8, $D_{\mathrm{8}}$) contains $V$.

2 Klein 4-group as a vector space

As $V$ is isomorphic to ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$ it is a 2-dimensional vector space over the Galois field ${\mathbb{Z}}_2$. 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 group and so in our context $\mathrm{Aut}V\cong GL(2,2)$. As we can interchange any basis of a vector space we can label the elements $e_1=(12)(34)$, $e_2=(13)(24)$ and $e_3=(14)(23)$ so that we have the permutations $(e_1,e_2)$ and $(e_2,e_3)$ and so we generate all permutations on $\{e_1,e_2,e_3\}$. This proves:

Proposition 3.

$\mathrm{Aut}V\cong GL(2,2)\cong S_3$. Furthermore, the affine linear group of $V$ is $A\mathit{}G\mathit{}L\mathit{}\mathrm{(}\mathrm{2}\mathrm{,}\mathrm{2}\mathrm{)}\mathrm{=}V\mathrm{⋊}{S}_{\mathrm{3}}$.

3 Klein 4-group as a normal subgroup

Because $V$ is a subgroup of $S_4$ we can consider its conjugates. Because conjugation in $S_4$ respects the cycle structure. From this we see that the conjugacy class in $S_4$ of every element of $V$ lies again in $V$. Thus $V$ is normal. This now allows us to combine both of the previous sections to outline the exceptional nature (amongst $S_n$ families) of $S_4$. We collect these into

Theorem 4.

1. $V$ is a normal subgroup of $S_4$.

2. $V$ is contained in $A_4$ and so it is a normal subgroup of $A_4$.

3. $V$ is the Sylow 2-subgroup of $A_4$.

4. $V$ is the intersection of all Sylow 2-subgroups of $S_4$, that is, the $2$-core of $S_4$.

5. $S_4/V\cong S_3$.

6. $S_4\cong AGL(2,2)\cong V⋊S_3$.

Proof.

We have already argued that $V$ is normal in $S_4$. Upon inspecting the elements of $V$ we see $V$ contains only even permutations so $V\le A_4$ and consequently $V$ is normal in $A_4$ as well. As $|A_4|=12$ and $|V|=4$ we establish $V$ is a Sylow 2-subgroup of $A_4$. But $V$ is normal so it the Sylow 2-subgroup of $A_4$ (Sylow subgroups are conjugate.)

Now notice that the dihedral group $D_8$ acts on a square and so it is represented as a permutation group on 4 vertices, so $D_8$ embeds in $S_4$. As $|D_8|=8$ and $|S_4|=24$, $D_8$ is a Sylow 2-subgroup of $S_4$ and so all Sylow 2-subgroups of $S_4$ are embeddings of $D_8$ (in particular various relabellings of the vertices of the square.) But by Proposition 2 we know that each embedding contains $V$. As there are 3 non-equal embeddings of $D_8$ (think of the 3 non-equal labellings of a square) we know that the intersection of these $D_8$ is a proper subgroup of $D_8$. As $V$ is a maximal subgroup of each $D_8$ and contained in each, $V$ is the intersection of all these embeddings.

Now the action of $S_4$ by conjugation on the Sylow 2-subgroups $D_8$ permutes all 3 (again Sylow subgroups are conjugate) so $S_4\mapsto S_3$. Indeed, $V$ is in the kernel of this action as $V$ is in each $D_8$. Indeed a three cycle $(123)$ permutes the $D_8$’s with no fixed point (consider the relabellings) and $(12)$ fixes only one. So $S_4$ maps onto $S_3$ and so the kernel is precisely $V$. Thus $S_4/V=S_3$.

Now we can embed $S_3$ into $S_4$ as $⟨(123),(12)⟩$ so $V\cap S_3=1$, $V{S}_3=S_4$ so $S_4=V⋉S_3$. Finally, $AGL(2,2)$ acts transitively on the four points of the vector space $V$ so $AGL(2,2)$ embeds in $S_4$. And by Proposition 3 we conclude $S_4\cong AGL(2,2)$. ∎

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 $S_4$. For all other $S_n$’s, $A_n$ is the only proper normal subgroup.

We can view the properties of our theorem in a geometric way as follows: $S_4$ is the group of symmetries of a tetrahedron. There is an induced action of $S_4$ on the six edges of the tetrahedron. Observing that this action preserves incidence relations one gets an action of $S_4$ 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 of ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$. The other 3 subgroups of $S_4$ which are isomorphic to ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$ 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