Klein 4group
The Klein 4group 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://wwwgap.dcs.stand.ac.uk/ history/Mathematicians/Klein.htmlFelix Klein, a pioneering figure in the field of geometric group theory.
1 Klein 4group 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:
$$ 
In order the automorphism groups of these graphs are ${S}_{4}$, $\u27e8(12),(34)\u27e9$, $\u27e8(12),(1234)\u27e9$ and ${S}_{4}$. None of these are $V$, though the second is isomorphic to $V$. ∎
Though $V$ cannot be realized as an automorphism group of a planar graph it can be realized as the set of symmetries^{} of a polygon^{}, in particular, a nonsquare rectangle^{}.
$$ 
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 $(12)(34)$.
$$ 
An important corollary to this realization is
Proposition 2.
Given a square with vertices labeled in any way by $\mathrm{\{}\mathrm{1}\mathrm{,}\mathrm{2}\mathrm{,}\mathrm{3}\mathrm{,}\mathrm{4}\mathrm{\}}$, then the full symmetry group (the dihedral group^{} of order 8, ${D}_{\mathrm{8}}$) contains $V$.
2 Klein 4group as a vector space
As $V$ is isomorphic to ${\mathbb{Z}}_{2}\oplus {\mathbb{Z}}_{2}$ it is a 2dimensional vector space over the Galois field ${\mathbb{Z}}_{2}$. The projective geometry of $V$ – equivalently, the lattice of subgroups – is given in the following Hasse diagam:
$$ 
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 $A\mathit{}G\mathit{}L\mathit{}\mathrm{(}\mathrm{2}\mathrm{,}\mathrm{2}\mathrm{)}\mathrm{=}V\mathrm{\u22ca}{S}_{\mathrm{3}}$.
3 Klein 4group 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 2subgroup of ${A}_{4}$.

4.
$V$ is the intersection^{} of all Sylow 2subgroups of ${S}_{4}$, that is, the $2$core of ${S}_{4}$.

5.
${S}_{4}/V\cong {S}_{3}$.

6.
${S}_{4}\cong AGL(2,2)\cong V\u22ca{S}_{3}$.
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 2subgroup of ${A}_{4}$. But $V$ is normal so it the Sylow 2subgroup 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 2subgroup of ${S}_{4}$ and so all Sylow 2subgroups 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 nonequal embeddings of ${D}_{8}$ (think of the 3 nonequal 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 2subgroups ${D}_{8}$ 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 $\u27e8(123),(12)\u27e9$ so $V\cap {S}_{3}=1$, $V{S}_{3}={S}_{4}$ so ${S}_{4}=V\u22c9{S}_{3}$. 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 2dimensional 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 noncyclic 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 4group 
Canonical name  Klein4group 
Date of creation  20130322 12:49:02 
Last modified on  20130322 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 fourgroup 
Synonym  Viergruppe 
Related topic  GroupsInField 
Related topic  Klein4Ring 
Related topic  PrimeResidueClass 
Related topic  AbelianGroup2 