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Klein 4-group (Topic)

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

$\displaystyle (),\; (12)(34),\; (13)(24),\; (14)(23).$
(see cycle notation). This is an abelian group, isomorphic to the product $ \mathbb{Z}_2\oplus \mathbb{Z}_2$. The group is named after Felix Klein, a pioneering figure in the field of geometric group theory.

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:
$\displaystyle \begin{xy}<10mm,0mm>:<0mm,10mm>:: ( 0, 0) *+{1} = ''1''; ( 1, 0) ... ...@{-}; ''2''; ''3'' **@{-}; ''2''; ''4'' **@{-}; ''3''; ''4'' **@{-}; \end{xy}. $
In order the automorphism groups of these graphs are $ S_4$, $ \langle (12),(34)\rangle$, $ \langle (12),(1234)\rangle$ and $ S_4$. None of these are $ V$, though the second is isomorphic to $ V$. $ \qedsymbol$

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.

$\displaystyle \begin{xy}<10mm,0mm>:<0mm,10mm>:: ( 0, 0) *+{1} = ''1''; ( 2, 0) ... ...**@{-}; ''2''; ''3'' **@{-}; ''3''; ''4'' **@{-}; ''4''; ''1'' **@{-}; \end{xy}$
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)$.
$\displaystyle \begin{xy}<10mm,0mm>:<0mm,10mm>:: ( 0, 0) *+{3} = ''1''; ( 2, 0) ... ...@{-}; ''2''; ''3'' **@{-}; ''3''; ''4'' **@{-}; ''4''; ''1'' **@{-}; \end{xy}. $
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, $ D_8$) contains $ V$.

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:

$\displaystyle \begin{xy}<10mm,0mm>:<0mm,10mm>:: (0,0) *+{\langle ()\rangle} =''... ...1''; ''4.1'' **@{-}; ''5.1''; ''3.1'' **@{-}; ''5.1''; ''2.1'' **@{-}; \end{xy}$
The automorphism group of a vector space is called the general linear group and so in our context $ \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   $ \Aut V\cong GL(2,2)\cong S_3$. Furthermore, the affine linear group of $ V$ is $ AGL(2,2)=V\rtimes S_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\rtimes 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\leq A_4$ and consequently $ V$ is normal in $ A_4$ as well. As $ \vert A_4\vert=12$ and $ \vert V\vert=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 $ \vert D_8\vert=8$ and $ \vert S_4\vert=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 $ \langle (123),(12)\rangle$ so $ V\intersect S_3=1$, $ VS_3=S_4$ so $ S_4=V\ltimes 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)$. $ \qedsymbol$

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.

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.



"Klein 4-group" is owned by Algeboy. [ full author list (4) | owner history (3) ]
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See Also: groups in field, Klein 4-ring, prime residue class, abelian group

Other names:  Klein four-group
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Cross-references: tensor, curvature, regular representation, transitive, opposite, incidence relations, preserves, edges, induced, tetrahedron, parallel, properties, regular polygons, arguments, similar, points, onto, fixed point, kernel, action, maximal subgroup, proper subgroup, labellings, proposition, embeddings, permutation group, acts on, Sylow subgroups, even permutations, intersection, contained, sections, normal, structure, cycle, conjugation, conjugates, generate, label, basis, general linear group, lattice of subgroups, projective geometry, Galois field, vector space, dihedral group, square, diagonal, rotate, rectangle, polygon, symmetries, order, regular graph, map, vertex, degree, contains, simple graph, vertices, graphs, planar graphs, automorphism group, theory, field, group, product, isomorphic, abelian group, cycle notation, permutations, symmetric group, subgroup
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This is version 21 of Klein 4-group, born on 2002-06-27, modified 2007-02-12.
Object id is 3139, canonical name is Klein4Group.
Accessed 14890 times total.

Classification:
AMS MSC: 20K99 (Group theory and generalizations :: Abelian groups :: Miscellaneous)
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The automorphism group of V is S3 by jgade on 2003-01-11 19:15:01
It might be nice to add that the automorphism group of the Klein vierergroup actually is S3
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