Klein 4-group
The Klein 4-group is the subgroup (Vierergruppe) of (see symmetric group
) consisting of the following4 permutations
:
(see cycle notation). This is anabelian group, isomorphic
to the product
.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 is isomorphic to the automorphism group of various planargraphs
, including graphs of 4 vertices. Yet we have
Proposition 1.
is not the automorphism group of a simple graph.
Proof.
Suppose is the automorphism group of a simple graph .Because contains the permutations , and it follows the degree of every vertex is the same – we can mapevery vertex to every other. So is a regular graph on 4 vertices.This makes isomorphic to one of the following 4 graphs:
In order the automorphism groups of these graphs are ,, and .None of these are , though the second is isomorphic to .∎
Though cannot be realized as an automorphism group of a planar graphit can be realized as the set of symmetries of a polygon
, in particular,a non-square rectangle
.
We can rotate by which corresponds to the permutation. We can also flip the rectangle over the horizontal diagonalwhich gives the permutation , and finally also over the verticaldiagonal which gives the permutation .
An important corollary to this realization is
Proposition 2.
Given a square with vertices labeled in any way by , then thefull symmetry group (the dihedral group of order 8, ) contains .
2 Klein 4-group as a vector space
As is isomorphic to it is a 2-dimensionalvector space over the Galois field . The projective geometryof – equivalently, the lattice of subgroups – is given in the followingHasse diagam:
The automorphism group of a vector space is called the general lineargroup and so in our context . As we can interchangeany basis of a vector space we can label the elements , and so that we have the permutations and and so we generate all permutations on. This proves:
Proposition 3.
. Furthermore, the affine linear groupof is .
3 Klein 4-group as a normal subgroup
Because is a subgroup of we can consider its conjugates. Becauseconjugation in respects the cycle structure. From this we see thatthe conjugacy class
in of every element of lies again in . Thus is normal. This now allows us to combine both of the previous sections
to outline the exceptional nature (amongst families) of . Wecollect these into
Theorem 4.
- 1.
is a normal subgroup
of .
- 2.
is contained in and so it is a normal subgroup of .
- 3.
is the Sylow 2-subgroup of .
- 4.
is the intersection
of all Sylow 2-subgroups of , that is,the -core of .
- 5.
.
- 6.
.
Proof.
We have already argued that is normal in . Upon inspecting theelements of we see contains only even permutations so and consequently is normal in as well. As and we establish is a Sylow 2-subgroup of . But is normal so itthe Sylow 2-subgroup of (Sylow subgroups are conjugate.)
Now notice that the dihedral group acts on a square and so it isrepresented as a permutation group on 4 vertices, so embeds in .As and , is a Sylow 2-subgroup of and soall Sylow 2-subgroups of are embeddings
of (in particular variousrelabellings of the vertices of the square.) But by Proposition
2we know that each embedding contains . As there are 3 non-equalembeddings of (think of the 3 non-equal labellings of a square) weknow that the intersection of these is a proper subgroup
of .As is a maximal subgroup of each and contained in each, isthe intersection of all these embeddings.
Now the action of by conjugation on the Sylow 2-subgroups permutes all 3 (again Sylow subgroups are conjugate) so .Indeed, is in the kernel of this action as is in each .Indeed a three cycle permutes the ’s with no fixed point(consider the relabellings) and fixes only one. So mapsonto and so the kernel is precisely . Thus .
Now we can embed into as so, so . Finally, acts transitively on the four points of the vector space so embeds in . And by Proposition 3 we conclude.∎
We can make similar arguments about subgroups of symmetries forlarger regular polygons
. Likewise for other 2-dimensional vector spaceswe can establish similar structural properties. However it is onlywhen we study we involve that we find these two methods intersectin a this exceptionally parallel
fashion. Thus we establish the exceptionalstructure of . For all other ’s, is the only proper normalsubgroup.
We can view the properties of our theorem in a geometric way as follows: is the group of symmetries of a tetrahedron. There is an induced actionof on the six edges of the tetrahedron. Observing that this action preserves incidence relations
one gets an action of on the three pairsof opposite edges.
4 Other properties
is non-cyclic and of smallest possible order with this property.
is transitive and regular
. Indeed is the (unique) regular representation of. The other 3 subgroups of which are isomorphic to are not transitive.
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 |