(see cycle notation). This is an abelian 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
is not the automorphism group of a simple graph.
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 map every 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 . ∎
We can rotate by which corresponds to the permutation . We can also flip the rectangle over the horizontal diagonal which gives the permutation , and finally also over the vertical diagonal which gives the permutation .
An important corollary to this realization is
2 Klein 4-group as a vector space
The automorphism group of a vector space is called the general linear group and so in our context . As we can interchange any 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:
. Furthermore, the affine linear group of is .
3 Klein 4-group as a normal subgroup
Because is a subgroup of we can consider its conjugates. Because conjugation in respects the cycle structure. From this we see that the 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 . We collect these into
We have already argued that is normal in . Upon inspecting the elements 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 it the Sylow 2-subgroup of (Sylow subgroups are conjugate.)
Now notice that the dihedral group acts on a square and so it is represented as a permutation group on 4 vertices, so embeds in . As and , is a Sylow 2-subgroup of and so all Sylow 2-subgroups of are embeddings of (in particular various relabellings of the vertices of the square.) But by Proposition 2 we know that each embedding contains . As there are 3 non-equal embeddings of (think of the 3 non-equal labellings of a square) we know that the intersection of these is a proper subgroup of . As is a maximal subgroup of each and contained in each, is the 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 maps onto 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 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 that we find these two methods intersect in a this exceptionally parallel fashion. Thus we establish the exceptional structure of . For all other ’s, is the only proper normal subgroup.
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 action of on the six edges of the tetrahedron. Observing that this action preserves incidence relations one gets an action of on the three pairs of opposite edges.
4 Other properties
is non-cyclic and of smallest possible order with this property.
is the symmetry group of the Riemannian curvature tensor.
|Date of creation||2013-03-22 12:49:02|
|Last modified on||2013-03-22 12:49:02|
|Last modified by||Algeboy (12884)|