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单词 Klein4group
释义

Klein 4-group


The Klein 4-group is the subgroupMathworldPlanetmathPlanetmath V (Vierergruppe) ofS4 (see symmetric groupMathworldPlanetmathPlanetmath) consisting of the following4 permutationsMathworldPlanetmath:

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

(see cycle notation). This is anabelian groupMathworldPlanetmath, isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to the productPlanetmathPlanetmathPlanetmath 22.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 groupMathworldPlanetmath of various planargraphsMathworldPlanetmath, including graphs of 4 vertices. Yet we have

Proposition 1.

V is not the automorphism group of a simple graphMathworldPlanetmath.

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 mapevery vertex to every other. So G is a regular graphMathworldPlanetmath on 4 vertices.This makes G isomorphic to one of the following 4 graphs:

{xy}<10mm,0mm>:<0mm,10mm>::(0,0)*+1="1";(1,0)*+2="2";(1,1)*+3="3";(0,1)*+4="4";{xy}<10mm,0mm>:<0mm,10mm>::(0,0)*+1="1";(1,0)*+2="2";(1,1)*+3="3";(0,1)*+4="4";"1";"2"**@-;"3";"4"**@-;{xy}<10mm,0mm>:<0mm,10mm>::(0,0)*+1="1";(1,0)*+2="2";(1,1)*+3="3";(0,1)*+4="4";"1";"2"**@-;"2";"3"**@-;"3";"4"**@-;"4";"1"**@-;{xy}<10mm,0mm>:<0mm,10mm>::(0,0)*+1="1";(1,0)*+2="2";(-0.5,0.86)*+3="3";(-0.5,-0.86)*+4="4";"1";"2"**@-;"1";"3"**@-;"1";"4"**@-;"2";"3"**@-;"2";"4"**@-;"3";"4"**@-;.

In order the automorphism groups of these graphs are S4,(12),(34), (12),(1234) and S4.None of these are V, though the second is isomorphic to V.∎

Though V cannot be realized as an automorphism group of a planar graphit can be realized as the set of symmetriesMathworldPlanetmathPlanetmathPlanetmath of a polygonMathworldPlanetmathPlanetmath, in particular,a non-square rectangleMathworldPlanetmathPlanetmath.

{xy}<10mm,0mm>:<0mm,10mm>::(0,0)*+1="1";(2,0)*+2="2";(2,1)*+3="3";(0,1)*+4="4";"1";"2"**@-;"2";"3"**@-;"3";"4"**@-;"4";"1"**@-;

We can rotate by 180 which corresponds to the permutation(13)(24). We can also flip the rectangle over the horizontal diagonalwhich gives the permutation (14)(23), and finally also over the verticaldiagonal which gives the permutation (12)(34).

{xy}<10mm,0mm>:<0mm,10mm>::(0,0)*+3="1";(2,0)*+4="2";(2,1)*+1="3";(0,1)*+2="4";"1";"2"**@-;"2";"3"**@-;"3";"4"**@-;"4";"1"**@-;,{xy}<10mm,0mm>:<0mm,10mm>::(0,0)*+4="1";(2,0)*+3="2";(2,1)*+2="3";(0,1)*+1="4";"1";"2"**@-;"2";"3"**@-;"3";"4"**@-;"4";"1"**@-;,{xy}<10mm,0mm>:<0mm,10mm>::(0,0)*+2="1";(2,0)*+1="2";(2,1)*+4="3";(0,1)*+3="4";"1";"2"**@-;"2";"3"**@-;"3";"4"**@-;"4";"1"**@-;.

An important corollary to this realization is

Proposition 2.

Given a square with vertices labeled in any way by {1,2,3,4}, then thefull symmetry group (the dihedral groupMathworldPlanetmath of order 8, D8) contains V.

2 Klein 4-group as a vector space

As V is isomorphic to 22 it is a 2-dimensionalvector space over the Galois field 2. The projective geometryof V – equivalently, the lattice of subgroups – is given in the followingHasse diagam:

{xy}<10mm,0mm>:<0mm,10mm>::(0,0)*+()="1.1";(-2,1)*+(12)(34)="2.1";(0,1)*+(13)(24)="3.1";(2,1)*+(14)(23)="4.1";(0,2)*+V="5.1";"2.1";"1.1"**@-;"3.1";"1.1"**@-;"4.1";"1.1"**@-;"5.1";"4.1"**@-;"5.1";"3.1"**@-;"5.1";"2.1"**@-;

The automorphism group of a vector space is called the general lineargroupMathworldPlanetmath and so in our context AutVGL(2,2). As we can interchangeany basis of a vector space we can label the elements e1=(12)(34),e2=(13)(24) and e3=(14)(23) so that we have the permutations(e1,e2) and (e2,e3) and so we generate all permutations on{e1,e2,e3}. This proves:

Proposition 3.

AutVGL(2,2)S3. Furthermore, the affine linear groupof V is AGL(2,2)=VS3.

3 Klein 4-group as a normal subgroup

Because V is a subgroup of S4 we can consider its conjugates. Becauseconjugation in S4 respects the cycle structureMathworldPlanetmath. From this we see thatthe conjugacy classMathworldPlanetmath in S4 of every element of V lies again in V. ThusV is normal. This now allows us to combine both of the previous sectionsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathto outline the exceptional nature (amongst Sn families) of S4. Wecollect these into

Theorem 4.
  1. 1.

    V is a normal subgroupMathworldPlanetmath of S4.

  2. 2.

    V is contained in A4 and so it is a normal subgroup of A4.

  3. 3.

    V is the Sylow 2-subgroup of A4.

  4. 4.

    V is the intersectionMathworldPlanetmathPlanetmath of all Sylow 2-subgroups of S4, that is,the 2-core of S4.

  5. 5.

    S4/VS3.

  6. 6.

    S4AGL(2,2)VS3.

Proof.

We have already argued that V is normal in S4. Upon inspecting theelements of V we see V contains only even permutationsMathworldPlanetmath so VA4and consequently V is normal in A4 as well. As |A4|=12 and |V|=4we establish V is a Sylow 2-subgroup of A4. But V is normal so itthe Sylow 2-subgroup of A4 (Sylow subgroups are conjugate.)

Now notice that the dihedral group D8 acts on a square and so it isrepresented as a permutation groupMathworldPlanetmath on 4 vertices, so D8 embeds in S4.As |D8|=8 and |S4|=24, D8 is a Sylow 2-subgroup of S4 and soall Sylow 2-subgroups of S4 are embeddingsPlanetmathPlanetmath of D8 (in particular variousrelabellings of the vertices of the square.) But by PropositionPlanetmathPlanetmath 2we know that each embedding contains V. As there are 3 non-equalembeddings of D8 (think of the 3 non-equal labellings of a square) weknow that the intersection of these D8 is a proper subgroupMathworldPlanetmath of D8.As V is a maximal subgroup of each D8 and contained in each, V isthe intersection of all these embeddings.

Now the action of S4 by conjugation on the Sylow 2-subgroups D8permutes all 3 (again Sylow subgroups are conjugate) so S4S3.Indeed, V is in the kernel of this action as V is in each D8.Indeed a three cycle (123) permutes the D8’s with no fixed pointPlanetmathPlanetmath(consider the relabellings) and (12) fixes only one. So S4 mapsonto S3 and so the kernel is precisely V. Thus S4/V=S3.

Now we can embed S3 into S4 as (123),(12) soVS3=1, VS3=S4 so S4=VS3. Finally, AGL(2,2)acts transitively on the four points of the vector space V soAGL(2,2) embeds in S4. And by Proposition 3 we concludeS4AGL(2,2).∎

We can make similarMathworldPlanetmathPlanetmath arguments about subgroups of symmetries forlarger regular polygonsMathworldPlanetmath. Likewise for other 2-dimensional vector spaceswe can establish similar structural properties. However it is onlywhen we study we involve V that we find these two methods intersectin a this exceptionally parallelMathworldPlanetmathPlanetmath fashion. Thus we establish the exceptionalstructure of S4. For all other Sn’s, An is the only proper normalsubgroup.

We can view the properties of our theorem in a geometric way as follows:S4 is the group of symmetries of a tetrahedronMathworldPlanetmathPlanetmath. There is an induced actionof S4 on the six edges of the tetrahedron. Observing that this action preserves incidence relationsPlanetmathPlanetmath one gets an action of S4 on the three pairsof opposite edges.

4 Other properties

V is non-cyclic and of smallest possible order with this property.

V is transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and regularPlanetmathPlanetmath. Indeed V is the (unique) regular representation of22. The other 3 subgroups of S4 which are isomorphic to22 are not transitive.

V is the symmetry group of the Riemannian curvature tensor.

TitleKlein 4-group
Canonical nameKlein4group
Date of creation2013-03-22 12:49:02
Last modified on2013-03-22 12:49:02
OwnerAlgeboy (12884)
Last modified byAlgeboy (12884)
Numerical id26
AuthorAlgeboy (12884)
Entry typeTopic
Classificationmsc 20K99
SynonymKlein four-group
SynonymViergruppe
Related topicGroupsInField
Related topicKlein4Ring
Related topicPrimeResidueClass
Related topicAbelianGroup2
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