nonabelian group


A group is said to be nonabelianPlanetmathPlanetmathPlanetmath, or noncommutative, if has elements which do not commute, that is, if there exist a and b in the group such that abba. Equivalently, a group is nonabelian if there exist a and b in the group such that the commutatorPlanetmathPlanetmath [a,b] is not equal to the identityPlanetmathPlanetmathPlanetmathPlanetmath of the group. There exist many natural nonabelian groups, with order as small as 6. While any group for which the square map is a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is abelianMathworldPlanetmath, there exist nonabelian groups of order as small as 27 for which the cube map is a homomorphism.

In the first sectionMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath we give a way to visualize the group of rotationsMathworldPlanetmath of a sphere and prove that it is nonabelian. This should be readable by an undergraduate student in algebraPlanetmathPlanetmathPlanetmath. In the second section, we discuss groups admitting a cube map and show that there are small nonabelian examples. The second section is somewhat more technical than the first and will require more facility with group theory, especially working with finitely presented groups and the commutator calculus.

1 Concrete examples of nonabelian groups

Although most number systems we use are abelian by design, there exist quite natural nonabelian groups. Perhaps the simplest example to visualize is given by the group of rotations of a sphere.11The treatment we give here is informal. For a more formal treatment of the group of rotations, consult the entries “Rotation matrixMathworldPlanetmath (http://planetmath.org/RotationMatrix)” and “DimensionPlanetmathPlanetmath of the special orthogonal groupMathworldPlanetmath (http://planetmath.org/DimensionOfTheSpecialOrthogonalGroup)”. We can compose two rotations by performing them in sequencePlanetmathPlanetmath, and we can invert a rotation by rotating in the opposite direction, so rotations do form a group. To follow what rotation does to the sphere, imagine that inside is suspended a copy of Marshall Hall’s classic text The Theory of Groups. We will keep track of three pieces of information, namely, the directions that the front cover, the spine, and the bottom of the book face. When the sphere is in the identity position, the front cover faces the reader, the spine faces the left, and the bottom of the book is oriented downward.

In preparation for verifying that the group is not abelian, we define two rotations, F and R. First, let F (for “flip”) be the rotation which takes the point at the very top of the sphere and moves it forward through an angle of π. For example, if we start with the sphere in the identity position and then perform F, the front cover will face away from the reader, the spine will remain to the left, and the bottom of the book will be oriented upward. Second, let R (for “rotate”) be the rotation which takes the point at the very top of the sphere and moves it left through an angle of π2. If we start with the sphere in the identity position and then perform R, the front cover will continue to face the reader, the spine will face downward, and the bottom of the book will be oriented to the right.

We now verify that the group of rotations is not abelian. If we start with the sphere in the identity position and perform FR, that is, first F, then R, then the front cover will face away from the reader, the spine will face downward, and the bottom will be oriented to the left. On the other hand, if we start with the sphere in the identity position and perform RF, then while the front cover will face away from the reader, the spine will face upward, and the bottom will be oriented to the right. So it matters in which order we perform F and R, that is, FRRF, proving that the group is not abelian.

Since every rotation in three-dimensional Euclidean space can be decomposed as a finite sequence of reflectionsMathworldPlanetmath and rotations in the Euclidean plane, one might hope that we can find finite nonabelian groups arising from objects in the plane, and in fact we can. For each regular polygonMathworldPlanetmath, there is an associated group, the dihedral groupMathworldPlanetmath D2n, which is the group of symmetriesMathworldPlanetmathPlanetmathPlanetmath of the polygonMathworldPlanetmathPlanetmath. (Here n denotes the number of the sides of the polygon, and 2n gives the number of elements of the group of symmetries.) It is generated by two elements, F (for “flip”) and R (for “rotate”). These elements can be defined by analogyMathworldPlanetmath with the F and R above; for full details, consult the entry “Dihedral group (http://planetmath.org/DihedralGroup)”, where flips are labelled instead by M (for “mirror”). If n3 (so we are dealing with an actual polygon here), it is possible to show that FRRF. Moreover, every group with order 1, p, or p2, where p is a prime, is abelian. Thus the smallest possible order for a nonabelian group is 6. But D23 has 6 elements and is nonabelian, so it is the smallest possible nonabelian group.

2 Small nonabelian groups admitting a cube map

When we say that a group admits xxn, we mean that the function φ defined on the group by the formulaMathworldPlanetmathPlanetmath φ(x)=xn is a homomorphism, that is, that is, that for any x and y in the group,

(xy)n=φ(xy)=φ(x)φ(y)=xnyn.

If a group admits xx2, then for any x and y we have that (xy)2=x2y2. Multiplying on the left by x-1 and on the right by y-1 yields the identity yx=xy. Thus all such groups are abelian. Moreover, the generalized commutativity and associativity laws for abelian groups imply that an abelian group admits all maps xxn. It is therefore reasonable to wonder whether the converseMathworldPlanetmath holds. In fact it is possible for a nonabelian group to admit xx3. The smallest order for such a group is 27. It is beyond the scope of this entry to prove that 27 is the smallest possible order, but we will give an explicit example.

Let G be the group with presentationMathworldPlanetmathPlanetmath

G=a,b,ca3,b3,c3,[a,c],[b,c],[a,b]c.

This can be realized concretely as the group of upper-triangular matrices over /3 with 1s on the diagonalMathworldPlanetmath, but for simplicity we shall work directly with the presentation.

The first three relators tell us that each generatorPlanetmathPlanetmathPlanetmath of the group has order 3. The next two tell us that c is central — since it commutes with the other two generators and commutes with itself, it must therefore commute with everything. The final relator is perhaps the most interesting. We can interpret it as the rewrite rule

baabc,

that is,

“when b moves past a it turns into bc.”

Thus given an element of G we can always write it in the normal form ajbkc, where 0j,k,<3, and all such elements are distinct. This proves that the cardinality of G is 27. Moreover, we also observe that

ba=abcab,

so G is not abelian.

It remains to check that G admits the cube map. We will prove the simpler statement that G has exponent 3. Given x in G, we first normalize it, so x=ajbkc. Since c is in the center of G,

x3=(ajbk)3c3=(ajbk)3=aj(bkaj)2bk.

To normalize the word bkaj, we push each b past all of the as. Since pushing b past a single a turns it into bc, pushing it past aj turns it into bcj, that is,

bkaj=bk-1ajbcj.

By inductionMathworldPlanetmath it follows that

bkaj=ajbkcjk.

Applying this result to x3, we get that

x3=aj(ajbkcjk)2bk=a2j(bkaj)b2kc2jk=a3jb3kc3jk=1.

Since x3 is trivial for any x, it follows that G admits the cube map.

The other nonabelian group of order 27 has exponent 9 and also admits the cube map. This will be described in an attached entry.

Title nonabelian group
Canonical name NonabelianGroup
Date of creation 2013-03-22 14:02:04
Last modified on 2013-03-22 14:02:04
Owner drini (3)
Last modified by drini (3)
Numerical id 10
Author drini (3)
Entry type Definition
Classification msc 20A05
Synonym non-abelian group
Synonym noncommutative group
Synonym non-commutative group
Related topic AbelianGroup2