nonabelian group

nonabelian group

A group is said to be

nonabelian

, or

noncommutative

, if
has

elements

which do not commute, that is, if there exist

a

and

b

in the group such that

a

⁢

b

≠

b

⁢

a

. Equivalently, a group is
nonabelian if there exist

a

and

b

in the group such that the

commutator

[

a

,

b

]

is not equal to the

identity

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

homomorphism

is

abelian

,
there exist nonabelian groups of order as small as

27

for which the
cube map is a homomorphism.

In the first

section

we give a way to visualize the group of

rotations

of a sphere and prove that it is nonabelian. This should be readable
by an undergraduate student in

algebra

. 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.

1

1

The
treatment we give here is informal. For a more formal treatment of
the group of rotations, consult the entries “

Rotation
matrix

(

http://planetmath.org/RotationMatrix

)” and “

Dimension

of the

special
orthogonal group

(

http://planetmath.org/DimensionOfTheSpecialOrthogonalGroup

)”.

We can
compose two rotations by performing them in

sequence

, 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

π

2

. 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

F

⁢

R

, 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

R

⁢

F

, 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,

F

⁢

R

≠

R

⁢

F

, proving that the
group is not abelian.

Since every rotation in three-dimensional

Euclidean space

can be
decomposed as a

finite sequence

of

reflections

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 polygon

, there is an associated group, the

dihedral group

D

2

⁢

n

, which is the group of

symmetries

of the

polygon

. (Here

n

denotes the number of the sides of the polygon, and

2

⁢

n

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

analogy

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

n

≥

3

(so we are dealing with an actual polygon
here), it is possible to show that

F

⁢

R

≠

R

⁢

F

. Moreover, every

group
with

order

1

,

p

, or

p

2

, where

p

is a prime, is abelian. Thus
the smallest possible order for a nonabelian group is

6

. But

D

2

⋅

3

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

x

↦

x

n

, we mean that the

function

φ

defined on the group by the

formula

φ

⁢

(

x

)

=

x

n

is a homomorphism, that is, that is, that for any

x

and

y

in
the group,

(

x

⁢

y

)

n

=

φ

⁢

(

x

⁢

y

)

=

φ

⁢

(

x

)

⁢

φ

⁢

(

y

)

=

x

n

⁢

y

n

.

If a group admits

x

↦

x

2

, then for any

x

and

y

we have
that

(

x

⁢

y

)

2

=

x

2

⁢

y

2

. Multiplying on the left by

x

-

1

and on
the right by

y

-

1

yields the identity

y

⁢

x

=

x

⁢

y

. Thus all such
groups are abelian. Moreover, the generalized

commutativity

and

associativity

laws for

abelian groups

imply that an abelian group
admits all maps

x

↦

x

n

. It is therefore reasonable to wonder
whether the

converse

holds. In fact it is possible for a nonabelian
group to admit

x

↦

x

3

. 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

presentation

G

=

⟨

a

,

b

,

c

∣

a

3

,

b

3

,

c

3

,

[

a

,

c

]

,

[

b

,

c

]

,

[

a

,

b

]

⁢

c

⟩

.

This can be realized concretely as the group of upper-triangular
matrices over

ℤ

/

3

⁢

ℤ

with

1

s on the

diagonal

, but for

simplicity

we shall work directly with the presentation.

The first three relators tell us that each

generator

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

b

⁢

a

↦

a

⁢

b

⁢

c

,

that is,

“when

b

moves past

a

it turns into

b

⁢

c

.”

Thus given an element of

G

we can always write it in the normal form

a

j

⁢

b

k

⁢

c

ℓ

, where

0

≤

j

,

k

,

ℓ

<

3

, and all such elements
are distinct. This proves that the

cardinality

of

G

is

27

.
Moreover, we also observe that

b

⁢

a

=

a

⁢

b

⁢

c

≠

a

⁢

b

,

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

=

a

j

⁢

b

k

⁢

c

ℓ

. Since

c

is in the
center of

G

,

x

3

=

(

a

j

⁢

b

k

)

3

⁢

c

3

⁢

ℓ

=

(

a

j

⁢

b

k

)

3

=

a

j

⁢

(

b

k

⁢

a

j

)

2

⁢

b

k

.

To normalize the word

b

k

⁢

a

j

, we push each

b

past all of the

a

s. Since pushing

b

past a

single

a

turns it into

b

⁢

c

, pushing
it past

a

j

turns it into

b

⁢

c

j

, that is,

b

k

⁢

a

j

=

b

k

-

1

⁢

a

j

⁢

b

⁢

c

j

.

By

induction

it follows that

b

k

⁢

a

j

=

a

j

⁢

b

k

⁢

c

j

⁢

k

.

Applying this result to

x

3

, we get that

x

3

=

a

j

⁢

(

a

j

⁢

b

k

⁢

c

j

⁢

k

)

2

⁢

b

k

=

a

2

⁢

j

⁢

(

b

k

⁢

a

j

)

⁢

b

2

⁢

k

⁢

c

2

⁢

j

⁢

k

=

a

3

⁢

j

⁢

b

3

⁢

k

⁢

c

3

⁢

j

⁢

k

=

1

.

Since

x

3

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

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