examples of groups

examples of groups

Groups (

http://planetmath.org/Group

) are ubiquitous throughout mathematics. Many “naturally
occurring” groups are either groups of numbers (typically

Abelian

)
or groups of

symmetries

(typically

non-Abelian

).

Groups of numbers

•

The most important group is the group of

integers

ℤ

with

addition

as

operation

and zero as

identity element

.

•

The integers modulo

n

, often denoted by

ℤ

n

, form a
group under addition. Like

ℤ

itself, this is a

cyclic group

; any
cyclic group is

isomorphic

to one of these.

•

The

rational

(or real, or

complex

) numbers form a group under
addition.

•

The

positive

rationals form a group under

multiplication

with identity element 1, and so
do the non-zero rationals. The same is true for the reals and real

algebraic numbers

.

•

The non-zero

complex numbers

form a group under multiplication.
So do the non-zero

quaternions

. The latter is our first example of a

non-Abelian group

.

•

More generally, any (skew) field gives rise to two groups: the

additive
group

of all field

elements

with 0 as identity element, and the

multiplicative group

of all
non-zero field elements with 1 as identity element.

•

The complex numbers of

absolute value

1 form a group under
multiplication, best thought of as the

unit circle

. The quaternions of
absolute value 1 form a group under multiplication, best thought of as
the three-dimensional

unit sphere

S

3

. The two-dimensional sphere

S

2

however
is not a group in any natural way.

•

The

positive integers

less than

n

which are

coprime

to

n

form a group if the operation is defined as
multiplication modulo

n

. This is an

Abelian group

whose order is given by the
Euler phi-function

ϕ

⁢

(

n

)

.

•

The units of the

number ring

ℤ

⁢

[

3

]

form the multiplicative group consisting of all integer powers of

2

+

3

and their negatives (see

units of quadratic fields

).

•

Generalizing the last two examples, if

R

is a ring with

multiplicative identity

1, then the units of

R

(

http://planetmath.org/GroupOfUnits

) (the elements

invertible

with respect to multiplication) form a

group with

respect to ring multiplication and with identity element 1. See

examples of rings

.

Most groups of numbers carry natural topologies turning them into
topological groups.

Symmetry groups

•

The

symmetric group

of degree

n

, denoted by

S

n

, consists of
all

permutations

of

n

items and has

n

!

elements. Every

finite
group

is isomorphic to a

subgroup

of some

S

n

(Cayley’s

theorem

).

•

An important subgroup of the symmetric group of degree

n

is the

alternating group

, denoted

A

n

. This consists of all

even

permutations on

n

items. A permutation is said to be even if it can
be written as the

product

of an

even number

of

transpositions

. The alternating
group is normal in

S

n

, of index

2

, and it is an interesting fact that

A

n

is simple for

n

≥

5

. See the proof on the

simplicity

of the alternating
groups. See also examples of finite simple groups.

•

If any geometrical object is given, one can consider its

symmetry group

consisting of all

rotations

and

reflections

which leave
the object unchanged. For example, the symmetry group of a cone is
isomorphic to

S

1

; the symmetry group of a square has eight elements and is isomorphic to the

dihedral group

D

4

.

•

The set of all

automorphisms

of a given group (or field,
or graph, or topological space, or object in any

category

) forms a group with operation
given by the

composition

of

homomorphisms

. This is the

automorphism group

of the given object and captures its “internal symmetries”.

•

In

Galois theory

, the symmetry groups of

field extensions

(or
equivalently: the symmetry groups of

solutions

to

polynomial
equations

) are the

central

object of study; they are called

Galois groups

. One version of the

inverse Galois problem

asks whether every finite group can arise as the symmetry group of some

algebraic extension

of the rational numbers. The answer is

unknown

.

•

Several

matrix groups

describe various aspects of the symmetry of

n

-space:

–

The

general linear group

GL

⁡

(

n

,

ℝ

)

of all real invertible

n

×

n

matrices (with

matrix multiplication

as operation) contains
rotations, reflections,

dilations

, shear transformations, and their

combinations

.

–

The

orthogonal group

O

⁡

(

n

,

ℝ

)

of all real

orthogonal

n

×

n

matrices contains the rotations and reflections of

n

-space.

–

The

special orthogonal group

SO

⁡

(

n

,

ℝ

)

of all real orthogonal

n

×

n

matrices with

determinant

1 contains the rotations of

n

-space.

All these matrix groups are Lie groups: groups which are differentiable manifolds
such that the

group operations

are

smooth maps

.

Other groups

•

The

trivial group

consists only of its identity element.

•

The

Klein 4-group

is a non-cyclic abelian group with four elements. For other small groups, see

groups of small order

.

•

If

X

is a topological space and

x

is a point of

X

, we can
define the

fundamental group

of

X

at

x

. It consists of
(homotopy classes of)

continuous

paths starting and ending at

x

and describes the

structure

of the “holes” in

X

accessible

from

x

. The fundamental group is generalized by the higher homotopy groups.

•

Other groups studied in algebraic topology are the

homology groups

of a topological space. In a different way, they also provide information about the “holes” of the space.

•

The

free groups

are important in algebraic topology. In a sense,
they are the most general groups, having only those

relations

among
their elements that are absolutely required by the group axioms. The free group on the set

S

has as members all the finite strings that can be formed from elements of

S

and their

inverses

; the operation comes from string concatenation.

•

If

A

and

B

are two Abelian groups (or modules over the same ring), then the set

Hom

⁡

(

A

,

B

)

of all homomorphisms from

A

to

B

is an Abelian group. Note that the

commutativity

of

B

is crucial here: without it, one couldn’t prove that the sum of two homomorphisms is again a homomorphism.

•

Given any set

X

, the

powerset

𝒫

⁢

(

X

)

of

X

becomes an abelian group if we use the

symmetric difference

as operation. In this group, any element is its own inverse, which makes it into a

vector space

over

ℤ

2

.

•

If

R

is a ring with multiplicative identity, then the set of all invertible

n

×

n

matrices over

R

forms a group under matrix multiplication with the

identity matrix

as identity element; this group is denoted by

GL

⁡

(

n

,

R

)

. It is the

group of units

of the ring of all

n

×

n

matrices over

R

. For a given

n

, the groups

GL

⁡

(

n

,

R

)

with

commutative ring

R

can be viewed as the points on the

general linear group scheme

GL

n

.

•

If

K

is a

number field

, then multiplication of (

equivalence classes

of) non-zero ideals in the

ring of algebraic integers

𝒪

𝒦

gives rise to the

ideal class group

of

K

.

•

The set of the equivalence classes of

commensurability

of the positive real numbers is an Abelian group with respect to the defined operation.

•

The set of

arithmetic functions

that take a value other than 0 at 1 form an Abelian group under Dirichlet convolution. They include as a subgroup the set of

multiplicative functions

.

•

Consider the curve

C

=

{

(

x

,

y

)

∈

K

2

∣

y

2

=

x

3

-

x

}

, where

K

is any field. Every

straight line

intersects

this set in three points (counting a point twice if the line is

tangent

, and allowing for a

point at infinity

). If we require that those three points add up to zero for any straight line, then we have defined an abelian group structure on

C

. Groups like these are called

abelian varieties

.

•

Let

E

be an

elliptic curve

defined over any field

F

. Then the set of

F

-

rational points

in the curve

E

, denoted by

E

⁢

(

F

)

, can be given the structure of abelian group. If

F

is a number field, then

E

⁢

(

F

)

is a

finitely generated

abelian group. The curve

C

in the example above is an elliptic curve defined over

ℚ

, thus

C

⁢

(

ℚ

)

is a finitely generated abelian group.

•

In the

classification

of all finite simple groups, several
“sporadic” groups occur which don’t follow any discernable

pattern

.
The largest of these is the

monster group

with about

8

⋅

10

53

elements.

Title

examples of groups

Canonical name

ExamplesOfGroups

Date of creation

2013-03-22 12:49:19

Last modified on

2013-03-22 12:49:19

Owner

AxelBoldt (56)

Last modified by

AxelBoldt (56)

Numerical id

34

Author

AxelBoldt (56)

Entry type

Example

Classification

msc 20-00

Classification

msc 20A05

Related topic

ExamplesOfFiniteSimpleGroups

Related topic

SpinGroup

Related topic

ExamplesOfAlgebraicKTheoryGroups

Related topic

QuantumGroups

Related topic

GroupsOfSmallOrder

Related topic

TriangleGroups

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