free group

free group

\PMlinkescapephrase

multiplicative group

Definition

If

F

is a

group with

a subset

A

such that for every group

G

every

function

ψ

:

A

→

G

extends to a unique

homomorphism

(

http://planetmath.org/GroupHomomorphism

)

ϕ

:

F

→

G

, then

F

is said to be a

free group

of

rank

|

A

|

,
and we say that

A

freely generates

F

.

Examples

A group with only one

element

is a free group of rank

0

, freely generated by the

empty set

.

The

infinite cyclic group

ℤ

is a free group of rank

1

, freely generated by either

{

1

}

or

{

-

1

}

.

An example of a free group of rank

2

is the multiplicative group of

2

×

2

integer matrices

generated by

(

1

2

0

1

)

and

(

1

0

2

1

)

.

If

F

is a free group freely generated by a set

A

, where

|

A

|

>

1

, then for distinct

a

,

b

∈

A

the set

{

a

n

⁢

b

n

∣

0

<

n

∈

ℤ

}

generates a free group of

countably infinite

rank.

Properties

If a free group

F

is freely generated by

A

, then

A

is a

minimal

generating set

for

F

, and no set of smaller

cardinality

than

A

can generate

F

.
It follows that if

F

is freely generated by both

A

and

B

, then

|

A

|

=

|

B

|

. So the rank of a free group is a well-defined

concept

, and free groups of different ranks are non-isomorphic.

For every

cardinal number

κ

there is, up to

isomorphism

, exactly one free group of rank

κ

.
The

abelianization

of a free group of rank

κ

is a

free abelian group

of rank

κ

.

Every group is a homomorphic image of some free group.
More precisely, if

G

is a group generated by a set of cardinality

κ

, then

G

is a homomorphic image of every free group of rank

κ

or more.

The

Nielsen-Schreier Theorem

(

http://planetmath.org/NielsenSchreierTheorem

) states that every

subgroup

(

http://planetmath.org/Subgroup

) of a free group is itself free.

Construction

For any set

A

, the following construction gives a free group of rank

|

A

|

.

Let

A

be a set with elements

a

i

for

i

in some

index set

I

.
We refer to

A

as an

alphabet

and the elements of

A

as

letters

.
A

syllable

is a symbol of the form

a

i

n

for

n

∈

ℤ

.
It is customary to write

a

for

a

1

. Define a

word

to be a

finite sequence

of syllables. For example,

a

2

3

⁢

a

1

⁢

a

4

-

1

⁢

a

3

2

⁢

a

2

-

3

is a five-syllable word. Notice that there exists a unique

empty word

, i.e., the word with no syllables, usually written simply as

1

.
Denote the set of all words formed from elements of

A

by

𝒲

⁢

[

A

]

.

Define a

binary operation

, called the

product

, on

𝒲

⁢

[

A

]

by

concatenation

of words. To illustrate, if

a

2

3

⁢

a

1

and

a

1

-

1

⁢

a

3

4

are elements of

𝒲

⁢

[

A

]

then their product is simply

a

2

3

⁢

a

1

⁢

a

1

-

1

⁢

a

3

4

.
This gives

𝒲

⁢

[

A

]

the

structure

of a monoid.
The empty word

1

acts as a right and

left identity

(

http://planetmath.org/LeftIdentityAndRightIdentity

) in

𝒲

⁢

[

A

]

, and is the only element which has an

inverse

. In order to give

𝒲

⁢

[

A

]

the structure of a group, two more ideas are needed.

If

v

=

u

1

⁢

a

i

0

⁢

u

2

is a word where

u

1

,

u

2

are also words and

a

i

is some element of

A

, an

elementary

contraction

of type I

replaces the occurrence of

a

0

by

1

. Thus, after this type of contraction we get another word

w

=

u

1

⁢

u

2

. If

v

=

u

1

⁢

a

i

p

⁢

a

i

q

⁢

u

2

is a word, an

elementary contraction of type II

replaces the occurrence of

a

i

p

⁢

a

i

q

by

a

i

p

+

q

which results in

w

=

u

1

⁢

a

i

p

+

q

⁢

u

2

. In either of these cases, we also say that

w

is obtained from

v

by an elementary contraction, or that

v

is obtained from

w

by an elementary

expansion

.

Call two words

u

,

v

equivalent

(denoted

u

∼

v

) if one can be obtained from the other by a finite sequence of elementary contractions or expansions. This is an

equivalence relation

on

𝒲

⁢

[

A

]

. Let

ℱ

⁢

[

A

]

be the set of

equivalence classes

of words in

𝒲

⁢

[

A

]

. Then

ℱ

⁢

[

A

]

is group under the

operation

[

u

]

⁢

[

v

]

=

[

u

⁢

v

]

where

[

u

]

∈

ℱ

⁢

[

A

]

. The inverse

[

u

]

-

1

of an element

[

u

]

is obtained by reversing the order of the syllables of

[

u

]

and changing the sign of each syllable. For example, if

[

u

]

=

[

a

1

⁢

a

3

2

]

, then

[

u

]

-

1

=

[

a

3

-

2

⁢

a

1

-

1

]

.

It can be shown that

ℱ

⁢

[

A

]

is a free group freely generated by the set

{

[

x

]

∣

x

∈

A

}

. Moreover, a group is free if and only if it is isomorphic to

ℱ

⁢

[

A

]

for some set

A

.

Title

free group

Canonical name

FreeGroup

Date of creation

2013-03-22 12:28:39

Last modified on

2013-03-22 12:28:39

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

43

Author

yark (2760)

Entry type

Definition

Classification

msc 20E05

Related topic

FreeModule

Related topic

Subgroup

Related topic

Group

Related topic

FreeProduct

Defines

freely generates

Defines

freely generated

Defines

rank

Defines

empty word

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