group extension

group extension

Let

G

and

H

be groups. A group

E

is called an

extension

of

G

by

H

if

1.

G

is

isomorphic

to a

normal subgroup

N

of

E

, and

2.

H

is isomorphic to the

quotient group

E

/

N

.

The definition is well-defined and it is convenient sometimes to
regard

G

as a normal subgroup of

E

. The definition can be
alternatively defined:

E

is an extension of

G

by

H

if there is
a

short exact sequence

of groups:

1

⟶

G

⟶

E

⟶

H

⟶

1

.

In fact, some authors define an extension (of a group by a group) to
be a short exact sequence of groups described above. Also, many authors
prefer the reverse terminology, calling the group

E

an extension of

H

by

G

.

Remarks

•

Given any groups

G

and

H

, an extension of

G

by

H

exists: take the

direct product

of

G

and

H

.

•

An intermediate

concept

between an extension a direct product is that of a

semidirect product

of two groups: If

G

and

H

are groups, and

E

is an extension of

G

by

H

(identifying

G

with a normal subgroup
of

E

), then

E

is called a semidirect product of

G

by

H

if

(a)

H

is isomorphic to a

subgroup

of

E

, thus viewing

H

as a
subgroup of

E

,

(b)

E

=

G

⁢

H

, and

(c)

G

∩

H

=

⟨

1

⟩

.

Equivalently,

E

is a semidirect product of

G

and

H

if the
short exact sequence

1

⟶

G

⟶

E

⟶

α

H

⟶

1

splits. That is, there is a

group homomorphism

ϕ

:

H

→

E

such
that the

composition

H

⟶

ϕ

E

⟶

α

H

gives the

identity map

.
Thus, a semidirect product is also known as a

split
extension

. That a semidirect product

E

of

G

by

H

is also an extension of

G

by

H

can be seen via the

isomorphism

h

↦

h

⁢

G

.

Furthermore, if

H

happens to be normal in

E

, then

E

is isomorphic to the direct product of

G

and

H

. (We need to show that

(

g

,

h

)

↦

g

⁢

h

is an isomorphism. It is not hard to see that the map is a

bijection

. The trick is to show that it is a

homomorphism

, which boils down to showing that every element of

G

commutes with every element of

H

. To show the last step, suppose

g

⁢

h

⁢

g

-

1

=

h

¯

∈

H

. Then

g

⁢

h

=

h

¯

⁢

g

, so

g

⁢

h

⁢

h

¯

-

1

=

h

¯

⁢

g

⁢

h

¯

-

1

=

g

¯

∈

G

, or that

h

⁢

h

¯

-

1

=

g

-

1

⁢

g

¯

. Therefore,

h

=

h

¯

.)

•

The

extension problem

in

group theory

is the
classification of all extension groups of a given group

G

by a
given group

H

. Specifically, it is a problem of finding all
“inequivalent” extensions of

G

by

H

. Two extensions

E

1

and

E

2

of

G

by

H

are

equivalent

if there is a homomorphism

e

:

E

1

→

E

2

such that the following

diagram

of two short exact
sequences is

commutative

:

\xymatrix

⁢

1

⁢

\ar

⁢

@

=

[

d

]

⁢

\ar

⁢

[

r

]

⁢

&

⁢

G

⁢

\ar

⁢

@

=

[

d

]

⁢

\ar

⁢

[

r

]

⁢

&

⁢

E

1

⁢

\ar

⁢

[

d

]

e

⁢

\ar

⁢

[

r

]

⁢

&

⁢

H

⁢

\ar

⁢

@

=

[

d

]

⁢

\ar

⁢

[

r

]

⁢

&

⁢

1

⁢

\ar

⁢

@

=

[

d

]

⁢

1

⁢

\ar

⁢

[

r

]

⁢

&

⁢

G

⁢

\ar

⁢

[

r

]

⁢

&

⁢

E

2

⁢

\ar

⁢

[

r

]

⁢

&

⁢

H

⁢

\ar

⁢

[

r

]

⁢

&

⁢

1

.

According to the

5-lemma

,

e

is actually an isomorphism. Thus equivalences
of extensions are well-defined.

•

Like split extensions, special extensions are formed when
certain conditions are imposed on

G

,

H

, or even

E

:

(a)

If all the groups involved are

abelian

(only that

E

is abelian is
necessary here), then we have an

abelian extension

.

(b)

If

G

, considered as a normal subgroup of

E

, actually lies within
the center of

E

, then

E

is called a

central extension

. A central extension that is also a semidirect product is a direct product. Indeed, if

E

is both a central extension and a semidirect product of

G

by

H

, we observe that

(

g

⁢

h

¯

)

⁢

h

⁢

(

g

⁢

h

¯

)

-

1

=

h

¯

⁢

h

⁢

h

¯

-

1

∈

H

so that

H

is normal in

E

. Applying this result to the previous discussion and we have

E

≅

G

×

H

.

(c)

If

G

is a

cyclic group

, then the extensions in question are
called

cyclic extensions

.

Title

group extension

Canonical name

GroupExtension

Date of creation

2013-03-22 15:24:25

Last modified on

2013-03-22 15:24:25

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 20J05

Related topic

HNNExtension

Defines

split extension

Defines

abelian extension

Defines

central extension

Defines

cyclic extension

Defines

extension problem

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