aarjaneiro/planetmath huggingface dataset:
the kernel of a group homomorphism is a normal subgroup

the kernel of a group homomorphism is a normal subgroup

In this entry we show the following simple lemma:

Lemma 1

.

Let

G

and

H

be groups (with

group operations

∗

G

,

∗

H

and

identity elements

e

G

and

e

H

, respectively) and let

Φ

:

G

→

H

be a

group homomorphism

. Then, the kernel of

Φ

, i.e.

Ker

⁡

(

Φ

)

=

{

g

∈

G

:

Φ

⁢

(

g

)

=

e

H

}

,

is a

normal subgroup

of

G

.

Proof.

Let

G

,

H

and

Φ

be as in the statement of the lemma and let

g

∈

G

and

k

∈

Ker

⁡

(

Φ

)

. Then,

Φ

⁢

(

k

)

=

e

H

by definition and:

Φ

⁢

(

g

∗

G

k

∗

G

g

-

1

)

=

Φ

⁢

(

g

)

∗

H

Φ

⁢

(

k

)

∗

H

Φ

⁢

(

g

-

1

)

=

Φ

⁢

(

g

)

∗

H

(

e

H

)

∗

H

Φ

⁢

(

g

-

1

)

=

Φ

⁢

(

g

)

∗

H

Φ

⁢

(

g

-

1

)

=

Φ

⁢

(

g

)

∗

H

Φ

⁢

(

g

)

-

1

=

e

H

,

where we have used several times the properties of group homomorphisms and the properties of the identity element

e

H

. Thus,

Φ

⁢

(

g

⁢

k

⁢

g

-

1

)

=

e

H

and

g

⁢

k

⁢

g

-

1

∈

G

is also an element of the kernel of

Φ

. Since

g

∈

G

and

k

∈

Ker

⁡

(

Φ

)

were arbitrary, it follows that

Ker

⁡

(

Φ

)

is normal in

G

.
∎

Conversely

:

Lemma 2

.

Let

G

be a group and let

K

be a normal subgroup of

G

. Then there exists a group homomorphism

Φ

:

G

→

H

, for some group

H

, such that the kernel of

Φ

is precisely

K

.

Proof.

Simply set

H

equal to the

quotient group

G

/

K

and define

Φ

:

G

→

G

/

K

to be the

natural projection

from

G

to

G

/

K

(i.e.

Φ

sends

g

∈

G

to the coset

g

⁢

K

). Then it is clear that the kernel of

Φ

is precisely formed by those elements of

K

.
∎

Although the first lemma is very simple, it is very useful when one tries to prove that a

subgroup

is normal.

Example

.

Let

F

be a field. Let us prove that the

special linear group

SL

⁡

(

n

,

F

)

is normal inside the

general linear group

GL

⁡

(

n

,

F

)

, for all

n

≥

1

. By the lemmas, it suffices to construct a

homomorphism

of

GL

⁡

(

n

,

F

)

with

SL

⁡

(

n

,

F

)

as kernel. The

determinant

of matrices is the homomorphism we are looking for. Indeed:

det

:

GL

⁡

(

n

,

F

)

→

F

×

is a group homomorphism from

GL

⁡

(

n

,

F

)

to the

multiplicative group

F

×

and, by definition, the kernel is precisely

SL

⁡

(

n

,

F

)

, i.e. the matrices with determinant

=

1

. Hence,

SL

⁡

(

n

,

F

)

is normal in

GL

⁡

(

n

,

F

)

.

Title

the kernel of a group homomorphism is a normal subgroup

Canonical name

TheKernelOfAGroupHomomorphismIsANormalSubgroup

Date of creation

2013-03-22 17:20:34

Last modified on

2013-03-22 17:20:34

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

8

Author

alozano (2414)

Entry type

Theorem

Classification

msc 20A05

Related topic

KernelOfAGroupHomomorphism

Related topic

NaturalProjection

Generated on Fri Feb 9 19:19:21 2018 by

LaTeXML
third isomorphism theorem

third isomorphism theorem

If

G

is a group (or ring, or module) and

H

and

K

are

normal subgroups

(or ideals, or

submodules

, respectively) of

G

, with

H

⊆

K

, then there is a

natural isomorphism

(

G

/

H

)

/

(

K

/

H

)

≅

G

/

K

.

This is usually known either as the

Third Isomorphism Theorem

, or as the

Second Isomorphism Theorem

(depending on the order in which the theorems are introduced). It is also occasionally called the Freshman Theorem.

Title

third isomorphism theorem

Canonical name

ThirdIsomorphismTheorem

Date of creation

2013-03-22 12:04:03

Last modified on

2013-03-22 12:04:03

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

12

Author

yark (2760)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 13A15

Classification

msc 16D10

Synonym

freshman theorem

Generated on Fri Feb 9 19:19:23 2018 by

LaTeXML
Cayley table

Cayley table

A

Cayley table

for a group is essentially the “multiplication table” of the group.

1

1

A caveat to novices in

group theory

:

multiplication

is usually used notationally to represent the

group operation

, but the operation needn’t resemble multiplication in the reals. Hence, you should take “multiplication table” with a grain or two of salt.

The columns and rows of the table (or matrix) are labeled with the elements of the group, and the cells represent the result of applying the group operation to the row-th and column-th elements.

Formally, let

G

be our group, with operation

∘

the group operation. Let

C

be the Cayley table for the group, with

C

⁢

(

i

,

j

)

denoting the element at row

i

and column

j

. Then

C

⁢

(

i

,

j

)

=

e

i

∘

e

j

where

e

i

is the

i

th element of the group, and

e

j

is the

j

th element.

Note that for an

Abelian group

, we have

e

i

∘

e

j

=

e

j

∘

e

i

, hence the Cayley table is a

symmetric matrix

.

All Cayley tables for

isomorphic groups

are

isomorphic

(that is, the same,

invariant

of the

labeling

and

ordering

of group elements).

0.1

Examples.

•

The Cayley table for

ℤ

4

, the group of

integers

modulo 4 (under

addition

), would be

(

[

0

]

[

1

]

[

2

]

[

3

]

[

0

]

[

0

]

[

1

]

[

2

]

[

3

]

[

1

]

[

1

]

[

2

]

[

3

]

[

0

]

[

2

]

[

2

]

[

3

]

[

0

]

[

1

]

[

3

]

[

3

]

[

0

]

[

1

]

[

2

]

)

•

The Cayley table for

S

3

, the

permutation group

of order 3, is

(

(

1

)

(

123

)

(

132

)

(

12

)

(

13

)

(

23

)

(

1

)

(

1

)

(

123

)

(

132

)

(

12

)

(

13

)

(

23

)

(

123

)

(

123

)

(

132

)

(

1

)

(

13

)

(

23

)

(

12

)

(

132

)

(

132

)

(

1

)

(

123

)

(

23

)

(

12

)

(

13

)

(

12

)

(

12

)

(

23

)

(

13

)

(

1

)

(

132

)

(

123

)

(

13

)

(

13

)

(

12

)

(

23

)

(

123

)

(

1

)

(

132

)

(

23

)

(

23

)

(

13

)

(

12

)

(

132

)

(

123

)

(

1

)

)

Title

Cayley table

Canonical name

CayleyTable

Date of creation

2013-03-22 13:06:44

Last modified on

2013-03-22 13:06:44

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

11

Author

akrowne (2)

Entry type

Definition

Classification

msc 20A99

Synonym

Cayley-table

Generated on Fri Feb 9 19:19:25 2018 by

LaTeXML
characterization of full families of groups

characterization of full families of groups

Proposition

.

Let

𝒢

=

{

G

k

}

k

∈

I

be a family of groups. Then

𝒢

is full if and only if for any

i

,

j

∈

I

such that

i

≠

j

we have that any

homomorphism

f

:

G

i

→

G

j

is

trivial

.

Proof.

,,

⇒

” Assume that

f

:

G

i

→

G

j

is a nontrivial

group homomorphism

. Then define

h

:

⊕

k

∈

I

G

k

→

⊕

k

∈

I

G

k

as follows: if

t

∈

I

is such that

t

≠

i

and

g

∈

⊕

k

∈

I

G

k

is such that

g

∈

G

t

, then

h

⁢

(

g

)

=

g

. If

g

∈

⊕

k

∈

I

G

k

is such that

g

∈

G

i

, then

h

⁢

(

g

)

⁢

(

j

)

=

f

⁢

(

g

⁢

(

i

)

)

and

h

⁢

(

g

)

⁢

(

k

)

=

0

for

k

≠

j

. This values uniquely define

h

and one can easily check that

h

is not

decomposable

.

□

,,

⇐

” Assume that for any

i

,

j

∈

I

such that

i

≠

j

we have that any homomorphism

f

:

G

i

→

G

j

is trivial. Let

h

:

⊕

k

∈

I

G

k

→

⊕

k

∈

I

G

k

be any homomorphism. Moreover, let

i

∈

I

and

g

∈

⊕

k

∈

I

G

k

be such that

g

∈

G

i

. We wish to show that

h

⁢

(

g

)

∈

G

i

.

So assume that

h

⁢

(

g

)

∉

G

i

. Then there exists

j

≠

i

such that

0

≠

h

⁢

(

g

)

⁢

(

j

)

∈

G

j

. Let

π

:

⊕

k

∈

I

G

k

→

G

j

be the

projection

and let

u

:

G

i

→

⊕

k

∈

I

G

k

be the natural

inclusion

homomorphism. Then

π

∘

u

:

G

i

→

G

j

is a nontrivial group homomorphism.

Contradiction

.

□

Corollary.

Assume that

{

G

k

}

k

∈

I

is a family of nontrivial groups such that

G

i

is

periodic

for each

i

∈

I

. Moreover assume that for any

i

,

j

∈

I

such that

i

≠

j

and any

g

∈

G

i

,

h

∈

G

j

orders

|

g

|

and

|

h

|

are realitvely prime (which implies that

I

is

countable

). Then

{

G

k

}

k

∈

I

is full.

Proof.

Assume that

i

≠

j

and

f

:

G

i

→

G

j

is a group homomorphism. Then

|

f

⁢

(

g

)

|

divides

|

g

|

for any

g

∈

G

i

. But

f

⁢

(

g

)

∈

G

j

, so

|

g

|

and

|

f

⁢

(

g

)

|

are relatively prime. Thus

|

f

⁢

(

g

)

|

=

1

, so

f

⁢

(

g

)

=

0

. Therefore

f

is trivial, which (due to proposition)

completes

the proof.

□

Title

characterization of full families of groups

Canonical name

CharacterizationOfFullFamiliesOfGroups

Date of creation

2013-03-22 18:36:08

Last modified on

2013-03-22 18:36:08

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

9

Author

joking (16130)

Entry type

Derivation

Classification

msc 20A99

Generated on Fri Feb 9 19:19:28 2018 by

LaTeXML
decomposable homomorphisms and full families of groups

decomposable homomorphisms and full families of groups

Let

{

G

i

}

i

∈

I

,

{

H

i

}

i

∈

I

be two families of groups (indexed with the same set

I

).

Definition.

We will say that a

homomorphism

f

:

⊕

i

∈

I

G

i

→

⊕

i

∈

I

H

i

is

decomposable

if there exists a family of homomorphisms

{

f

i

:

G

i

→

H

i

}

i

∈

I

such that

f

=

⊕

i

∈

I

f

i

.

Remarks.

For each

j

∈

I

and

g

∈

⊕

i

∈

I

G

i

we will say that

g

∈

G

j

if

g

⁢

(

i

)

=

0

for any

i

≠

j

. One can easily show that any homomorphism

f

:

⊕

i

∈

I

G

i

→

⊕

i

∈

I

H

i

is decomposable if and only if for any

j

∈

I

and any

g

∈

⊕

i

∈

I

G

i

such that

g

∈

G

j

we have

f

⁢

(

g

)

∈

H

j

. This implies that if

f

is an

isomorphism

and

f

is decomposable, then each homomorphism in decomposition is an isomorphism and

(

⊕

i

∈

I

f

i

)

-

1

=

⊕

i

∈

I

f

i

-

1

.

Also it is worthy to note that

composition

of two decomposable homomorphisms is also decomposable and

(

⊕

i

∈

I

f

i

)

∘

(

⊕

i

∈

I

g

i

)

=

⊕

i

∈

I

f

i

∘

g

i

.

Definition.

We will say that family of groups

{

G

i

}

i

∈

I

is

full

if each homomorphism

f

:

⊕

i

∈

I

G

i

→

⊕

i

∈

I

G

i

is decomposable.

Remark.

It is easy to see that if

{

G

i

}

i

∈

I

is a full family of groups and

I

0

⊆

I

, then

{

G

i

}

i

∈

I

0

is also a full family of groups.

Example.

Let

𝒫

=

{

p

∈

ℕ

|

p

⁢

is prime

}

. Then

{

ℤ

p

}

p

∈

𝒫

is full. Indeed, let

f

:

⊕

p

∈

𝒫

ℤ

p

→

⊕

p

∈

𝒫

ℤ

p

be a

group homomorphism

. Then, for any

q

∈

𝒫

and

a

∈

⊕

p

∈

𝒫

ℤ

p

such that

a

∈

ℤ

q

we have that

|

a

|

divides

q

and thus

|

f

⁢

(

a

)

|

divides

q

, so it is easy to see that

f

⁢

(

a

)

∈

ℤ

q

. Therefore (due to first remark)

f

is decomposable.

Counterexample.

Let

G

1

,

G

2

be two copies of

ℤ

. Then

{

G

1

,

G

2

}

is not full. Indeed, let

f

:

ℤ

⊕

ℤ

→

ℤ

⊕

ℤ

be a group homomorphism defined by

f

⁢

(

x

,

y

)

=

(

0

,

x

+

y

)

.

Now assume that

f

=

f

1

⊕

f

2

. Then we have:

(

0

,

1

)

=

f

⁢

(

1

,

0

)

=

(

f

1

⁢

(

1

)

,

f

2

⁢

(

0

)

)

and so

f

2

⁢

(

0

)

=

1

.

Contradiction

, since group homomorphisms preserve

neutral elements

.

Title

decomposable homomorphisms and full families of groups

Canonical name

DecomposableHomomorphismsAndFullFamiliesOfGroups

Date of creation

2013-03-22 18:36:03

Last modified on

2013-03-22 18:36:03

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

7

Author

joking (16130)

Entry type

Definition

Classification

msc 20A99

Generated on Fri Feb 9 19:19:32 2018 by

LaTeXML
direct products of groups

direct products of groups

\PMlinkescapephrase

cartesian product

Let

(

G

i

)

i

∈

I

be a family of groups.

The

unrestricted direct product

(or

complete direct product

, or

Cartesian product

)

∏

i

∈

I

G

i

is the Cartesian product (

http://planetmath.org/GeneralizedCartesianProduct

)

∏

i

∈

I

G

i

with

multiplication

defined

pointwise

,
that is, for all

f

,

g

∈

∏

i

∈

I

G

i

and all

i

∈

I

we have

(

f

⁢

g

)

⁢

(

i

)

=

f

⁢

(

i

)

⁢

g

⁢

(

i

)

.
It is easily verified that this multiplication
makes the Cartesian product into a group.
This construction is in fact the

categorical direct product

(

http://planetmath.org/CategoricalDirectProduct

) in the category of groups.

The

restricted direct product

⊕

i

∈

I

G

i

is the

subgroup

of

∏

i

∈

I

G

i

consisting of all those

elements

with

finite support

. That is,

⊕

i

∈

I

G

i

=

{

f

∈

∏

i

∈

I

G

i

|

f

⁢

(

i

)

=

1

⁢

for all but finitely many

⁢

i

∈

I

}

.

The restricted direct product is also called the

direct sum

, although this usage is usually reserved for the case where all the

G

i

are

abelian

(see direct sum of modules (

http://planetmath.org/DirectSum

) and categorical direct sum (

http://planetmath.org/CategoricalDirectSum

)).

The unqualified term

direct product

can refer either to the unrestricted direct product or to the restricted direct product, depending on the author.
Note that if

I

is finite then the unrestricted direct product and the restricted direct product are in fact the same.
The direct product of two groups

G

and

H

is usually written

G

×

H

,
or sometimes

G

⊕

H

(or

G

⁢

∐

H

) if

G

and

H

are both abelian.

Title

direct products of groups

Canonical name

DirectProductsOfGroups

Date of creation

2013-03-22 14:52:54

Last modified on

2013-03-22 14:52:54

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

17

Author

yark (2760)

Entry type

Definition

Classification

msc 20A99

Related topic

SubdirectProductOfGroups

Related topic

DirectProduct2

Defines

direct product

Defines

unrestricted direct product

Defines

complete direct product

Defines

restricted direct product

Defines

direct sum

Defines

direct product of groups

Defines

unrestricted direct product of groups

Defines

restricted direct product of groups

Defines

direct sum of groups

Defines

Cartesian product of g

Generated on Fri Feb 9 19:19:35 2018 by

LaTeXML
direct products of homomorphisms

direct products of homomorphisms

Assume that

{

f

i

:

G

i

→

H

i

}

i

∈

I

is a family of

homomorphisms

between groups. Then we can define the

Cartesian product

(or

unrestricted direct product

) of this family as a homomorphism

∏

i

∈

I

f

i

:

∏

i

∈

I

G

i

→

∏

i

∈

I

H

i

such that

(

∏

i

∈

I

f

i

)

⁢

(

g

)

⁢

(

j

)

=

f

j

⁢

(

g

⁢

(

j

)

)

for each

g

∈

∏

i

∈

I

G

i

and

j

∈

I

.

One can easily show that

∏

i

∈

I

f

i

is a

group homomorphism

. Moreover it is clear that

(

∏

i

∈

I

f

i

)

⁢

(

⊕

i

∈

I

G

i

)

⊆

⊕

i

∈

I

H

i

,

so

∏

i

∈

I

f

i

induces a homomorphism

⊕

i

∈

I

f

i

:

⊕

i

∈

I

G

i

→

⊕

i

∈

I

H

i

,

which is a

restriction

of

∏

i

∈

I

f

i

to

⊕

i

∈

I

G

i

. This homomorphism is called the

direct product

(or

restricted direct product

) of

{

f

i

:

G

i

→

H

i

}

i

∈

I

.

Title

direct products of homomorphisms

Canonical name

DirectProductsOfHomomorphisms

Date of creation

2013-03-22 18:36:00

Last modified on

2013-03-22 18:36:00

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Definition

Classification

msc 20A99

Generated on Fri Feb 9 19:19:37 2018 by

LaTeXML
full families of Hopfian (co-Hopfian) groups

full families of Hopfian (co-Hopfian) groups

Proposition

.

Let

{

G

i

}

i

∈

I

be a full family of groups. Then each

G

i

is

Hopfian

(co-Hopfian) if and only if

⊕

i

∈

I

G

i

is Hopfian (co-Hopfian).

Proof.

,,

⇒

” Let

f

:

⊕

i

∈

I

G

i

→

⊕

i

∈

I

G

i

be a

surjective

(

injective

)

homomorphism

. Since

{

G

i

}

i

∈

I

is full, then there exists family of homomorphisms

{

f

i

:

G

i

→

G

i

}

i

∈

I

such that

f

=

⊕

i

∈

I

f

i

.

Of course since

f

is surjective (injective), then each

f

i

is surjective (injective). Thus each

f

i

is an

isomorphism

, because each

G

i

is Hopfian (co-Hopfian). Therefore

f

is an isomorphism, because

f

-

1

=

⊕

i

∈

I

f

i

-

1

.

□

,,

⇐

” Fix

j

∈

I

and assume that

f

j

:

G

j

→

G

j

is a surjective (injective) homomorphism. For

i

∈

I

such that

i

≠

j

define

f

i

:

G

i

→

G

i

to be any automorphism of

G

i

. Then

⊕

i

∈

I

f

i

:

⊕

i

∈

I

G

i

→

⊕

i

∈

I

G

i

is a surjective (injective)

group homomorphism

. Since

⊕

i

∈

I

G

i

is Hopfian (co-Hopfian) then

⊕

i

∈

I

f

i

is an isomorphism. Thus each

f

i

is an isomorphism. In particular

f

j

is an isomorphism, which

completes

the proof.

□

Example.

Let

𝒫

=

{

p

∈

ℕ

|

p

⁢

is prime

}

and

𝒫

0

be any subset of

𝒫

. Then

⊕

p

∈

P

0

ℤ

p

is both Hopfian and co-Hopfian.

Proof.

It is easy to see that

{

ℤ

p

}

p

∈

𝒫

is full, so

{

ℤ

p

}

p

∈

𝒫

0

is also full. Moreover for any

p

∈

P

0

the group

ℤ

p

is finite, so both Hopfian and co-Hopfian. Therefore (due to proposition)

⊕

p

∈

P

0

ℤ

p

is both Hopfian and co-Hopfian.

□

Title

full families of Hopfian (co-Hopfian) groups

Canonical

name

FullFamiliesOfHopfiancoHopfianGroups

Date of creation

2013-03-22 18:36:05

Last modified on

2013-03-22 18:36:05

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

7

Author

joking (16130)

Entry type

Theorem

Classification

msc 20A99

Generated on Fri Feb 9 19:19:43 2018 by

LaTeXML
proof that a finite abelian group has element with ERROR(\delimiter)⁢69640972⁢g⁢ERROR(\delimiter)⁢86418188=e⁢x⁢p(G)

proof that a finite abelian group has element with

\delimiter

⁢

69640972

⁢

g

⁢

\delimiter

⁢

86418188

=

e

⁢

x

⁢

p

(

G

)

Theorem 1

If

G

is a finite abelian group, then

G

has an element of order

exp

⁡

(

G

)

.

Proof.

Write

exp

⁡

(

G

)

=

∏

p

i

k

i

. Since

exp

⁡

(

G

)

is the least common multiple of the orders of each group element, it follows that for each

i

, there is an element whose order is a multiple of

p

i

k

i

, say

|

c

i

|

=

a

i

⁢

p

i

k

i

. Let

d

i

=

c

i

a

i

. Then

|

d

i

|

=

p

i

k

i

. The

d

i

thus have pairwise relatively prime orders, and thus

|

∏

d

i

|

=

∏

|

d

i

|

=

exp

⁡

(

G

)

so that

∏

d

i

is the desired element.

Title

proof that a finite abelian group has element with

\delimiter

⁢

69640972

⁢

g

⁢

\delimiter

⁢

86418188

=

e

⁢

x

⁢

p

(

G

)

Canonical name

ProofThatAFiniteAbelianGroupHasElementWithlvertGrvertexpG

Date of creation

2013-03-22 16:34:05

Last modified on

2013-03-22 16:34:05

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Proof

Classification

msc 20A99

Generated on Fri Feb 9 19:19:52 2018 by

LaTeXML
proof that G is cyclic if and only if ERROR(\delimiter)⁢69640972⁢G⁢ERROR(\delimiter)⁢86418188=e⁢x⁢p(G)

proof that

G

is cyclic if and only if

\delimiter

⁢

69640972

⁢

G

⁢

\delimiter

⁢

86418188

=

e

⁢

x

⁢

p

(

G

)

Theorem 1

A finite abelian group

G

is cyclic if and only if

|

G

|

=

exp

⁡

(

G

)

.

Proof.

G

is cyclic if and only if it has an element of order

|

G

|

. But

exp

⁡

(

G

)

is the maximum order of any element of

G

. Thus

G

is cyclic only if these two are equal.

Title

proof that

G

is cyclic if and only if

\delimiter

⁢

69640972

⁢

G

⁢

\delimiter

⁢

86418188

=

e

⁢

x

⁢

p

(

G

)

Canonical name

ProofThatGIsCyclicIfAndOnlyIflvertGrvertexpG

Date of creation

2013-03-22 16:34:14

Last modified on

2013-03-22 16:34:14

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Proof

Classification

msc 20A99

Generated on Fri Feb 9 19:19:54 2018 by

LaTeXML
proper subgroup

proper subgroup

A

proper subgroup

of a group

G

is a

subgroup

H

of

G

such that

H

≠

G

.

The only subgroup of

G

that is not a proper subgroup is

G

itself,
which may therefore occasionally be called an

improper subgroup

.

Title

proper subgroup

Canonical name

ProperSubgroup

Date of creation

2013-03-22 12:49:10

Last modified on

2013-03-22 12:49:10

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Definition

Classification

msc 20A99

Related topic

Subgroup

Related topic

Group

Defines

improper subgroup

Generated on Fri Feb 9 19:19:55 2018 by

LaTeXML
quaternion group

quaternion group

The

quaternion group

, or quaternionic group, is a

noncommutative
group

with eight elements. It is traditionally denoted by

Q

(not to be
confused with

ℚ

) or by

Q

8

. This group is defined by the

presentation

{

i

,

j

;

i

4

,

i

2

⁢

j

2

,

i

⁢

j

⁢

i

-

1

⁢

j

}

or, equivalently, defined by the multiplication table

⋅

1

i

j

k

-

i

-

j

-

k

-

1

1

1

i

j

k

-

i

-

j

-

k

-

1

i

i

-

1

k

-

j

1

-

k

j

-

i

j

j

-

k

-

1

i

k

1

-

i

-

j

k

k

j

-

i

-

1

-

j

i

1

-

k

-

i

-

i

1

-

k

j

-

1

k

-

j

i

-

j

-

j

k

1

-

i

-

k

-

1

i

j

-

k

-

k

-

j

i

1

j

-

i

-

1

k

-

1

-

1

-

i

-

j

-

k

i

j

k

1

where we have put each

product

x

⁢

y

into row

x

and column

y

.
The minus signs are justified by the fact that

{

1

,

-

1

}

is

subgroup

contained in the center of

Q

.
Every subgroup of

Q

is normal and, except for
the trivial subgroup

{

1

}

, contains

{

1

,

-

1

}

.
The

dihedral group

D

4

(the group of

symmetries

of a square) is the
only other noncommutative group of order 8.

Since

i

2

=

j

2

=

k

2

=

-

1

,
the elements

i

,

j

, and

k

are known as the

imaginary units

, by
analogy with

i

∈

ℂ

. Any pair of the imaginary units

generate

the group. Better, given

x

,

y

∈

{

i

,

j

,

k

}

, any element of

Q

is

expressible

in the form

x

m

⁢

y

n

.

Q

is identified with the

group of units

(

invertible elements

) of the
ring of

quaternions

over

ℤ

. That ring
is not identical to the

group ring

ℤ

⁢

[

Q

]

, which has dimension 8
(not 4) over

ℤ

. Likewise the usual

quaternion algebra

is not quite the same thing as the group algebra

ℝ

⁢

[

Q

]

.

Quaternions were known to Gauss in 1819 or 1820, but he did not
publicize this discovery, and quaternions weren’t rediscovered until
1843, with Hamilton.

Title

quaternion group

Canonical name

QuaternionGroup

Date of creation

2013-03-22 12:35:35

Last modified on

2013-03-22 12:35:35

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20A99

Synonym

quaternionic group

Related topic

Quaternions

Defines

quaternion group

Generated on Fri Feb 9 19:19:58 2018 by

LaTeXML