aarjaneiro/planetmath huggingface dataset:
group homomorphism

group homomorphism

Let

(

G

,

∗

)

and

(

K

,

⋆

)

be two groups. A

group homomorphism

is a

function

ϕ

:

G

→

K

such that

ϕ

⁢

(

s

∗

t

)

=

ϕ

⁢

(

s

)

⋆

ϕ

⁢

(

t

)

for all

s

,

t

∈

G

.

A

composition

of group homomorphisms is again a

homomorphism

.

Let

ϕ

:

G

→

K

a group homomorphism.
Then the kernel of

ϕ

is a

normal subgroup

of

G

,
and the image of

ϕ

is a

subgroup

of

K

.
Also,

ϕ

⁢

(

g

n

)

=

ϕ

⁢

(

g

)

n

for all

g

∈

G

and for all

n

∈

ℤ

.
In particular,
taking

n

=

-

1

we have

ϕ

⁢

(

g

-

1

)

=

ϕ

⁢

(

g

)

-

1

for all

g

∈

G

,
and taking

n

=

0

we have

ϕ

⁢

(

1

G

)

=

1

K

,
where

1

G

and

1

K

are the

identity elements

of

G

and

K

,
respectively.

Some special homomorphisms have special names.
If the homomorphism

ϕ

:

G

→

K

is

injective

,
we say that

ϕ

is a

monomorphism

,
and if

ϕ

is

surjective

we call it an

epimorphism

.
When

ϕ

is both injective and surjective (that is,

bijective

) we call it an

isomorphism

.
In the latter case we also say that

G

and

K

are

isomorphic

, meaning they are basically the same group (have the same

structure

).
A homomorphism from

G

on itself is called an

endomorphism

,
and if it is bijective then it is called an

automorphism

.

Title

group homomorphism

Canonical name

GroupHomomorphism

Date of creation

2013-05-17 17:53:21

Last modified on

2013-05-17 17:53:21

Owner

yark (2760)

Last modified by

unlord (1)

Numerical id

27

Author

yark (1)

Entry type

Definition

Classification

msc 20A05

Synonym

homomorphism

Synonym

homomorphism of groups

Related topic

Group

Related topic

Kernel

Related topic

Subgroup

Related topic

TypesOfHomomorphisms

Related topic

KernelOfAGroupHomomorphism

Related topic

GroupActionsAndHomomorphisms

Related topic

Endomorphism2

Related topic

GroupsOfRealNumbers

Related topic

HomomorphicImageOfGroup

Defines

epimorphism

Defines

monomorphism

Defines

automorphism

Defines

endomorphism

Defines

isomorphism

Defines

isomorphic

Defines

group epimorphism

Defines

group monomorphism

Defines

group automorphism

Defines

group endomorphism

Defines

group isomorphism

Defines

epimorphism of groups

Defines

monomorphism of groups

Defines

automorphism of a group

Defines

endom

Generated on Fri Feb 9 19:17:29 2018 by

LaTeXML
groups of small order

groups of small order

Below is a list of all possible groups per order up to

isomorphism

.

Groups of prime order:

•

All groups of prime order are isomorphic to a

cyclic group

of that order.

Groups of prime square order:

•

All groups of order

p

2

, where

p

is a prime, are isomorphic to one of the following:

–

C

p

2

(

Abelian

)

: cyclic group of order

p

2

.

–

C

p

×

C

p

(Abelian)

:

elementary abelian group

of order

p

2

.

Groups of order 1:

•

trivial group

(i.e.

{

e

}

)

.

Groups of order 6:

•

C

6

(Abelian)

: cyclic group of order 6.

•

S

3

(

non-Abelian

)

:

symmetric group

where

n

=

3

.

Groups of order 8:

•

C

8

(Abelian)

: cyclic group of order 8.

•

C

4

×

C

2

(Abelian)

:

direct product

of two groups of a cyclic group of order 4 and a cyclic group of order 2.

•

C

2

×

C

2

×

C

2

(Abelian)

: direct product of three groups of a cyclic group of order 2.

•

D

4

(non-Abelian)

:

octic group

;

dihedral group

of degree 4.

•

Q

8

(non-Abelian)

:

quaternion group

.

Groups of order 10:

•

C

10

(Abelian)

: cyclic group of order 10.

•

D

5

(non-Abelian)

: dihedral group of degree 5.

Groups of order 12:

•

C

12

(Abelian)

: cyclic group of order 12.

•

C

2

×

C

6

(Abelian)

.

•

A

4

(non-Abelian)

:

alternating group

of degree 4.

•

D

6

(non-Abelian)

: dihedral group of degree 6.

•

Dic

⁡

(

C

6

)

(non-Abelian)

: dicyclic group of order 12.
This is a

generalized quaternion group

Q

12

.

Groups of order 14:

•

C

14

(Abelian)

: cyclic group of order 14.

•

D

7

(non-Abelian)

: dihedral group of degree 7.

Groups of order 15:

•

C

15

(Abelian)

: cyclic group of order 15.

References

PJ

Pedersen, John:

Groups of small order

.

http://www.math.usf.edu/ eclark/algctlg/small_groups.html

http://www.math.usf.edu/ eclark/algctlg/small_groups.html

Title

groups of small order

Canonical name

GroupsOfSmallOrder

Date of creation

2013-03-22 14:47:54

Last modified on

2013-03-22 14:47:54

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

15

Author

Daume (40)

Entry type

Example

Classification

msc 20A05

Classification

msc 20-00

Related topic

ExamplesOfGroups

Generated on Fri Feb 9 19:17:31 2018 by

LaTeXML
groups with abelian inner automorphism group

groups with abelian inner automorphism group

The

inner automorphism group

Inn

⁡

(

G

)

is

isomorphic

to the

central quotient

of

G

,

G

/

Z

⁢

(

G

)

. if

Inn

⁡

(

G

)

is

abelian

, one cannot conclude that

G

itself is abelian. For example, let

G

=

𝒟

8

, the

dihedral group

of

symmetries

of the square.

G

=

⟨

r

,

s

∣

r

4

=

s

2

=

1

,

r

⁢

s

=

s

⁢

r

3

⟩

and

Z

⁢

(

G

)

=

{

1

,

r

2

}

. Representatives of the cosets of

Z

⁢

(

G

)

are

{

1

,

r

,

s

,

r

⁢

s

}

; these define a group of order

4

that is isomorphic to the Klein

4

-group (

http://planetmath.org/Klein4Group

)

V

4

. Thus the central quotient is abelian, but the group itself is not.

However, if the central quotient is

cyclic

, it does follow that

G

is abelian. For, choose a representative

x

in

G

of a

generator

for

G

/

Z

⁢

(

G

)

. Each element of

G

is thus of the form

x

a

⁢

z

for

z

∈

Z

⁢

(

G

)

. So given

g

,

h

∈

G

,

g

⁢

h

=

x

a

1

⁢

z

1

⁢

x

a

2

⁢

z

2

=

x

a

1

⁢

x

a

2

⁢

z

1

⁢

z

2

=

x

a

1

+

a

2

⁢

z

1

⁢

z

2

=

x

a

2

⁢

x

a

1

⁢

z

2

⁢

z

1

=

x

a

2

⁢

z

2

⁢

x

a

1

⁢

z

1

=

h

⁢

g

where the various manipulations are justified by the fact that the

z

i

∈

Z

⁢

(

G

)

and that powers of

x

commute with themselves.

Finally, note that if

Inn

⁡

(

G

)

is non-trivial, then

G

is

nonabelian

, for

Inn

⁡

(

G

)

nontrivial implies that for some

g

∈

G

,

conjugation

by

g

is not the

identity

, so there is some element of

G

with which

g

does not commute. So by the above

argument

,

Inn

⁡

(

G

)

, if non-trivial, cannot be cyclic (else

G

would be abelian).

Title

groups with abelian inner automorphism group

Canonical name

GroupsWithAbelianInnerAutomorphismGroup

Date of creation

2013-03-22 17:25:30

Last modified on

2013-03-22 17:25:30

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

7

Author

rm50 (10146)

Entry type

Topic

Classification

msc 20A05

Generated on Fri Feb 9 19:17:33 2018 by

LaTeXML
homogeneous group

homogeneous group

A

homogeneous group

is a set

G

together with a map

(

)

:

G

×

G

×

G

→

G

satisfying:

i)

(

a

,

a

,

b

)

=

b

ii)

(

a

,

b

,

b

)

=

a

iii)

(

(

a

,

b

,

c

)

,

d

,

e

)

=

(

a

,

b

,

(

c

,

d

,

e

)

)

for all

a

,

b

,

c

,

d

,

e

∈

G

.

A map

f

:

G

→

H

of homogeneous groups is a

homomorphism

if it

f

⁢

(

a

,

b

,

c

)

=

(

f

⁢

a

,

f

⁢

b

,

f

⁢

c

)

, for all

a

,

b

,

c

∈

G

.

A non-empty homogeneous group is essentially a group, as given any

x

∈

G

, we may define the following

product

on

G

:

a

⁢

b

=

(

a

,

x

,

b

)

.

This gives

G

the of a

group with

identity

x

. The choice of

x

does not affect the isomorphism class of the group obtained.

One may recover a homogeneous group from a group obtained this way, by setting

(

a

,

b

,

c

)

=

a

⁢

b

-

1

⁢

c

.

Also, every group may be obtained from a homogeneous group.

Homogeneous groups are

homogeneous

: Given

a

,

b

∈

G

we have a homomorphism

f

taking

a

to

b

, given by

f

⁢

x

=

(

x

,

a

,

b

)

.

In this way homogeneous groups differ from groups, as whilst often used to describe

symmetry

, groups themselves have a distinct

element

: the identity.

Also the

definition

of homogeneous groups is given purely in
of identities, and does not exclude the

empty set

.

Title

homogeneous group

Canonical

name

HomogeneousGroup

Date of creation

2013-03-22 16:12:12

Last modified on

2013-03-22 16:12:12

Owner

whm22 (2009)

Last modified by

whm22 (2009)

Numerical id

15

Author

whm22 (2009)

Entry type

Definition

Classification

msc 20A05

Defines

homomorphism of homogeneous groups

Generated on Fri Feb 9 19:17:34 2018 by

LaTeXML
homomorphic image of group

homomorphic image of group

Theorem

.

The

homomorphic image

of a group is a group.  More detailed, if

f

is a

homomorphism

from the group

(

G

,

∗

)

to the

groupoid

(

Γ

,

⋆

)

,  then the groupoid

(

f

⁢

(

G

)

,

⋆

)

also is a group.  Especially, the

isomorphic

image of a group is a group.

Proof.

Let

α

,

β

,

γ

be arbitrary elements of the image

f

⁢

(

G

)

and

a

,

b

,

c

some elements of

G

such that

f

⁢

(

a

)

=

α

,

f

⁢

(

b

)

=

β

,

f

⁢

(

c

)

=

γ

.  Then

α

⋆

β

=

f

⁢

(

a

)

⋆

f

⁢

(

b

)

=

f

⁢

(

a

∗

b

)

∈

f

⁢

(

G

)

,

whence

f

⁢

(

G

)

is

closed under

“

⋆

”, and we, in fact, can speak of a groupoid

(

f

⁢

(

G

)

,

⋆

)

.

Secondly, we can calculate

(

α

⋆

β

)

⋆

γ

=

(

f

⁢

(

a

)

⋆

f

⁢

(

b

)

)

⋆

f

⁢

(

c

)

=

f

⁢

(

a

∗

b

)

⋆

f

⁢

(

c

)

=

f

⁢

(

(

a

∗

b

)

∗

c

)

=

f

⁢

(

a

∗

(

b

∗

c

)

)

=

f

⁢

(

a

)

⋆

f

⁢

(

b

∗

c

)

=

f

⁢

(

a

)

⋆

(

f

⁢

(

b

)

⋆

f

⁢

(

c

)

)

=

α

⋆

(

β

⋆

γ

)

,

whence the

associativity

is in in the groupoid

(

f

⁢

(

G

)

,

⋆

)

.

Let

e

be the

identity element

of

(

G

,

∗

)

and

f

⁢

(

e

)

=

ε

.  Then

ε

⋆

α

=

f

⁢

(

e

)

⋆

f

⁢

(

a

)

=

f

⁢

(

e

∗

a

)

=

f

⁢

(

a

)

=

α

,

α

⋆

ε

=

f

⁢

(

a

)

⋆

f

⁢

(

e

)

=

f

⁢

(

a

∗

e

)

=

f

⁢

(

a

)

=

α

,

and therefore

ε

is an identity element in

f

⁢

(

G

)

.

If

f

⁢

(

a

-

1

)

=

α

′

, then

α

⋆

α

′

=

f

⁢

(

a

)

⋆

f

⁢

(

a

-

1

)

=

f

⁢

(

a

∗

a

-

1

)

=

f

⁢

(

e

)

=

ε

,

α

′

⋆

α

=

f

⁢

(

a

-

1

)

⋆

f

⁢

(

a

)

=

f

⁢

(

a

-

1

∗

a

)

=

f

⁢

(

e

)

=

ε

.

Thus any

element

α

of

f

⁢

(

G

)

has in

f

⁢

(

G

)

an

inverse

.

Accordingly,

(

f

⁢

(

G

)

,

⋆

)

is a group.

Remark 1.

If

(

G

,

∗

)

is

Abelian

, the same is true for

(

f

⁢

(

G

)

,

⋆

)

.

Remark 2.

Analogically, one may prove that the homomorphic image of a ring is a ring.

Example.

If we define the

mapping

f

from the group

(

ℤ

,

+

)

to the groupoid

(

ℤ

9

,

⋅

)

by

f

⁢

(

n

)

:=

⟨

4

⟩

n

,

then

f

is homomorphism:

f

⁢

(

m

+

n

)

=

⟨

4

⟩

m

+

n

=

⟨

4

⟩

m

⁢

⟨

4

⟩

n

=

f

⁢

(

m

)

⁢

f

⁢

(

n

)

.

The image

f

⁢

(

ℤ

)

consists of powers of the

residue class

(

http://planetmath.org/

Congruences

)

⟨

4

⟩

, which are

⟨

4

⟩

,

⟨

16

⟩

=

⟨

7

⟩

,

⟨

64

⟩

=

⟨

1

⟩

.

These apparently form the

cyclic group

of order 3.

Title

homomorphic image of group

Canonical name

HomomorphicImageOfGroup

Date of creation

2013-03-22 18:56:27

Last modified on

2013-03-22 18:56:27

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

14

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 08A05

Related topic

GroupHomomorphism

Related topic

CorrespondenceBetweenNormalSubgroupsAndHomomorphicImages

Generated on Fri Feb 9 19:17:40 2018 by

LaTeXML
ideal of elements with finite order

ideal of elements with finite order

Theorem.

The set of all elements of a ring, which have a finite order in the

additive group

of the ring, is a (two-sided) ideal of the ring.

Proof.

Let

S

be the set of the elements with finite order in the ring

R

.  Denote by

o

⁢

(

x

)

the order of

x

.  Take arbitrary elements

a

,

b

of the set

S

.

If

lcm

⁡

(

o

⁢

(

a

)

,

o

⁢

(

b

)

)

=

n

=

k

⁢

o

⁢

(

a

)

=

l

⁢

o

⁢

(

b

)

,  then

n

⁢

(

a

-

b

)

=

n

⁢

a

-

n

⁢

b

=

k

⁢

o

⁢

(

a

)

⁢

a

-

l

⁢

o

⁢

(

b

)

⁢

b

=

k

⋅

0

-

l

⋅

0

=

0

-

0

=

0

.

Thus

o

⁢

(

a

-

b

)

≦

n

<

∞

and so

a

-

b

∈

S

.

For any element

r

of

R

we have

o

⁢

(

a

)

⁢

(

r

⁢

a

)

=

r

⁢

a

+

r

⁢

a

+

…

+

r

⁢

a

⏟

o

⁢

(

a

)

=

r

⁢

(

a

+

a

+

…

+

a

⏟

o

⁢

(

a

)

)

=

r

⁢

(

o

⁢

(

a

)

⁢

a

)

=

r

⋅

0

=

0

.

Therefore,

o

⁢

(

r

⁢

a

)

≦

o

⁢

(

a

)

<

∞

and

r

⁢

a

∈

S

.  Similarly,

a

⁢

r

∈

S

.

Since

S

satisfies the conditions for an ideal, the theorem has been proven.

Title

ideal of elements with finite order

Canonical name

IdealOfElementsWithFiniteOrder

Date of creation

2013-03-22 17:52:30

Last modified on

2013-03-22 17:52:30

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 16D25

Related topic

OrderGroup

Related topic

Lcm

Related topic

Multiple

Related topic

OrdersOfElementsInIntegralDomain

Related topic

CharacteristicOfFiniteRing

Generated on Fri Feb 9 19:17:43 2018 by

LaTeXML
identity element

identity element

Let

G

be a

groupoid

, that is a set with a

binary operation

G

×

G

→

G

, written muliplicatively so that

(

x

,

y

)

↦

x

⁢

y

.

An

identity element

for

G

is an element

e

such that

g

⁢

e

=

e

⁢

g

=

g

for all

g

∈

G

.

The symbol

e

is most commonly used for identity elements. Another common symbol for an identity element is

1

, particularly in

semigroup

theory (and ring theory, considering the multiplicative structure as a semigroup).

Groups, monoids, and loops are classes of groupoids that, by definition, always have an identity element.

Title

identity element

Canonical name

IdentityElement

Date of creation

2013-03-22 12:49:07

Last modified on

2013-03-22 12:49:07

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

9

Author

mclase (549)

Entry type

Definition

Classification

msc 20A05

Classification

msc 20N02

Classification

msc 20N05

Classification

msc 20M99

Synonym

neutral element

Related topic

LeftIdentityAndRightIdentity

Related topic

Group

Generated on Fri Feb 9 19:17:44 2018 by

LaTeXML
injection can be extended to isomorphism

injection can be extended to isomorphism

Theorem

.

If

f

is an

injection

from a set

S

into a group

G

, then there exist a group

H

containing

S

and a

group isomorphism

φ

:

H

→

G

such that

φ

|

S

=

f

.

Proof.

Let

M

be a set such that

card

⁡

(

M

)

≧

card

⁡

(

G

)

.  Because

card

⁡

(

f

⁢

(

S

)

)

=

card

⁡

(

S

)

, we have

card

⁡

(

M

∖

S

)

≧

card

⁡

(

G

∖

f

⁢

(

S

)

)

,  and therefore there exists an injection

ψ

:

G

∖

f

⁢

(

S

)

→

M

∖

S

(provided that

G

∖

f

⁢

(

S

)

≠

∅

;  otherwise the

mapping

f

:

S

→

G

would be a

bijection

).  Define

H

:=

S

∪

ψ

⁢

(

G

∖

f

⁢

(

S

)

)

,

φ

⁢

(

h

)

:=

{

f

⁢

(

h

)

for

⁢

h

∈

S

,

ψ

-

1

⁢

(

h

)

for

⁢

h

∈

H

∖

S

.

Then apparently,

φ

:

H

→

G

is a bijection and

φ

|

S

=

f

.  Moreover, define the

binary operation

“

*

” of the set

H

by

h

1

∗

h

2

:=

φ

-

1

⁢

(

φ

⁢

(

h

1

)

⋅

φ

⁢

(

h

2

)

)

.

(1)

We see first that

(

h

1

∗

h

2

)

∗

h

3

=

φ

-

1

⁢

(

φ

⁢

(

φ

-

1

⁢

(

φ

⁢

(

h

1

)

⋅

φ

⁢

(

h

2

)

)

)

⋅

φ

⁢

(

h

3

)

)

=

φ

-

1

⁢

(

(

φ

⁢

(

h

1

)

⋅

φ

⁢

(

h

2

)

)

⋅

φ

⁢

(

h

3

)

)

=

φ

-

1

⁢

(

φ

⁢

(

h

1

)

⋅

(

φ

⁢

(

h

2

)

⋅

φ

⁢

(

h

3

)

)

)

=

φ

-

1

⁢

(

φ

⁢

(

h

1

)

⋅

φ

⁢

(

φ

-

1

⁢

(

φ

⁢

(

h

2

)

⋅

φ

⁢

(

h

3

)

)

)

)

=

h

1

∗

(

h

2

∗

h

3

)

.

Secondly,

h

∗

φ

-

1

⁢

(

e

)

=

φ

-

1

⁢

(

φ

⁢

(

h

)

⋅

φ

⁢

(

φ

-

1

⁢

(

e

)

)

)

=

φ

-

1

⁢

(

φ

⁢

(

h

)

)

=

h

,

whence

φ

-

1

⁢

(

e

)

is the

right identity

element of

H

.  Then,

h

∗

φ

-

1

⁢

(

(

φ

⁢

(

h

)

)

-

1

)

=

φ

-

1

⁢

(

φ

⁢

(

h

)

⋅

φ

⁢

(

φ

-

1

⁢

(

φ

⁢

(

h

)

-

1

)

)

)

=

φ

-

1

⁢

(

e

)

,

and accordingly

φ

-

1

⁢

(

(

φ

⁢

(

h

)

)

-

1

)

is the

right inverse

of

h

in

H

.  Consequently,

(

H

,

∗

)

is a group.  The

equation

(1) implies that

φ

⁢

(

h

1

∗

h

2

)

=

φ

⁢

(

h

1

)

⋅

φ

⁢

(

h

2

)

,

whence

φ

is an

isomorphism

from

H

onto

G

.  Q.E.D.

Title

injection can be extended to isomorphism

Canonical name

InjectionCanBeExtendedToIsomorphism

Date of creation

2013-03-22 18:56:50

Last modified on

2013-03-22 18:56:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

10

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 03E20

Related topic

Restriction

Related topic

Cardinality

Generated on Fri Feb 9 19:17:47 2018 by

LaTeXML
inner automorphism

inner automorphism

Let

G

be a group. For every

x

∈

G

, we define a

mapping

ϕ

x

:

G

→

G

,

y

↦

x

⁢

y

⁢

x

-

1

,

y

∈

G

,

called

conjugation

by

x

.
It is easy to show the conjugation map is in fact, a

group automorphism

.

An

automorphism

of

G

that corresponds to conjugation by some

x

∈

G

is called

inner

. An automorphism that isn’t inner is called
an

outer

automorphism.

The composition operation gives the set of all automorphisms of

G

the

structure

of a group,

Aut

⁡

(

G

)

. The

inner
automorphisms

also form a group,

Inn

⁡

(

G

)

, which is a

normal subgroup

of

Aut

⁡

(

G

)

. Indeed, if

ϕ

x

,

x

∈

G

is an inner automorphism and

π

:

G

→

G

an arbitrary
automorphism, then

π

∘

ϕ

x

∘

π

-

1

=

ϕ

π

⁢

(

x

)

.

Let us also note that the mapping

x

↦

ϕ

x

,

x

∈

G

is a

surjective

group homomorphism

with kernel

Z

⁡

(

G

)

, the centre

subgroup

. Consequently,

Inn

⁡

(

G

)

is naturally

isomorphic

to the

quotient

of

G

/

Z

⁡

(

G

)

.

Note: the above definitions and assertions hold, mutatis mutandi, if we define
the conjugation action of

x

∈

G

on

B

to be the

right action

y

↦

x

-

1

⁢

y

⁢

x

,

y

∈

G

,

rather than the left action given above.

Title

inner automorphism

Canonical name

InnerAutomorphism

Date of creation

2013-03-22 12:49:53

Last modified on

2013-03-22 12:49:53

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

12

Author

rmilson (146)

Entry type

Definition

Classification

msc 20A05

Synonym

inner

Defines

conjugation

Defines

outer

Defines

outer automorphism

Defines

automorphism group

Generated on Fri Feb 9 19:17:49 2018 by

LaTeXML
inverse of a product

inverse of a product

Theorem

.

If

a

and

b

are arbitrary elements of the group

(

G

,

*

)

, then the

inverse

of

a

*

b

is

(

a

*

b

)

-

1

=

b

-

1

*

a

-

1

.

(1)

Proof.

Let the

neutral element

of the group, which may be proved unique, be

e

.  Using only the group

postulates

we obtain

(

a

*

b

)

*

(

b

-

1

*

a

-

1

)

=

a

*

(

b

*

(

b

-

1

*

a

-

1

)

)

=

a

*

(

(

b

*

b

-

1

)

*

a

-

1

)

=

a

*

(

e

*

a

-

1

)

=

a

*

a

-

1

=

e

,

(

b

-

1

*

a

-

1

)

*

(

a

*

b

)

=

b

-

1

*

(

a

-

1

*

(

a

*

b

)

)

=

b

-

1

*

(

(

a

-

1

*

a

)

*

b

)

=

b

-

1

*

(

e

*

b

)

=

b

-

1

*

b

=

e

,

Q.E.D.

Note.

The (1) may be by

induction

extended to the form

(

a

1

*

⋯

*

a

n

)

-

1

=

a

n

-

1

*

⋯

*

a

1

-

1

.

Title

inverse of a

product

Canonical name

InverseOfAProduct

Date of creation

2015-01-30 21:19:19

Last modified on

2015-01-30 21:19:19

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

17

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 20-00

Synonym

inverse of a product in group

Synonym

inverse of product

Related topic

InverseOfCompositionOfFunctions

Related topic

GeneralAssociativity

Related topic

Division

Related topic

InverseNumber

Related topic

OrderOfProducts

Generated on Fri Feb 9 19:17:51 2018 by

LaTeXML
isomorphic groups

isomorphic groups

Two groups

(

X

1

,

*

1

)

and

(

X

2

,

*

2

)

are said to be

isomorphic

if there is a group isomorphism

ψ

:

X

1

→

X

2

.

Next we name a few necessary conditions for two groups

X

1

,

X

2

to be isomorphic (with

isomorphism

ψ

as above).

1.

If two groups are isomorphic, then they have the same

cardinality

. Indeed, an isomorphism is in particular a

bijection

of sets.

2.

If the group

X

1

has an

element

g

of order

n

, then the group

X

2

must have an element of the same order

. If there is an isomorphism

ψ

then

ψ

⁢

(

g

)

∈

X

2

and

(

ψ

⁢

(

g

)

)

n

=

ψ

⁢

(

g

n

)

=

ψ

⁢

(

e

1

)

=

e

2

where

e

i

is the

identity elements

of

X

i

. Moreover, if

(

ψ

⁢

(

g

)

)

m

=

e

2

then

ψ

⁢

(

g

m

)

=

e

2

and by the injectivity of

ψ

we must have

g

m

=

e

1

so

n

divides

m

. Therefore the order of

ψ

⁢

(

g

)

is

n

.

3.

If one group is cyclic, the other one must be cyclic too

. Suppose

X

1

is cyclic generated by an element

g

. Then it is easy to see that

X

2

is generated by the element

ψ

⁢

(

g

)

. Also

if

X

1

is

finitely generated

, then

X

2

is finitely generated as well

.

4.

If one group is

abelian

, the other one must be abelian as well

. Indeed, suppose

X

2

is abelian. Then

ψ

⁢

(

g

*

1

h

)

=

ψ

⁢

(

g

)

*

2

ψ

⁢

(

h

)

=

ψ

⁢

(

h

)

*

2

ψ

⁢

(

g

)

=

ψ

⁢

(

h

*

1

g

)

and using the injectivity of

ψ

we conclude

g

*

1

h

=

h

*

1

g

.

Note.

Isomorphic groups

are sometimes said to be

abstractly identical

, because their “abstract” are completely

similar

— one may think that their elements are the same but have only different names.

Title

isomorphic groups

Canonical name

IsomorphicGroups

Date of creation

2013-03-22 14:01:58

Last modified on

2013-03-22 14:01:58

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

10

Author

alozano (2414)

Entry type

Definition

Classification

msc 20A05

Defines

isomorphic

Defines

abstractly identical

Generated on Fri Feb 9 19:17:53 2018 by

LaTeXML
kernel

kernel

Let

ρ

:

G

→

K

be a

group homomorphism

. The

preimage

of the

codomain

identity element

e

K

∈

K

forms a

subgroup

of the domain

G

, called the

kernel

of the

homomorphism

;

ker

⁡

(

ρ

)

=

{

s

∈

G

∣

ρ

⁢

(

s

)

=

e

K

}

The kernel is a

normal subgroup

. It is the trivial subgroup if and
only if

ρ

is a

monomorphism

.

Title

kernel

Canonical name

Kernel

Date of creation

2013-03-22 11:58:24

Last modified on

2013-03-22 11:58:24

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

14

Author

rmilson (146)

Entry type

Definition

Classification

msc 20A05

Synonym

kernel of a group homomorphism

Related topic

GroupHomomorphism

Related topic

Kernel

Related topic

AHomomorphismIsInjectiveIffTheKernelIsTrivial

Generated on Fri Feb 9 19:17:55 2018 by

LaTeXML