aarjaneiro/planetmath huggingface dataset:
BN-pair

BN-pair

Let

G

be a group. Then

G

has a

B

⁢

N

-pair or a

Tits system

if the following conditions hold:

1.

B

and

N

are

subgroups

of

G

such that

G

=

<

B

,

N

>

.

2.

B

∩

N

=

T

◁

N

and

N

/

T

=

W

is a group generated by a set

S

.

3.

s

⁢

B

⁢

w

⊆

B

⁢

w

⁢

B

∪

B

⁢

s

⁢

w

⁢

B

for all

s

∈

S

and

w

∈

W

.

4.

s

⁢

B

⁢

s

-

1

⊈

B

for all

s

∈

S

.

Where

B

⁢

w

⁢

B

is a

double coset

with respect to

B

. It can be proven that

S

is in fact made up of elements of order 2, and that

W

is a

Coxeter group

.

Example:

Let

G

=

G

⁢

L

n

⁢

(

𝕂

)

where

𝕂

is some field. Then, if we let

B

be the subgroup of

upper triangular matrices

and

N

be the subgroup of

monomial matrices

(i.e. matrices having one nonzero entry in each row and each column, or more precisely the

stabilizer

of the lines

{

[

e

1

]

,

…

,

[

e

n

]

}

). Then, it can be shown that

B

and

N

generate

G

and that

T

is the subgroup of

diagonal matrices

. In turn, it follows that

W

in this case is

isomorphic

to the

symmetric group

on

n

letters,

S

n

.

For more, consult chapter 5 in the book Buildings, by Kenneth Brown

Title

BN-pair

Canonical name

BNpair

Date of creation

2013-03-22 15:30:11

Last modified on

2013-03-22 15:30:11

Owner

tedgar (10630)

Last modified by

tedgar (10630)

Numerical id

11

Author

tedgar (10630)

Entry type

Definition

Classification

msc 20F55

Synonym

Tits Systems

Generated on Fri Feb 9 19:29:55 2018 by

LaTeXML
dihedral group

dihedral group

The

n

th

dihedral group

is the

symmetry group

of
the

regular

n

-sided polygon. The group consists of

n

reflections

,

n

-

1

rotations

, and the

identity transformation

. In this entry we will denote the group in question by

𝒟

n

.
An alternate notation is

𝒟

2

⁢

n

; in this approach, the subscript indicates the order of the group.

Letting

ω

=

exp

⁡

(

2

⁢

π

⁢

i

/

n

)

denote a

primitive

n

th

root of
unity, and assuming the polygon is centered at the origin, the
rotations

R

k

,

k

=

0

,

…

,

n

-

1

(Note:

R

0

denotes the

identity

)
are given by

R

k

:

z

↦

ω

k

⁢

z

,

z

∈

ℂ

,

and the reflections

M

k

,

k

=

0

,

…

,

n

-

1

by

M

k

:

z

↦

ω

k

⁢

z

¯

,

z

∈

ℂ

The

abstract group

structure is given by

R

k

⁢

R

l

=

R

k

+

l

,

R

k

⁢

M

l

=

M

k

+

l

M

k

⁢

M

l

=

R

k

-

l

,

M

k

⁢

R

l

=

M

k

-

l

,

where the addition and

subtraction

is carried out modulo

n

.

The group can also be described in terms of

generators and relations

as

(

M

0

)

2

=

(

M

1

)

2

=

(

M

1

⁢

M

0

)

n

=

id

.

This means that

𝒟

n

is a rank-1

Coxeter group

.

Since the group acts by

linear transformations

(

x

,

y

)

→

(

x

^

,

y

^

)

,

(

x

,

y

)

∈

ℝ

2

there is a
corresponding action on

polynomials

p

→

p

^

, defined by

p

^

⁢

(

x

^

,

y

^

)

=

p

⁢

(

x

,

y

)

,

p

∈

ℝ

⁢

[

x

,

y

]

.

The polynomials
left

invariant

by all the group

transformations

form an

algebra

. This
algebra is

freely generated

by the following two basic invariants:

x

2

+

y

2

,

x

n

-

(

n

2

)

⁢

x

n

-

2

⁢

y

2

+

⋯

,

the latter
polynomial being the

real part

of

(

x

+

i

⁢

y

)

n

. It is easy to check
that these two polynomials are invariant. The first polynomial
describes the

distance

of a point from the origin, and this is
unaltered by

Euclidean

reflections through the origin. The second
polynomial is unaltered by a rotation through

2

⁢

π

/

n

radians, and is
also invariant with respect to

complex conjugation

. These two
transformations generate the

n

th

dihedral group. Showing that
these two invariants polynomially generate the full algebra of
invariants is somewhat trickier, and is best done as an application of
Chevalley’s theorem regarding the invariants of a finite reflection
group.

Title

dihedral group

Canonical name

DihedralGroup

Date of creation

2013-03-22 12:22:53

Last modified on

2013-03-22 12:22:53

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

15

Author

rmilson (146)

Entry type

Definition

Classification

msc 20F55

Related topic

Symmetry2

Generated on Fri Feb 9 19:30:02 2018 by

LaTeXML
octic group

octic group

The

octic group

also known as the

4

t

⁢

h

dihedral group

, is a

non-Abelian group

with eight elements. It is traditionally denoted by

D

4

. This group is defined by the

presentation

<

s

,

t

∣

s

4

=

t

2

=

e

,

s

⁢

t

=

t

⁢

s

-

1

>

or, equivalently, defined by the multiplication table

⋅

e

s

s

2

s

3

t

t

⁢

s

t

⁢

s

2

t

⁢

s

3

e

e

s

s

2

s

3

t

t

⁢

s

t

⁢

s

2

t

⁢

s

3

s

s

s

2

s

3

e

t

⁢

s

3

t

t

⁢

s

t

⁢

s

2

s

2

s

2

s

3

e

s

t

⁢

s

2

t

⁢

s

3

t

t

⁢

s

s

3

s

3

e

s

s

2

t

⁢

s

t

⁢

s

2

t

⁢

s

3

t

t

t

t

⁢

s

t

⁢

s

2

t

⁢

s

3

e

s

s

2

s

3

t

⁢

s

t

⁢

s

t

⁢

s

2

t

⁢

s

3

t

s

3

e

s

s

2

t

⁢

s

2

t

⁢

s

2

t

⁢

s

3

t

t

⁢

s

s

2

s

3

e

s

t

⁢

s

3

t

⁢

s

3

t

t

⁢

s

t

⁢

s

2

s

s

2

s

3

e

where we have put each

product

x

⁢

y

into row

x

and column

y

. The lattice of the

subgroups

is given below:

where

<

a

1

,

…

,

a

n

>

denotes the subgroup generated by

a

1

,

…

,

a

n

and

{

b

1

,

…

,

b

n

}

denotes the subgroup. Of those subgroups, the following are its proper

normal subgroup

:

{

e

,

s

2

,

t

,

t

⁢

s

2

}

,

<

s

>

,

{

e

,

s

2

,

s

⁢

t

,

t

⁢

s

}

, and

<

s

2

>

. In addition the center and

commutator subgroup

of the octic group is

<

s

2

>

. It can also be shown that the

automorphism

of the octic group

(

Aut

⁡

(

D

4

)

)

is

isomorphic

to itself

(

D

4

)

.

[

PJ

]

An additional property is that the subgroup of the

general linear group

of

dimension

2 over the

real numbers

generated by:
[

0

1

-1

0

],[

0

1

1

0

]
is isomorphic to the octic group.

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

octic group

Canonical name

OcticGroup

Date of creation

2013-03-22 14:47:59

Last modified on

2013-03-22 14:47:59

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

5

Author

Daume (40)

Entry type

Example

Classification

msc 20F55

Synonym

D

4

Generated on Fri Feb 9 19:30:17 2018 by

LaTeXML
corollaries of basic theorem on ordered groups

corollaries of basic theorem on ordered groups

Corollary 1

Let

G

be an

ordered group

. For all

x

∈

G

, either

x

≤

1

≤

x

-

1

or

x

-

1

≤

1

≤

x

.

Proof:

By

conclusion

1, either

x

<

1

or

x

=

1

or

1

<

x

. If

x

<

1

, then, by conclusion 5,

1

-

1

<

x

-

1

, so

x

<

1

<

x

-

1

. If

x

=

1

, the conclusion is

trivial

. If

1

<

x

, then, by conclusion 5,

x

-

1

<

1

-

1

, so

x

-

1

<

1

<

x

.

Q.E.D.

Corollary 2

Let

G

be an ordered group and

n

a strictly positive integer. Then, for all

x

,

y

∈

G

, we have

x

<

y

if and only if

x

n

<

y

n

.

Proof:

We shall first prove that

x

<

y

implies

x

n

<

y

n

by

induction

. If

n

=

1

, this is a simple

tautology

. Assume the conclusion is true for a certain value of

n

. Then, conclusion 4 allows us to multiply the

inequalities

x

<

y

and

x

n

<

y

n

to obtain

x

n

+

1

<

y

n

+

1

.

As for the proof that

x

n

<

y

n

implies

x

<

y

, we shall prove the

contrapositive

statement. Assume that

x

<

y

is false. By conclusion 1, it follows that either

x

=

y

or

x

>

y

. If

x

=

y

, then

x

n

=

y

n

so, by conclusion 1

x

n

<

y

n

is false. If

x

>

y

then, by what we have already shown,

x

n

>

y

n

so

x

n

<

y

n

is also false in this case for the same reason.

Q.E.D.

Corollary 3

Let

G

be an ordered group and

n

a strictly positive integer. Then, for all

x

,

y

∈

G

, we have

x

=

y

if and only if

x

n

=

y

n

.

Proof:

It is trivial that, if

x

=

y

, then

x

n

=

y

n

. Assume that

x

n

=

y

n

. By conclusion 1 of the main

theorem

, it is the case that either

x

<

y

or

x

=

y

or

y

<

x

. If

x

<

y

then, by the preceding corollary,

x

n

<

y

n

, which is not possible. Likewise, if

y

<

x

, then we would have

y

n

<

x

n

, which is also impossible. The only remaining possibility is

x

=

y

.

Q.E.D.

Corollary 4

An ordered group cannot contain any elements of
finite order.

Let

x

be an element of an ordered group distinct from the

identity

.
By definition, if

x

were of finite order, there would exist an
integer such that

x

n

=

1

. Since

1

=

1

n

, we would have

x

n

=

1

n

but, by Corollary 3, this would imply

x

=

1

, which contradicts
our

hypothesis

.

Q.E.D.

It is worth noting that, in the

context

of

additive groups

of rings,
this result states that

ordered rings

have characteristic zero.

Title

corollaries of basic theorem on ordered groups

Canonical name

CorollariesOfBasicTheoremOnOrderedGroups

Date of creation

2013-03-22 14:55:12

Last modified on

2013-03-22 14:55:12

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Corollary

Classification

msc 20F60

Classification

msc 06A05

Generated on Fri Feb 9 19:30:20 2018 by

LaTeXML
isolated subgroup

isolated subgroup

Let

G

be a

ordered group

and

F

its

subgroup

.  We call this subgroup if every element

f

of

F

and every element

g

of

G

satisfy

f

≦

g

≦

1

⇒

g

∈

F

.

If an ordered group

G

has only a finite number of

isolated subgroups

, then the number of proper (

≠

G

) isolated subgroups of

G

is the of

G

.

Theorem

.

Let

G

be an

abelian

ordered group with order (

http://planetmath.org/OrderGroup

) at least 2.  The of

G

equals one iff there is an

order-preserving

isomorphism

from

G

onto some subgroup of the

multiplicative group

of

real numbers

.

References

1

M. Larsen & P. McCarthy

:

Multiplicative theory of ideals

.  Academic Press. New York (1971).

Title

isolated subgroup

Canonical name

IsolatedSubgroup

Date of creation

2013-03-22 14:55:08

Last modified on

2013-03-22 14:55:08

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 20F60

Classification

msc 06A05

Related topic

RankOfValuation

Related topic

KrullValuation

Defines

rank of ordered group

Generated on Fri Feb 9 19:30:22 2018 by

LaTeXML
proof of basic theorem about ordered groups

proof of basic theorem about ordered groups

Property 1:

Consider

a

⁢

b

-

1

∈

G

. Since

G

can be written as a

pairwise disjoint

union, exactly one of the following conditions must hold:

a

⁢

b

-

1

∈

S

a

⁢

b

-

1

=

1

a

⁢

b

-

1

∈

S

-

1

By

definition

of the

ordering relation

,

a

<

b

if the first condition holds. If the second condition holds, then

a

=

b

. If the third condition holds, then we must have

a

⁢

b

-

1

=

s

-

1

for some

s

∈

S

. Taking

inverses

, this means that

b

⁢

a

-

1

=

s

, so

b

<

a

, or equivalently

a

>

b

. Hence, one of the following three conditions must hold:

a

<

b

a

=

b

b

<

a

Property 2:

The hypotheses can be rewritten as

a

⁢

b

-

1

∈

S

b

⁢

c

-

1

∈

S

Multiplying, and remembering that

S

is

closed under

multiplication

,

a

⁢

c

-

1

=

(

a

⁢

b

-

1

)

⁢

(

b

⁢

c

-

1

)

∈

S

.

In other words,

a

<

c

.

Property 3:

Suppose that

a

<

b

, so

a

⁢

b

-

1

=

s

∈

S

. Then

s

=

a

⁢

b

-

1

=

a

⁢

1

⁢

b

-

1

=

a

⁢

c

⁢

c

-

1

⁢

b

-

1

=

(

a

⁢

c

)

⁢

(

b

⁢

c

)

-

1

so

a

⁢

c

<

b

⁢

c

.

By the

defining property

of

S

, we have

c

⁢

s

⁢

c

-

1

∈

S

. Also,

c

⁢

s

⁢

c

-

1

=

c

⁢

a

⁢

b

-

1

⁢

c

-

1

=

(

c

⁢

a

)

⁢

(

c

⁢

b

)

-

1

,

hence

(

c

⁢

a

)

⁢

(

c

⁢

b

)

-

1

∈

S

, so

c

⁢

a

<

c

⁢

b

Property 4:

By

property

3,

a

<

b

implies

a

⁢

c

<

b

⁢

c

and likewise

c

<

d

implies

b

⁢

c

<

b

⁢

d

. Then, by property 2, we conclude

a

⁢

c

<

b

⁢

d

.

Property 5:

By the

hypothesis

,

a

⁢

b

-

1

=

s

∈

S

. By the defining property,

b

-

1

⁢

s

⁢

b

∈

S

. Since

b

-

1

⁢

s

⁢

b

=

b

-

1

⁢

a

, we have

b

-

1

⁢

a

∈

S

. In other words,

b

-

1

<

a

-

1

.

Property 6:

By definition,

a

<

1

means that

a

⁢

1

-

1

∈

S

. Since

1

-

1

=

1

and

a

⁢

1

=

a

, this is

equivalent

to stating that

a

∈

S

.

Title

proof of basic theorem about ordered groups

Canonical

name

ProofOfBasicTheoremAboutOrderedGroups

Date of creation

2013-03-22 14:54:46

Last modified on

2013-03-22 14:54:46

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

14

Author

rspuzio (6075)

Entry type

Proof

Classification

msc 20F60

Classification

msc 06A05

Generated on Fri Feb 9 19:30:25 2018 by

LaTeXML
graph product of groups

graph product of groups

Let

Γ

be a finite

undirected graph

and let

{

G

v

:

v

∈

V

⁢

(

Γ

)

}

be a

collection

of groups associated with the vertices of

Γ

. Then the

graph product

of the groups

G

v

is the group

G

=

F

/

R

, where

F

is the

free product

of the

G

v

and

R

is generated by the

relations

that elements of

G

u

commute with elements of

G

v

whenever

u

and

v

are

adjacent

in

Γ

.

The free product and the

direct product

are the extreme examples of the graph product. To obtain the free product, let

Γ

be an

anticlique

, and to obtain the direct product, let

Γ

be a clique.

References

1

E.R. Green,

Graph products of groups

, Doctoral thesis, The University of Leeds, 1990.

2

S. Hermiller and J. Meier,

Algorithms and

geometry

for graph products of groups

, Journal of Algebra 117 (1995), 230–257.

3

M. Lohrey and G. Sénizergues,

When is a graph product of groups virtually-free?

, to appear in Communications in

Algebra

. 2006 preprint available online at

http://inf.informatik.uni-stuttgart.de/fmi/ti/personen/Lohrey/05-Graphprod.pdf

.

4

R.Brown, M. Bullejos, and T. Porter,‘Crossed complexes, free
crossed

resolutions

and graph products of groups’, Proceedings
Workshop Korea 2000, J. Mennicke, Moo Ha Woo (eds.) Recent
Advances in

Group Theory

, Heldermann Verlag Research and
Exposition in Mathematics 27 (2002) 8–23. arXiv:math/0101220

Title

graph product of groups

Canonical

name

GraphProductOfGroups

Date of creation

2013-03-22 16:10:36

Last modified on

2013-03-22 16:10:36

Owner

mps (409)

Last modified by

mps (409)

Numerical id

8

Author

mps (409)

Entry type

Definition

Classification

msc 20F65

Defines

graph product of groups

Defines

graph product

Generated on Fri Feb 9 19:30:32 2018 by

LaTeXML
groups that act freely on trees are free

groups that act freely on trees are free

Theorem

.

Groups that act freely and without

inversions

on trees are free.

Proof.

Let

Γ

be a group acting freely and without inversions by

graph automorphisms

on a tree

T

.
Since

Γ

acts freely on

T

, the quotient graph

T

/

Γ

is well-defined, and

T

is the

universal cover

of

T

/

Γ

since

T

is

contractible

. Thus by faithfulness

Γ

≅

π

1

⁢

(

X

/

Γ

)

. Since any graph is

homotopy equivalent

to a wedge of circles, and the

fundamental group

of such a space is free by Van Kampen’s theorem,

Γ

is free.
∎

Title

groups that act freely on trees are free

Canonical name

GroupsThatActFreelyOnTreesAreFree

Date of creation

2013-03-22 13:54:23

Last modified on

2013-03-22 13:54:23

Owner

mps (409)

Last modified by

mps (409)

Numerical id

10

Author

mps (409)

Entry type

Theorem

Classification

msc 20F65

Generated on Fri Feb 9 19:30:34 2018 by

LaTeXML
essential subgroup

essential subgroup

A

subgroup

H

of a group

G

is

essential

if

H

∩

K

≠

{

e

}

for any subgroup

K

of

G

,

K

≠

{

e

}

(i.e., a subgroup

H

of

G

is called essential if it intersects non-trivially every
non-trivial subgroup of

G

).

Title

essential subgroup

Canonical

name

EssentialSubgroup

Date of creation

2013-03-22 15:31:05

Last modified on

2013-03-22 15:31:05

Owner

gh0st (10693)

Last modified by

gh0st (10693)

Numerical id

5

Author

gh0st (10693)

Entry type

Definition

Classification

msc 20F99

Related topic

Group

Related topic

Subgroup

Related topic

EssentialSubmodule

Generated on Fri Feb 9 19:30:41 2018 by

LaTeXML
growth

growth

Let

G

be a

finitely generated group

with generating set

A

(

closed under

inverses

).

For

g

=

a

1

⁢

a

2

⁢

…

⁢

a

m

∈

G

,

a

i

∈

A

, let

l

⁢

(

g

)

be the minimum value of

m

.

Define

γ

⁢

(

n

)

=

∣

{

g

∈

G

:

l

⁢

(

g

)

≤

n

}

∣

.

The

function

γ

is called the

growth function

for

G

with generating set

A

. If

γ

is either

(a) bounded above by a polynomial function,

(b) bounded below by an

exponential function

, or

(c) neither,

then this condition is preserved under changing the generating set for

G

. Respectively, then,

G

is said to have

(a)

polynomial growth

,

(b)

exponential growth

, or

(c)

intermediate growth

.

For a survey on the topic, see:
R. I. Grigorchuk, On growth in

group theory

, Proceedings of the

International Congress of Mathematicians

, Kyoto 1990, Volume I,
II (Math. Soc. Japan, 1991), pages 325 to 338.

Note that, as the generating set is assumed to be closed under inverses, we need only have

G

as a

semigroup

- as such, the above applies equally well in semigroup theory.

Title

growth

Canonical name

Growth

Date of creation

2013-03-22 14:36:09

Last modified on

2013-03-22 14:36:09

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20F99

Classification

msc 20E99

Related topic

GrowthOfExponentialFunction

Defines

polynomial growth

Defines

intermediate growth

Defines

exponential growth

Generated on Fri Feb 9 19:30:44 2018 by

LaTeXML
Hopfian group

Hopfian group

\PMlinkescapephrase

finite rank

A group is said to be

Hopfian

if it is not

isomorphic

to any of its proper

quotients

(

http://planetmath.org/QuotientGroup

).
A group

G

is Hopfian if and only if every surjective

endomorphism

G

→

G

is an automorphism.

A group is said to be

co-Hopfian

if it is not isomorphic to any of its

proper subgroups

.
A group

G

is co-Hopfian if and only if every injective endomorphism

G

→

G

is an automorphism.

Examples

Every

finite group

is obviously Hopfian and co-Hopfian.

The group of

rationals

is an example of an

infinite group

that is both Hopfian and co-Hopfian.

The group of integers is Hopfian, but not co-Hopfian.
More generally, every

finitely generated

abelian group

is Hopfian,
but is not co-Hopfian unless it is finite.

Quasicyclic groups

are co-Hopfian, but not Hopfian.

Free groups

of infinite rank are neither Hopfian nor co-Hopfian.
By contrast, free groups of finite rank are Hopfian (though not co-Hopfian unless of rank zero).

By a theorem of Mal’cev, every finitely generated residually finite group is Hopfian.

The

Baumslag-Solitar group

with

presentation

⟨

b

,

t

∣

t

-

1

⁢

b

2

⁢

t

=

b

3

⟩

is an example of a finitely generated group that is not Hopfian.

Title

Hopfian group

Canonical name

HopfianGroup

Date of creation

2013-03-22 15:36:03

Last modified on

2013-03-22 15:36:03

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 20F99

Related topic

HopfianModule

Defines

Hopfian

Defines

co-Hopfian

Defines

cohopfian

Defines

co-Hopfian group

Defines

cohopfian group

Generated on Fri Feb 9 19:30:46 2018 by

LaTeXML
perfect group

perfect group

A group

G

is called

perfect

if

G

=

[

G

,

G

]

, where

[

G

,

G

]

is the

derived subgroup

of

G

, or equivalently, if the

abelianization

of

G

is trivial.

Title

perfect group

Canonical name

PerfectGroup

Date of creation

2013-03-22 13:56:24

Last modified on

2013-03-22 13:56:24

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 20F99

Synonym

perfect

Generated on Fri Feb 9 19:30:47 2018 by

LaTeXML
proof of properties of Hopfian and co-Hopfian groups

proof of properties of Hopfian and co-Hopfian groups

Proposition

. A group

G

is

Hopfian

if and only if every

surjective

homomorphism

G

→

G

is an automorphism.

Proof

. “

⇒

” Assume that

ψ

:

G

→

G

is a surjective homomorpism such that

ψ

is not an automorphism, which means that

Ker

⁢

(

ψ

)

is nontrivial. Then (due to the

First Isomorphism Theorem

)

G

/

Ker

⁢

(

ψ

)

is

isomorphic

to

Im

⁢

(

ψ

)

=

G

.

Contradiction

, since

G

is Hopfian.

“

⇐

” Assume that

G

is not Hopfian. Then there exists nontrivial

normal subgroup

H

of

G

and an

isomorphism

ϕ

:

G

/

H

→

G

. Let

π

:

G

→

G

/

H

be the

quotient

homomorphism. Then obviously

π

∘

ϕ

:

G

→

G

is a surjective homomorphism, but

Ker

⁢

(

π

∘

ϕ

)

=

H

is nontrivial, therefore

π

∘

ϕ

is not an automorphism. Contradiction.

□

Proposition

. A group

G

is co-Hopfian if and only if every

injective

homomorphism

G

→

G

is an automorphism.

Proof

. “

⇒

” Assume that

ψ

:

G

→

G

is an injective homomorphism which is not an automorphism. Therefore

Im

⁢

(

ψ

)

is a

proper subgroup

of

G

, therefore (since

Ker

⁢

(

ψ

)

=

{

e

}

and due to the First Isomorphism Theorem)

G

is isomorphic to its proper subgroup, namely

Im

⁢

(

ψ

)

. Contradiction, since

G

is co-Hopfian.

“

⇐

” Assume that

G

is not co-Hopfian. Then there exists a proper subgroup

H

of

G

and an isomorphism

ϕ

:

G

→

H

. Let

i

:

H

→

G

be an

inclusion

homomorphism. Then

i

∘

ϕ

:

G

→

G

is an injective homomorphism which is not onto (because

i

is not). Contradiction.

□

Title

proof of properties of Hopfian and co-Hopfian groups

Canonical name

ProofOfPropertiesOfHopfianAndCoHopfianGroups

Date of creation

2013-03-22 18:31:17

Last modified on

2013-03-22 18:31:17

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

6

Author

joking (16130)

Entry type

Proof

Classification

msc 20F99

Generated on Fri Feb 9 19:30:49 2018 by

LaTeXML
section of a group

section of a group

A

section

of a group

G

is
a

quotient

(

http://planetmath.org/QuotientGroup

) of a

subgroup

of

G

.
That is, a section of

G

is a group of the form

H

/

N

,
where

H

is a subgroup of

G

, and

N

is a

normal subgroup

of

H

.

A group

G

is said to be

involved in

a group

K

if

G

is

isomorphic

to a section of

K

.

The

relation

‘is involved in’ is

transitive

(

http://planetmath.org/Transitive3

),
that is, if

G

is involved in

K

and

K

is involved in

L

,
then

G

is involved in

L

.

Intuitively, ‘

G

is involved in

K

’
means that all of the

structure

of

G

can be found inside

K

.

Title

section of a group

Canonical

name

SectionOfAGroup

Date of creation

2013-03-22 17:15:04

Last modified on

2013-03-22 17:15:04

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 20F99

Synonym

section

Synonym

quotient of a subgroup

Defines

involved in

Generated on Fri Feb 9 19:30:51 2018 by

LaTeXML