aarjaneiro/planetmath huggingface dataset:
a group embeds into its profinite completion if and only if it is residually finite

a group embeds into its profinite completion if and only if it is residually finite

Let

G

be a group.

First suppose that

G

is

residually finite

, that is,

R

⁢

(

G

)

:=

⋂

N

⊴

f

G

N

=

1

(where

N

⊴

f

G

denotes that

N

is a

normal subgroup

of finite index in

G

).
Consider the natural

mapping

of

G

into its

profinite completion

G

^

given by

g

↦

(

N

⁢

g

)

N

⊴

f

G

.
It is clear that the kernel of this map is precisely

R

⁢

(

G

)

,
so that it is a

monomorphism

when

G

is residually finite.

Now suppose that

G

embeds into its profinite completion

G

^

and identify

G

with a

subgroup

of

G

^

. Now, a

theorem

on

profinite groups

tells us that

⋂

N

⊴

o

G

^

N

=

1

,

(where

N

⊴

o

G

denotes that

N

is an open (

http://planetmath.org/TopologicalSpace

) normal subgroup of

G

) and since

open subgroups

of a profinite group have finite index, we
have that

R

⁢

(

G

^

)

=

1

,

so

G

^

is residually finite. Then

G

is a subgroup of a
residually finite group, so is itself residually finite, as required.

References

1

J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal,

Analytic

pro-

p

groups

, 2nd ed., Cambridge studies in advanced mathematics,
Cambridge University Press, 1999.

Title

a

group embeds into its profinite completion if and only if it is residually finite

Canonical name

AGroupEmbedsIntoItsProfiniteCompletionIfAndOnlyIfItIsResiduallyFinite

Date of creation

2013-03-22 15:15:56

Last modified on

2013-03-22 15:15:56

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Theorem

Classification

msc 20E18

Related topic

ProfiniteCompletion

Related topic

ProfiniteGroup

Related topic

ResiduallyCalP

Generated on Fri Feb 9 19:26:49 2018 by

LaTeXML
order of a profinite group

order of a profinite group

Let

G

be a

profinite group

, and let

H

be any

closed subgroup

. We define the

of

H

in

G

by

[

G

:

H

]

=

lcm

(

{

[

G

/

N

:

H

N

/

N

]

}

)

,

where

N

runs over all open (and hence of finite index)

subgroups

of

G

, and where

lcm

is taken in the sense of the least common

multiple

of

supernatural numbers

.

In particular, we can define the

order of a profinite group

to be the index of the

identity

subgroup in

G

:

|

G

|

:=

[

G

:

{

e

}

]

.

Some examples of

orders of profinite groups

:

•

G

=

ℤ

p

, the ring of

p

-adic

integers

. Since every finite

quotient

of

ℤ

p

is cyclic of

p

n

elements

(for some

n

), and every such group occurs as a quotient, we have

|

G

|

=

lcm

⁡

(

p

n

)

,

where

n

runs over all

natural numbers

. Thus

|

G

|

=

p

∞

.

•

G

=

ℤ

^

.

Since

G

≈

∏

p

ℤ

p

, we have

|

G

|

=

∏

p

|

ℤ

p

|

=

∏

p

p

∞

. This example illustrates the limitations of this

concept

: Despite being “relatively small” in of profinite groups,

ℤ

^

has the largest possible profinite order.

References

1

[Ram] Ramakrishnan, Dinikara and Valenza, Robert.

Fourier Analysis on

Number Fields

. Graduate Texts in Mathematics, volume 186. Springer-Verlag, New York, NY. 1989.

2

[Ser] Serre, J.-P. (Ion, P., translator)

Galois Cohomology

. Springer, New York, NY. 1997

Title

order of a profinite group

Canonical

name

OrderOfAProfiniteGroup

Date of creation

2013-03-22 15:23:39

Last modified on

2013-03-22 15:23:39

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20E18

Defines

index of a profinite subgroup

Defines

index of a profinite group

Generated on Fri Feb 9 19:26:52 2018 by

LaTeXML
procyclic group

procyclic group

Definition

.

A group

G

is a

procyclic group

if

G

is a

profinite group

which is

isomorphic

to the

inverse limit

of some

projective system

of

cyclic groups

.

Example

.

The

p

-adic integers (

http://planetmath.org/PAdicIntegers

)

ℤ

p

form a procyclic group since:

ℤ

p

=

lim

←

⁡

ℤ

/

p

n

⁢

ℤ

.

Title

procyclic group

Canonical name

ProcyclicGroup

Date of creation

2013-03-22 15:09:14

Last modified on

2013-03-22 15:09:14

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 20E18

Synonym

pro-cyclic group

Synonym

pro-cyclic

Synonym

procyclic

Generated on Fri Feb 9 19:26:54 2018 by

LaTeXML
profinite completion

profinite completion

The

profinite completion

of a group

G

is defined to be the

profinite group

G

^

=

lim

←

N

⊴

f

G

⁡

G

/

N

,

where

N

⊴

f

G

means that

N

is a

normal subgroup

of finite index in

G

.

A

group embeds into its profinite completion
if and only if it is residually finite

.

Title

profinite completion

Canonical name

ProfiniteCompletion

Date of creation

2013-03-22 15:15:54

Last modified on

2013-03-22 15:15:54

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

10

Author

yark (2760)

Entry type

Definition

Classification

msc 20E18

Related topic

AGroupsEmbedsIntoItsProfiniteCompletionIfAndOnlyIfItIsResiduallyFinite

Generated on Fri Feb 9 19:26:58 2018 by

LaTeXML
profinite group

profinite group

1

Definition

A

topological group

G

is

profinite

if it is

isomorphic

to the

inverse limit

of some

projective system

of

finite groups

. In other words,

G

is profinite if there exists a

directed set

I

, a

collection

of finite groups

{

H

i

}

i

∈

I

, and

homomorphisms

α

i

⁢

j

:

H

j

→

H

i

for each pair

i

,

j

∈

I

with

i

≤

j

, satisfying

1.

α

i

⁢

i

=

1

for all

i

∈

I

,

2.

α

i

⁢

j

∘

α

j

⁢

k

=

α

i

⁢

k

for all

i

,

j

,

k

∈

I

with

i

≤

j

≤

k

,

with the property that:

•

G

is isomorphic as a group to the projective limit

lim

⟵

⁢

H

i

:=

{

(

h

i

)

∈

∏

i

∈

I

H

i

|

α

i

⁢

j

⁢

(

h

j

)

=

h

i

⁢

for all

⁢

i

≤

j

}

under componentwise

multiplication

.

•

The

isomorphism

from

G

to

lim

⟵

⁢

H

i

(considered as a

subspace

of

∏

H

i

) is a

homeomorphism

of

topological spaces

, where each

H

i

is given the

discrete topology

and

∏

H

i

is given the

product topology

.

The

topology

on a profinite group is called the

profinite topology

.

2

Properties

One can show that a topological group is profinite if and only if it is

compact

and

totally disconnected

. Moreover, every profinite group is

residually finite

.

Title

profinite group

Canonical name

ProfiniteGroup

Date of creation

2013-03-22 12:48:50

Last modified on

2013-03-22 12:48:50

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Definition

Classification

msc 20E18

Classification

msc 22C05

Synonym

profinite

Related topic

InverseLimit

Defines

profinite topology

Generated on Fri Feb 9 19:27:00 2018 by

LaTeXML
pro-p group

pro-

p

group

Definition

.

Let

p

be a prime number. A group

G

is a pro-

p

group if

G

is a

profinite group

which is

isomorphic

to the

inverse limit

of some

projective system

of

p

-groups (

http://planetmath.org/PGroup4

).

Example

.

The

p

-adic integers (

http://planetmath.org/PAdicIntegers

)

ℤ

p

form a pro-

p

group since:

ℤ

p

=

lim

←

⁡

ℤ

/

p

n

⁢

ℤ

.

Title

pro-

p

group

Canonical

name

PropGroup

Date of creation

2013-03-22 15:09:11

Last modified on

2013-03-22 15:09:11

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Definition

Classification

msc 20E18

Synonym

pro p group

Synonym

pro-p group

Synonym

pro

p

group

Related topic

PGroup4

Generated on Fri Feb 9 19:27:02 2018 by

LaTeXML
supernatural number

supernatural number

A

supernatural number

ω

is a formal

product

ω

=

∏

p

p

n

p

,

where

p

runs over all (

rational

)

prime numbers

, and the values

n

p

are each either

natural numbers

or the symbol

∞

.

We note first that by the

fundamental theorem of arithmetic

, we can view any natural number as a supernatural number. Supernatural numbers form a

generalization

of natural numbers in two ways: First, by allowing the possibility of infinitely many

prime factors

, and second, by allowing any given prime to divide

ω

“infinitely often,” by taking that prime’s corresponding

exponent

to be the symbol

∞

.

We can extend the usual

p

-adic to these supernatural numbers by defining, for

ω

as above,

v

p

⁢

(

ω

)

=

n

p

for each

p

. We can then extend the notion of

divisibility

to supernatural numbers by declaring

ω

1

∣

ω

2

if

v

p

⁢

(

ω

1

)

≤

v

p

⁢

(

ω

2

)

for all

p

(where, by

definition

, the symbol

∞

is considered greater than any natural number). Finally, we can also generalize the notion of the least common

multiple

(lcm) and

greatest common divisor

(gcd) for (arbitrarily many) supernatural numbers, by defining

lcm

⁡

(

{

ω

i

}

)

=

∏

p

p

sup

⁡

(

v

p

⁢

(

ω

i

)

)

gcd

⁡

(

{

ω

i

}

)

=

∏

p

p

inf

⁡

(

v

p

⁢

(

ω

i

)

)

Note that the supernatural version of the definitions of divisibility,

lcm

, and

gcd

carry over exactly from their corresponding notions for natural numbers, though we can now take the gcd or lcm of infinitely many natural numbers (to get a supernatural number).

Supernatural numbers are used to define orders and indices of

profinite groups

and

subgroups

, in which case many of the

theorems

from

finite group

theory carry over verbatim.

References

1

[Ram] Ramakrishnan, Dinikara and Valenza, Robert.

Fourier Analysis on

Number Fields

. Graduate Texts in Mathematics, volume 186. Springer-Verlag, New York, NY. 1989.

2

[Ser] Serre, J.-P. (Ion, P., translator)

Galois Cohomology

. Springer, New York, NY. 1997

Title

supernatural number

Canonical

name

SupernaturalNumber

Date of creation

2013-03-22 15:23:36

Last modified on

2013-03-22 15:23:36

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 20E18

Defines

gcd of supernatural numbers

Defines

greatest commond divisor of supernatural numbers

Defines

lcm of supernatural numbers

Defines

least common multiple of supernatural numbers

Generated on Fri Feb 9 19:27:05 2018 by

LaTeXML
examples of semidirect products of groups

examples of semidirect products of groups

Suppose

H

=

ℤ

/

n

⁢

ℤ

and let

r

be a

generator

for

H

. Let

Q

=

ℤ

/

2

⁢

ℤ

=

<

s

>

. Define

θ

:

Q

→

Aut

⁡

(

H

)

by

θ

⁢

(

s

)

⁢

(

r

)

=

r

-

1

. Let

G

=

H

⋊

θ

Q

. Then in

G

,

s

⁢

r

⁢

s

=

s

⁢

r

⁢

s

-

1

=

θ

⁢

(

s

)

⁢

(

r

)

=

r

-

1

by the canonical equivalence of inner and outer

semidirect products

. So

G

has

2

⁢

n

elements, two generators

r

,

s

satisfying

r

n

=

s

2

=

1

s

⁢

r

⁢

s

=

r

-

1

and thus

G

=

𝒟

2

⁢

n

, the

n

th

dihedral group

.

If instead

H

=

ℤ

, the result is the

infinite dihedral group

.

As another example, if

G

is a group, then the

holomorph

of

G

is

G

⋊

Aut

⁡

(

G

)

under the identity map from

Aut

⁡

(

G

)

to itself.

Title

examples of semidirect products of groups

Canonical name

ExamplesOfSemidirectProductsOfGroups

Date of creation

2013-03-22 17:22:52

Last modified on

2013-03-22 17:22:52

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Example

Classification

msc 20E22

Generated on Fri Feb 9 19:27:07 2018 by

LaTeXML
extension

extension

A

short exact sequence

0

→

A

→

B

→

C

→

0

is sometimes called an extension of

C

by

A

. This is also applied to an object

B

which fits into such an

exact sequence

.

Title

extension

Canonical name

Extension

Date of creation

2013-03-22 13:51:10

Last modified on

2013-03-22 13:51:10

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 20E22

Generated on Fri Feb 9 19:27:09 2018 by

LaTeXML
generalized dihedral group

generalized dihedral group

Let

A

be an

abelian group

.
The

generalized dihedral group

Dih

⁡

(

A

)

is the

semidirect product

A

⋊

C

2

,
where

C

2

is the

cyclic group

of order

2

,
and the

generator

(

http://planetmath.org/Generator

) of

C

2

maps elements of

A

to their

inverses

.

If

A

is cyclic, then

Dih

⁡

(

A

)

is called a

dihedral group

.
The finite dihedral group

Dih

⁡

(

C

n

)

is commonly denoted by

D

n

or

D

2

⁢

n

(the differing conventions being a source of confusion).
The infinite dihedral group

Dih

⁡

(

C

∞

)

is denoted by

D

∞

,
and is

isomorphic

to
the

free product

C

2

*

C

2

of two cyclic groups of order

2

.

If

A

is an

elementary abelian

2

-group, then so is

Dih

⁡

(

A

)

.
If

A

is not an elementary abelian

2

-group, then

Dih

⁡

(

A

)

is

non-abelian

.

The

subgroup

A

×

{

1

}

of

Dih

⁡

(

A

)

is of index

2

,
and every element of

Dih

⁡

(

A

)

that is not in this subgroup has order

2

.
This

property

in fact characterizes generalized dihedral groups,
in the sense that if a group

G

has a subgroup

N

of index

2

such that all elements of the

complement

G

∖

N

are of order

2

,
then

N

is

abelian

and

G

≅

Dih

⁡

(

N

)

.

Title

generalized dihedral group

Canonical

name

GeneralizedDihedralGroup

Date of creation

2013-03-22 14:53:28

Last modified on

2013-03-22 14:53:28

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 20E22

Synonym

generalised dihedral group

Related topic

DihedralGroup

Defines

infinite dihedral group

Defines

infinite dihedral

Generated on Fri Feb 9 19:27:11 2018 by

LaTeXML
holomorph of a group

holomorph of a group

If

G

is a group, the

holomorph

of

G

is the group

G

⋊

θ

Aut

⁡

(

G

)

under the

identity map

θ

:

Aut

⁡

(

G

)

→

Aut

⁡

(

G

)

.

Title

holomorph of a group

Canonical name

HolomorphOfAGroup

Date of creation

2013-03-22 17:22:50

Last modified on

2013-03-22 17:22:50

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

4

Author

rm50 (10146)

Entry type

Definition

Classification

msc 20E22

Synonym

holomorph

Generated on Fri Feb 9 19:27:13 2018 by

LaTeXML
proof of the Jordan Hölder decomposition theorem

proof of the Jordan Hölder decomposition theorem

Let

|

G

|

=

N

. We first prove

existence

, using

induction

on

N

. If

N

=

1

(or, more generally, if

G

is simple) the result is clear. Now suppose

G

is not simple. Choose a

maximal proper normal subgroup

G

1

of

G

. Then

G

1

has a Jordan–Hölder decomposition by induction, which produces a Jordan–Hölder decomposition for

G

.

To prove uniqueness, we use induction on the length

n

of the decomposition series. If

n

=

1

then

G

is simple and we are done. For

n

>

1

, suppose that

G

⊃

G

1

⊃

G

2

⊃

⋯

⊃

G

n

=

{

1

}

and

G

⊃

G

1

′

⊃

G

2

′

⊃

⋯

⊃

G

m

′

=

{

1

}

are two decompositions of

G

. If

G

1

=

G

1

′

then we’re done (apply the

induction hypothesis

to

G

1

), so assume

G

1

≠

G

1

′

. Set

H

:=

G

1

∩

G

1

′

and choose a decomposition series

H

⊃

H

1

⊃

⋯

⊃

H

k

=

{

1

}

for

H

. By the

second isomorphism theorem

,

G

1

/

H

=

G

1

⁢

G

1

′

/

G

1

′

=

G

/

G

1

′

(the last equality is because

G

1

⁢

G

1

′

is a

normal subgroup

of

G

properly containing

G

1

). In particular,

H

is a normal subgroup of

G

1

with simple

quotient

. But then

G

1

⊃

G

2

⊃

⋯

⊃

G

n

and

G

1

⊃

H

⊃

⋯

⊃

H

k

are two decomposition series for

G

1

, and hence have the same simple quotients by the induction hypothesis; likewise for the

G

1

′

series. Therefore

n

=

m

. Moreover, since

G

/

G

1

=

G

1

′

/

H

and

G

/

G

1

′

=

G

1

/

H

(by the second isomorphism theorem), we have now accounted for all of the simple quotients, and shown that they are the same.

Title

proof of the Jordan Hölder decomposition

theorem

Canonical

name

ProofOfTheJordanHolderDecompositionTheorem

Date of creation

2013-03-22 12:08:49

Last modified on

2013-03-22 12:08:49

Owner

djao (24)

Last modified by

djao (24)

Numerical id

9

Author

djao (24)

Entry type

Proof

Classification

msc 20E22

Generated on Fri Feb 9 19:27:16 2018 by

LaTeXML
semi-direct factor and quotient group

semi-direct factor and quotient group

Theorem

.

If the group

G

is a

semi-direct product

of its

subgroups

H

and

Q

,
then the semi-direct

Q

is

isomorphic

to the

quotient group

G

/

H

.

Proof.

Every

element

g

of

G

has the unique

representation

g

=

h

⁢

q

with

h

∈

H

and

q

∈

Q

.
We therefore can define the

mapping

g

↦

q

from

G

to

Q

.
The mapping is

surjective

since any element

y

of

Q

is the image of

e

⁢

y

.
The mapping is also a

homomorphism

since if

g

1

=

h

1

⁢

q

1

and

g

2

=

h

2

⁢

q

2

, then we obtain

f

⁢

(

g

1

⁢

g

2

)

=

f

⁢

(

h

1

⁢

q

1

⁢

h

2

⁢

q

2

)

=

f

⁢

(

h

1

⁢

h

2

⁢

q

1

⁢

q

2

)

=

q

1

⁢

q

2

=

f

⁢

(

g

1

)

⁢

f

⁢

(

g

2

)

.

Then we see that

ker

⁡

f

=

H

because all elements

h

=

h

⁢

e

of

H

are mapped to the

identity element

e

of

Q

.
Consequently we get, according to the

first isomorphism theorem

, the result

G

/

H

≅

Q

.

Example.

The

multiplicative group

ℝ

×

of reals
is the semi-direct product of the subgroups

{

1

,

-

1

}

=

{

±

1

}

and

ℝ

+

.
The quotient group

ℝ

×

/

{

±

1

}

consists of all cosets

x

⁢

{

±

1

}

=

{

x

,

-

x

}

where

x

≠

0

, and is obviously isomorphic with

ℝ

+

=

{

x

⁢

∣

x

>

⁢

0

}

.

Title

semi-direct factor and quotient group

Canonical name

SemidirectFactorAndQuotientGroup

Date of creation

2013-03-22 15:10:22

Last modified on

2013-03-22 15:10:22

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Theorem

Classification

msc 20E22

Generated on Fri Feb 9 19:27:18 2018 by

LaTeXML
wreath product

wreath product

Let

A

and

B

be groups, and let

B

act on the set

Γ

.
Define the action of

B

on the

direct product

A

Γ

by

b

⁢

f

⁢

(

γ

)

:=

f

⁢

(

b

-

1

⁢

γ

)

,

for any

f

∈

A

Γ

and

γ

∈

Γ

.
The

wreath product

of

A

and

B

according to the action of

B

on

Γ

, denoted

A

≀

Γ

B

, is the

semidirect product of groups

A

Γ

⋊

B

.

Let us pause to unwind this definition. The elements of

A

≀

Γ

B

are

ordered pairs

(

f

,

b

)

, where

f

∈

A

Γ

and

b

∈

B

. The

group operation

is given by

(

f

,

b

)

⁢

(

f

′

,

b

′

)

=

(

f

⁢

b

⁢

f

′

,

b

⁢

b

′

)

.

Note that by definition of the action of

B

on

A

Γ

,

(

f

⁢

b

⁢

f

′

)

⁢

(

γ

)

=

f

⁢

(

γ

)

⁢

f

′

⁢

(

b

-

1

⁢

γ

)

.

The action of

B

on

Γ

in the

semidirect product

permutes the elements of a tuple

f

∈

A

Γ

, and the group operation defined on

A

Γ

gives

pointwise

multiplication

. To be explicit, suppose

Γ

is an

n

-tuple, and let

(

f

,

b

)

,

(

f

′

,

b

′

)

∈

A

≀

Γ

B

.
Let

b

i

denote

b

-

1

⁢

(

i

)

. Then

(

f

,

b

)

⁢

(

f

′

,

b

′

)

=

(

(

f

⁢

(

1

)

,

f

⁢

(

2

)

,

…

,

f

⁢

(

n

)

)

,

b

)

⁢

(

(

f

′

⁢

(

1

)

,

f

′

⁢

(

2

)

,

…

,

f

′

⁢

(

n

)

)

,

b

′

)

=

(

(

f

⁢

(

1

)

,

f

⁢

(

2

)

,

…

,

f

⁢

(

n

)

)

⁢

(

f

′

⁢

(

b

1

)

,

f

′

⁢

(

b

2

)

,

…

,

f

′

⁢

(

b

n

)

)

,

b

⁢

b

′

)

⁢

(*)

=

(

(

f

⁢

(

1

)

⁢

f

′

⁢

(

b

1

)

,

f

⁢

(

2

)

⁢

f

′

⁢

(

b

2

)

,

…

,

f

⁢

(

n

)

⁢

f

′

⁢

(

b

n

)

)

,

b

⁢

b

′

)

.

Notice the

permutation

of the indices in (*).

A bit amount of thought to understand this slightly messy notation will be illuminating, and might also shed some light on the choice of terminology.

Title

wreath product

Canonical name

WreathProduct

Date of creation

2014-12-30 11:23:20

Last modified on

2014-12-30 11:23:20

Owner

mps (409)

Last modified by

juanman (12619)

Numerical id

21

Author

mps (12619)

Entry type

Definition

Classification

msc 20E22

Defines

wreath product

Generated on Fri Feb 9 19:27:27 2018 by

LaTeXML
locally 𝒫

locally

𝒫

Let

𝒫

be a property of groups.
A group

G

is said to be

locally

P

if every nontrivial

finitely generated subgroup

of

G

has property

𝒫

.

For example, the locally

infinite groups

are precisely the

torsion-free

groups.
Other classes of groups defined this way include

locally finite groups

and

locally cyclic groups

.

Title

locally

𝒫

Canonical name

LocallycalP

Date of creation

2013-03-22 14:18:57

Last modified on

2013-03-22 14:18:57

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

5

Author

yark (2760)

Entry type

Definition

Classification

msc 20E25

Related topic

GeneralizedCyclicGroup

Related topic

LocallyFiniteGroup

Related topic

LocallyNilpotentGroup

Generated on Fri Feb 9 19:27:29 2018 by

LaTeXML
subgoups of locally cyclic groups are locally cyclic

subgoups of locally cyclic groups are locally cyclic

Theorem 1

.

A group

G

is

locally cyclic

iff every

subgroup

H

≤

G

is locally cyclic.

Proof.

Let

G

be a locally cyclic group and

H

a subgroup of

G

. Let

S

be a finite subset of

H

. Then the group

⟨

S

⟩

generated by

S

is a

cyclic subgroup

of

G

, by assumption. Since every element

a

of

⟨

S

⟩

is a

product

of elements or

inverses

of elements of

S

, and

S

is a subset of group

H

,

a

∈

H

. Hence

⟨

S

⟩

is a cyclic subgroup of

H

, so

H

is locally cyclic.

Conversely, suppose for every subgroup of

G

is locally cyclic. Let

H

be a subgroup generated by a finite subset of

G

. Since

H

is locally cyclic, and

H

itself is

finitely generated

,

H

is cyclic, and therefore

G

is locally cyclic.
∎

Title

subgoups of locally cyclic groups are locally cyclic

Canonical name

SubgoupsOfLocallyCyclicGroupsAreLocallyCyclic

Date of creation

2013-03-22 17:14:46

Last modified on

2013-03-22 17:14:46

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

12

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 20E25

Classification

msc 20K99

Generated on Fri Feb 9 19:27:30 2018 by

LaTeXML