aarjaneiro/planetmath huggingface dataset:
cyclic group

cyclic group

A group is said to be

cyclic

if it is generated by a single element.

Suppose

G

is a

cyclic group

generated by

x

∈

G

.
Then every element of

G

is equal to

x

k

for some

k

∈

ℤ

.
If

G

is

infinite

, then these

x

k

are all distinct,
and

G

is

isomorphic

to the group

ℤ

.
If

G

has finite order (

http://planetmath.org/OrderGroup

)

n

,
then every element of

G

can be expressed as

x

k

with

k

∈

{

0

,

…

,

n

-

1

}

,
and

G

is isomorphic to the

quotient group

ℤ

/

n

⁢

ℤ

.

Note that the

isomorphisms

mentioned in the previous paragraph
imply that all cyclic groups of the same order are isomorphic to one another.
The infinite cyclic group is sometimes written

C

∞

,
and the finite cyclic group of order

n

is sometimes written

C

n

.
However, when the cyclic groups are written additively,
they are commonly represented by

ℤ

and

ℤ

/

n

⁢

ℤ

.

While a cyclic group can, by

definition

, be generated by a single element,
there are often a number of different elements that can be used as the

generator

: an infinite cyclic group has

2

generators,
and a finite cyclic group of order

n

has

ϕ

⁢

(

n

)

generators,
where

ϕ

is the

Euler totient function

.

Some basic facts about cyclic groups:

•

Every cyclic group is

abelian

.

•

Every

subgroup

of a cyclic group is cyclic.

•

Every

quotient

of a cyclic group is cyclic.

•

Every group of prime order is cyclic. (This follows immediately from Lagrange’s

Theorem

.)

•

Every

finite subgroup

of the

multiplicative group

of a field is cyclic.

Title

cyclic group

Canonical

name

CyclicGroup

Date of creation

2013-03-22 12:23:27

Last modified on

2013-03-22 12:23:27

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

21

Author

yark (2760)

Entry type

Definition

Classification

msc 20A05

Related topic

GeneralizedCyclicGroup

Related topic

PolycyclicGroup

Related topic

VirtuallyCyclicGroup

Related topic

CyclicRing3

Defines

cyclic

Defines

cyclic subgroup

Defines

infinite cyclic

Defines

infinite cyclic group

Defines

infinite cyclic subgroup

Generated on Fri Feb 9 19:17:00 2018 by

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double coset

double coset

Let

H

and

K

be

subgroups

of a group

G

.
An

(

H

,

K

)

-

double coset

is a set of the form

H

⁢

x

⁢

K

for some

x

∈

G

.
Here

H

⁢

x

⁢

K

is defined in the

obvious

way as

H

⁢

x

⁢

K

=

{

h

⁢

x

⁢

k

∣

h

∈

H

⁢

and

⁢

k

∈

K

}

.

Note that the

(

H

,

{

1

}

)

-double cosets are just the

right cosets

of

H

,
and the

(

{

1

}

,

K

)

-double cosets are just the

left cosets

of

K

.
In general, every

(

H

,

K

)

-double coset is a union of right cosets of

H

,
and also a union of left cosets of

K

.

The set of all

(

H

,

K

)

-double cosets is denoted

H

\

G

/

K

.
It is straightforward to show that

H

\

G

/

K

is a

partition

(

http://planetmath.org/Partition

) of

G

,
that is, every element of

G

lies in exactly one

(

H

,

K

)

-double coset.

In contrast to the situation with ordinary cosets (

http://planetmath.org/Coset

),
the

(

H

,

K

)

-double cosets need not all be of the same cardinality.
For example, if

G

is the

symmetric group

(

http://planetmath.org/SymmetricGroup

)

S

3

,
and

H

=

⟨

(

1

,

2

)

⟩

and

K

=

⟨

(

1

,

3

)

⟩

,
then the two

(

H

,

K

)

-double cosets
are

{

e

,

(

1

,

2

)

,

(

1

,

3

)

,

(

1

,

3

,

2

)

}

and

{

(

2

,

3

)

,

(

1

,

2

,

3

)

}

.

Title

double coset

Canonical

name

DoubleCoset

Date of creation

2013-03-22 16:17:28

Last modified on

2013-03-22 16:17:28

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

8

Author

yark (2760)

Entry type

Definition

Classification

msc 20A05

Generated on Fri Feb 9 19:17:02 2018 by

LaTeXML
equivariant

equivariant

Let

G

be a group, and

X

and

Y

left (resp. right)

homogeneous spaces

of

G

. Then a map

f

:

X

→

Y

is called

equivariant

if

g

⁢

(

f

⁢

(

x

)

)

=

f

⁢

(

g

⁢

x

)

(resp.

(

f

⁢

(

x

)

)

⁢

g

=

f

⁢

(

x

⁢

g

)

) for all

g

∈

G

.

Title

equivariant

Canonical name

Equivariant

Date of creation

2013-03-22 13:56:45

Last modified on

2013-03-22 13:56:45

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

4

Author

bwebste (988)

Entry type

Definition

Classification

msc 20A05

Generated on Fri Feb 9 19:17:03 2018 by

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Euler-Fermat theorem

Euler-Fermat theorem

If

a

,

n

∈

ℤ

such that

gcd

⁡

(

a

,

n

)

=

1

, then

a

φ

⁢

(

n

)

≡

1

⁢

mod

⁡

n

, where

φ

is the

Euler totient function

.

Title

Euler-Fermat theorem

Canonical name

EulerFermatTheorem

Date of creation

2013-03-22 11:45:15

Last modified on

2013-03-22 11:45:15

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 20-01

Classification

msc 11-00

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Classification

msc 55-00

Classification

msc 18F99

Classification

msc 58Z05

Synonym

Euler’s theorem

Related topic

FermatsLittleTheorem

Related topic

FermatsTheoremProof

Generated on Fri Feb 9 19:17:05 2018 by

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example of straight-line program

example of straight-line program

SLPs can be used to shorten

computations

of algebraic expressions.

For example: The

product

5

19

may be evaluated in fewer than 19 multiplication.

Let

f

0

⁢

(

x

)

=

x

and

f

i

+

1

⁢

(

x

)

=

f

i

⁢

(

x

)

⋅

f

i

⁢

(

x

)

, for

i

∈

ℕ

.
Evidently

f

i

⁢

(

x

)

=

x

2

i

,

∀

i

∈

ℕ

.

However, we do not evaluate

f

i

⁢

(

5

)

as

5

i

. For instance, to compute

f

3

⁢

(

5

)

=

5

8

we
follow the

program

. Expand the first the expression from the left first until we
reach an

f

0

⁢

(

x

)

term:

f

3

⁢

(

5

)

=

f

2

⁢

(

5

)

⋅

f

2

⁢

(

5

)

=

(

f

1

⁢

(

5

)

⋅

f

1

⁢

(

5

)

)

⋅

f

2

⁢

(

5

)

=

(

(

f

0

⁢

(

5

)

⋅

f

0

⁢

(

5

)

)

⋅

f

1

⁢

(

5

)

)

⋅

f

2

⁢

(

5

)

.

Now evaluate the

terminal

parts and store all the intermediate results:

f

0

⁢

(

5

)

=

5

,

f

1

⁢

(

5

)

=

f

0

⁢

(

5

)

⋅

f

0

⁢

(

5

)

=

5

⋅

5

=

25

,

f

2

⁢

(

5

)

=

f

1

⁢

(

5

)

⋅

f

1

⁢

(

5

)

=

25

⋅

25

=

625

,

and

f

3

⁢

(

5

)

=

625

⋅

625

=

390625

.

The total number of multiplications here was 3, rather than 8.

Then

x

19

can be encoded as an SLP using the binary number for 19.

First, write 19 in binary (base 2):

19

=

1

⋅

2

4

+

0

⋅

2

3

+

0

⋅

2

2

+

1

⋅

2

1

+

1

⋅

2

0

,

or rather

10011

in the usual binary notation.

ℱ

⁢

(

x

)

=

f

4

⁢

(

x

)

1

⋅

f

3

⁢

(

x

)

0

⋅

f

2

⁢

(

x

)

0

⋅

f

1

⁢

(

x

)

1

⋅

f

0

⁢

(

x

)

1

=

f

4

⁢

(

x

)

⋅

f

1

⁢

(

x

)

⋅

f

0

⁢

(

x

)

.

Remark.

To evaluate this SLP we had to store intermediate values that
were ultimately not part of the final answer. In this example the final answer
was much larger than the intermediate steps and so we are free to assume that
any memory used in the process was far less than what was required for the final
outcome. However, it is entirely possible that evaluating an SLP will at times
use far more memory than required by the final product and may therefore be
infeasible.

Title

example of

straight-line program

Canonical name

ExampleOfStraightlineProgram

Date of creation

2013-03-22 16:16:18

Last modified on

2013-03-22 16:16:18

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

7

Author

Algeboy (12884)

Entry type

Example

Classification

msc 20A05

Classification

msc 20-00

Classification

msc 08A99

Generated on Fri Feb 9 19:17:08 2018 by

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examples of finite simple groups

examples of finite simple groups

This entry is under construction. If I take too long to finish it, nag me about it, or fill in the rest yourself.

All groups considered here are finite.

It is now widely believed that the

classification

of all finite simple groups up to

isomorphism

is finished. The proof runs for at least 10,000 printed pages, and as of the writing of this entry, has not yet been published in its entirety.

Abelian groups

•

The first

trivial

example of

simple groups

are the

cyclic groups

of prime (

http://planetmath.org/Prime

) order. It is not difficult to see (say, by Cauchy’s

theorem

) that these are the only

abelian

simple groups.

Alternating groups

•

The

alternating group

on

n

symbols is the set of all

even permutations

of

S

n

, the

symmetric group

on

n

symbols. It is usually denoted by

A

n

, or sometimes by

Alt

⁡

(

n

)

. This is a

normal subgroup

of

S

n

, namely the kernel of the

homomorphism

that sends every even permutation to

1

and the

odd permutations

to

-

1

. Because every

permutation

is either even or odd, and there is a

bijection

between the two (multiply every even permutation by a

transposition

), the index of

A

n

in

S

n

is

2

.

A

3

is simple because it only has three

elements

, and the simplicity of

A

n

for

n

≥

5

can be proved by an

elementary

argument

. The simplicity of the alternating groups is an important fact that Évariste Galois required in order to prove the insolubility by

radicals

of the general polynomial of degree higher than four. It is worth noting that some common sources of normal subgroups, namely centers and

commutators

, are therefore uninteresting in

A

n

for

n

≥

3

. Specifically,

[

A

n

,

A

n

]

=

A

n

and

A

n

has trivial center for

n

≥

3

.

Groups of Lie type

•

Projective special linear groups

•

Other groups of Lie type.

Sporadic groups

There are twenty-six

sporadic groups

(no more, no less!) that do not fit into any of the

infinite

sequences

of simple groups considered above. These often arise as the group of automorphisms of

strongly regular graphs

.

•

Mathieu groups

.

•

Janko groups

.

•

The baby monster.

•

The monster.

Title

examples of finite simple groups

Canonical name

ExamplesOfFiniteSimpleGroups

Date of creation

2013-03-22 13:07:54

Last modified on

2013-03-22 13:07:54

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

15

Author

mathcam (2727)

Entry type

Example

Classification

msc 20A05

Classification

msc 20E32

Related topic

ExamplesOfGroups

Related topic

SimplicityOfA_n

Related topic

JankoGroups

Defines

alternating group

Generated on Fri Feb 9 19:17:10 2018 by

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Feit-Thompson conjecture

Feit-Thompson conjecture

Conjecture (Walter Feit & John Thompson). There are no

prime numbers

p

and

q

(with

p

≠

q

) such that

p

q

-

1

p

-

1

is

divisible

by

q

p

-

1

q

-

1

.

Feit and Thompson, in regards to the

Feit-Thompson theorem

, have said that proving this conjecture would simplify their proof of their

theorem

, “rendering unnecessary the detailed use of

generators and relations

.” In 1971, Stephens strengthened the conjecture to state that

gcd

⁡

(

p

q

-

1

p

-

1

,

q

p

-

1

q

-

1

)

=

1

always, and then found the counterexample

p

=

17

,

q

=

3313

. The numbers

17

3313

-

1

17

-

1

and

3313

17

-

1

3313

-

1

do have 112643 as their

greatest common divisor

, but dividing the former by the latter leaves a

remainder

of 149073454345008273252753518779212742886488244343395482423. No other counterexamples have been found to Stephen’s stronger version of the conjecture.

References

1

N. M. Stephens, “On the

Feit-Thompson Conjecture

”

Math. of Computation

25

115 (1971): 625

Title

Feit-Thompson conjecture

Canonical name

FeitThompsonConjecture

Date of creation

2013-03-22 17:55:38

Last modified on

2013-03-22 17:55:38

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 20A05

Classification

msc 20E32

Generated on Fri Feb 9 19:17:12 2018 by

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Feit-Thompson theorem

Feit-Thompson theorem

An important result in the classification of all finite simple groups, the

Feit-Thompson theorem

states that every

non-Abelian

simple group

must have even order.

The only known proof requires 255 pages.

1

References

Feit, W. and Thompson, J. G.,

Solvability of groups of odd order

, Pacific J. Math. 13 (1963), 775-1029.

Title

Feit-Thompson theorem

Canonical name

FeitThompsonTheorem

Date of creation

2013-03-22 13:47:29

Last modified on

2013-03-22 13:47:29

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 20E32

Generated on Fri Feb 9 19:17:13 2018 by

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finite subgroup

finite subgroup

Theorem.

A non-empty finite subset

K

of a group

G

is a

subgroup

of

G

if and only if

x

⁢

y

∈

K

for all

x

,

y

∈

K

.

(1)

Proof.

The condition (1) is apparently true if

K

is a subgroup.  Conversely, suppose that a nonempty finite subset

K

of the group

G

satisfies (1).  Let

a

and

b

be arbitrary elements of

K

.  By (1), all () powers of

b

belong to

K

.  Because of the finiteness of

K

, there exist positive integers

r

,

s

such that

b

r

=

b

s

,

r

>

s

+

1

.

By (1),

K

∋

b

r

-

s

-

1

=

b

r

-

s

⁢

b

-

1

=

e

⁢

b

-

1

=

b

-

1

.

Thus also

a

⁢

b

-

1

∈

K

,  whence, by the theorem of the

http://planetmath.org/node/1045

parent entry,

K

is a subgroup of

G

.

Example.

The

multiplicative group

G

of all nonzero

complex numbers

has the finite

multiplicative subset

{

1

,

-

1

,

i

,

-

i

}

,  which has to be a subgroup of

G

.

Title

finite subgroup

Canonical name

FiniteSubgroup

Date of creation

2013-03-22 18:57:02

Last modified on

2013-03-22 18:57:02

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Theorem

Classification

msc 20A05

Synonym

criterion for finite subgroup

Synonym

finite subgroup criterion

Generated on Fri Feb 9 19:17:15 2018 by

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finitely generated group

finitely generated group

A

finitely generated group

is a group that has a finite

generating set

.

Every

finite group

is obviously

finitely generated

.
Every finitely generated group is countable.

Any

quotient

(

http://planetmath.org/QuotientGroup

)
of a finitely generated group is finitely generated.
However, a finitely generated group may have

subgroups

that are not finitely generated.
(For example, the

free group

of rank

2

is generated by just two elements,
but its

commutator subgroup

is not finitely generated.)
Nonetheless, a subgroup of finite index in a finitely generated group
is necessarily finitely generated;
a bound on the number of

generators

required for the subgroup is given by
the

Schreier index formula

(

http://planetmath.org/ScheierIndexFormula

).

The finitely generated groups
all of whose subgroups are also finitely generated
are precisely the groups satisfying the

maximal condition

.
This includes all finitely generated

nilpotent groups

and,
more generally, all

polycyclic groups

.

A group that is not finitely generated
is sometimes said to be

infinitely generated

.

Title

finitely generated group

Canonical name

FinitelyGeneratedGroup

Date of creation

2013-03-22 12:16:38

Last modified on

2013-03-22 12:16:38

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

24

Author

yark (2760)

Entry type

Definition

Classification

msc 20A05

Related topic

FundamentalTheoremOfFinitelyGeneratedAbelianGroups

Related topic

AFinitelyGeneratedGroupHasOnlyFinitelyManySubgroupsOfAGivenIndex

Defines

finitely generated

Defines

finitely generated subgroup

Defines

infinitely generated

Defines

infinitely generated group

Defines

infinitely generated subgroup

Generated on Fri Feb 9 19:17:16 2018 by

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fourth isomorphism theorem

fourth isomorphism theorem

Theorem 1

(The Fourth Isomorphism Theorem)

Let

G

be a group and

N

⊴

G

. There is a bijection between

G

⁢

(

N

)

, the set of

subgroups

of

G

containing

N

, and the set of subgroups of

G

/

N

defined by

A

→

A

/

N

. Moreover, for any two subgroups

A

,

B

in

G

⁢

(

N

)

, we have

1.

A

≤

B

if and only if

A

/

N

≤

B

/

N

,

2.

A

≤

B

implies

|

B

:

A

|

=

|

B

/

N

:

A

/

N

|

,

3.

⟨

A

,

B

⟩

/

N

=

⟨

A

/

N

,

B

/

N

⟩

,

4.

(

A

∩

B

)

/

N

=

(

A

/

N

)

∩

(

B

/

N

)

, and

5.

A

⊴

G

if and only if

(

A

/

N

)

⊴

(

G

/

N

)

.

Title

fourth isomorphism theorem

Canonical name

FourthIsomorphismTheorem

Date of creation

2013-03-22 14:00:52

Last modified on

2013-03-22 14:00:52

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

16

Author

bwebste (988)

Entry type

Theorem

Classification

msc 20A05

Synonym

correspondence theorem

Synonym

lattice isomorphism theorem

Generated on Fri Feb 9 19:17:18 2018 by

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fundamental homomorphism theorem

fundamental homomorphism theorem

The following

theorem

is also true for rings (with ideals instead of

normal subgroups

) or modules (with

submodules

instead of normal subgroups).

theorem 1

.

Let

G

,

H

be groups,

f

:

G

→

H

a

homomorphism

, and let

N

be a normal subgroup of

G

contained

in

ker

⁡

(

f

)

. Then there exists a unique homomorphism

h

:

G

/

N

→

H

so that

h

∘

φ

=

f

, where

φ

denotes the

canonical homomorphism

from

G

to

G

/

N

.

Furthermore, if

f

is onto, then so is

h

; and if

ker

⁡

(

f

)

=

N

, then

h

is

injective

.

Proof.

We’ll first show the uniqueness. Let

h

1

,

h

2

:

G

/

N

→

H

functions

such that

h

1

∘

φ

=

h

2

∘

φ

. For an

element

y

in

G

/

N

there exists an element

x

in

G

such that

φ

⁢

(

x

)

=

y

, so we have

h

1

⁢

(

y

)

=

(

h

1

∘

φ

)

⁢

(

x

)

=

(

h

2

∘

φ

)

⁢

(

x

)

=

h

2

⁢

(

y

)

for all

y

∈

G

/

N

, thus

h

1

=

h

2

.

Now we define

h

:

G

/

N

→

H

,

h

⁢

(

g

⁢

N

)

=

f

⁢

(

g

)

⁢

∀

g

∈

G

. We must check that the definition is of the given representative; so let

g

⁢

N

=

k

⁢

N

, or

k

∈

g

⁢

N

. Since

N

is a subset of

ker

⁡

(

f

)

,

g

-

1

⁢

k

∈

N

implies

g

-

1

⁢

k

∈

ker

⁡

(

f

)

, hence

f

⁢

(

g

)

=

f

⁢

(

k

)

. Clearly

h

∘

φ

=

f

.

Since

x

∈

ker

⁡

(

f

)

if and only if

h

⁢

(

x

⁢

N

)

=

1

H

, we have

ker

⁡

(

h

)

=

{

x

⁢

N

∣

x

∈

ker

⁡

(

f

)

}

=

ker

⁡

(

f

)

/

N

.

∎

A

consequence

of this is: If

f

:

G

→

H

is onto with

ker

⁡

(

f

)

=

N

, then

G

/

N

and

H

are

isomorphic

.

Title

fundamental homomorphism theorem

Canonical name

FundamentalHomomorphismTheorem

Date of creation

2013-03-22 15:35:06

Last modified on

2013-03-22 15:35:06

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Theorem

Classification

msc 20A05

Generated on Fri Feb 9 19:17:20 2018 by

LaTeXML
generating set of a group

generating set of a group

Let

G

be a group.

A subset

X

⊆

G

is said to

generate

G

(or to be a

generating set

of

G

)
if no

proper subgroup

of

G

contains

X

.

A subset

X

⊆

G

generates

G

if and only if
every element of

G

can be expressed as
a

product

of elements of

X

and

inverses

of elements of

X

(taking the empty product to be the

identity element

).
A subset

X

⊆

G

is said to be

closed under

inverses

if

x

-

1

∈

X

whenever

x

∈

X

;
if a generating set

X

of

G

is closed under inverses,
then every element of

G

is a product of elements of

X

.

A group that has a generating set with only one

element

is called a

cyclic group

.
A group that has a generating set with only finitely many elements
is called a

finitely generated group

.

If

X

is an arbitrary subset of

G

,
then the

subgroup

of

G

generated by

X

, denoted by

⟨

X

⟩

,
is the smallest subgroup of

G

that contains

X

.

The

generating rank

of

G

is
the minimum

cardinality

of a generating set of

G

.
(This is sometimes just called the

rank

of

G

, but this can
cause confusion with other meanings of the term

rank

.)
If

G

is

uncountable

, then its generating rank is simply

|

G

|

.

Title

generating set of a group

Canonical name

GeneratingSetOfAGroup

Date of creation

2013-03-22 15:37:14

Last modified on

2013-03-22 15:37:14

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

7

Author

yark (2760)

Entry type

Definition

Classification

msc 20A05

Classification

msc 20F05

Synonym

generating set

Related topic

Presentationgroup

Related topic

Generator

Defines

generate

Defines

generates

Defines

generated by

Defines

subgroup generated by

Defines

generating rank

Defines

closed under inverses

Defines

group generated by

Generated on Fri Feb 9 19:17:22 2018 by

LaTeXML
generator

generator

If

G

is a

cyclic group

and

g

∈

G

, then

g

is a

generator

of

G

if

⟨

g

⟩

=

G

.

All infinite cyclic groups have exactly

2

generators. To see this, let

G

be an infinite cyclic group and

g

be a generator of

G

. Let

z

∈

ℤ

such that

g

z

is a generator of

G

. Then

⟨

g

z

⟩

=

G

. Then

g

∈

G

=

⟨

g

z

⟩

. Thus, there exists

n

∈

ℤ

with

g

=

(

g

z

)

n

=

g

n

⁢

z

. Therefore,

g

n

⁢

z

-

1

=

e

G

. Since

G

is

infinite

and

|

g

|

=

|

⟨

g

⟩

|

=

|

G

|

must be

infinity

,

n

⁢

z

-

1

=

0

. Since

n

⁢

z

=

1

and

n

and

z

are

integers

, either

n

=

z

=

1

or

n

=

z

=

-

1

. It follows that the only generators of

G

are

g

and

g

-

1

.

A finite cyclic group of order

n

has exactly

φ

⁢

(

n

)

generators, where

φ

is the

Euler totient function

. To see this, let

G

be a finite cyclic group of order

n

and

g

be a generator of

G

. Then

|

g

|

=

|

⟨

g

⟩

|

=

|

G

|

=

n

. Let

z

∈

ℤ

such that

g

z

is a generator of

G

. By the

division algorithm

, there exist

q

,

r

∈

ℤ

with

0

≤

r

<

n

such that

z

=

q

⁢

n

+

r

. Thus,

g

z

=

g

q

⁢

n

+

r

=

g

q

⁢

n

⁢

g

r

=

(

g

n

)

q

⁢

g

r

=

(

e

G

)

q

⁢

g

r

=

e

G

⁢

g

r

=

g

r

. Since

g

r

is a generator of

G

, it must be the case that

⟨

g

r

⟩

=

G

. Thus,

n

=

|

G

|

=

|

⟨

g

r

⟩

|

=

|

g

r

|

=

|

g

|

gcd

⁡

(

r

,

|

g

|

)

=

n

gcd

⁡

(

r

,

n

)

. Therefore,

gcd

⁡

(

r

,

n

)

=

1

, and the result follows.

Title

generator

Canonical

name

Generator

Date of creation

2013-03-22 13:30:39

Last modified on

2013-03-22 13:30:39

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 20A05

Related topic

GeneratingSetOfAGroup

Related topic

ProperGeneratorTheorem

Generated on Fri Feb 9 19:17:24 2018 by

LaTeXML