exponent

exponent

Let

G

be a

group with

the that there exists a

positive integer

n

such that, for every

g

∈

G

,

g

n

=

e

G

. The

exponent

of

G

, denoted

exp

⁡

G

, is the smallest positive integer

m

such that, for every

g

∈

G

,

g

m

=

e

G

. Thus, for every

finite group

G

,

exp

⁡

G

divides

|

G

|

. Also, for every group

G

that has an exponent and for every

g

∈

G

,

|

g

|

divides

exp

⁡

G

.

The

concept

of exponent for finite groups is to that of characterisic (

http://planetmath.org/

Characteristic

) for

finite fields

.

If

G

is a finite abelian group, then there exists

g

∈

G

with

|

g

|

=

exp

⁡

G

. As a result of the fundamental theorem of finite abelian groups (

http://planetmath.org/FundamentalTheoremOfFinitelyGeneratedAbelianGroups

), there exist

a

1

,

…

,

a

n

with

a

i

dividing

a

i

+

1

for every

integer

i

between 1 and

n

such that

G

≅

ℤ

a

1

⊕

⋯

⊕

ℤ

a

n

. Since, for every

c

∈

G

,

c

a

n

=

e

G

, then

exp

⁡

G

≤

a

n

. Since

|

(

0

,

…

,

0

,

1

)

|

=

a

n

, it follows that

exp

⁡

G

=

a

n

.

Following are some examples of exponents of finite

nonabelian groups

.

Since

|

(

12

)

|

=

2

,

|

(

123

)

|

=

3

, and

|

S

3

|

=

6

, it follows that

exp

⁡

S

3

=

6

.

In

Q

8

=

{

1

,

-

1

,

i

,

-

i

,

j

,

-

j

,

k

,

-

k

}

, the ring of quaternions of order eight, since

|

i

|

=

|

-

i

|

=

|

j

|

=

|

-

j

|

=

|

k

|

=

|

-

k

|

=

4

and

1

4

=

(

-

1

)

4

=

1

, it follows that

exp

⁡

Q

8

=

4

.

Since the order of a

product

of two

disjoint

transpositions

is 2, the order of a three cycle is 3, and the only nonidentity elements of

A

4

are three cycles and products of two disjoint transpositions, it follows that

exp

⁡

A

4

=

6

.

Since

|

(

123

)

|

=

3

and

|

(

1234

)

|

=

4

,

exp

⁡

S

4

≥

12

. Since

S

4

has no elements of order 8, it cannot have an element of order 24. It follows that

exp

⁡

S

4

=

12

.

Following are some examples of exponents of

infinite groups

.

Clearly, any infinite group that has an element of infinite order does not have an exponent. On the other hand, just because an infinite group has the that every

element

has finite order does not that the group has an exponent. As an example, consider

G

=

ℚ

/

ℤ

, which is a group under

addition

. Despite that all of its elements have finite order,

G

does not have an exponent. This is because, for every positive integer

n

,

G

has an element of order

n

, namely

1

n

+

ℤ

.

On the other hand, some infinite groups have exponents. For example, let

𝔽

2

denote the field having two elements. Then

𝔽

2

⁢

[

x

]

, the ring of all

polynomials

in

x

with coefficients in

𝔽

2

, is an

abelian group

under addition. Moreover, it is an infinite group; however, every nonzero element has order

2

. Thus,

exp

⁡

𝔽

2

⁢

[

x

]

=

2

.

Title

exponent

Canonical

name

Exponent

Date of creation

2013-03-22 13:30:21

Last modified on

2013-03-22 13:30:21

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

23

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 20A99

Related topic

KummerTheory

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