Burnside ring

Burnside ring

Let

G

be a

finite group

. Recall that by

G

-set

we understand a pair

(

X

,

∘

)

, where

X

is a set and

∘

:

G

×

X

→

X

is a

group action

of

G

on

X

. For short notation the pair notation will be omitted and

G

-sets will be simply denoted by capital letters.

Recall that for each

subgroup

H

⊆

G

we have canonical

G

-set

G

/

H

=

{

g

H

;

g

∈

G

}

where group action is defined as follows: for any

g

,

k

∈

G

we have

(

g

,

k

⁢

H

)

⟼

g

⁢

k

⁢

H

.

Let

X

and

Y

be

G

-sets. Recall that by

G

-map

from

X

to

Y

we understand any

function

F

:

X

→

Y

such that for any

g

∈

G

and

x

∈

X

we have

F

⁢

(

g

⁢

x

)

=

g

⁢

F

⁢

(

x

)

.

It is easy to see that family of all

G

-sets and

G

-maps forms a

category

(with standard comoposition). We shall denote this category by

G

-

𝕊

. Moreover, by

G

-

𝕊

0

we shall denote

full subcategory

of

G

-

𝕊

whose objects are all finite

G

-sets.

From

G

-sets

X

and

Y

one can

construct

another

G

-set in two interesting (from our point of view) ways, i.e. by taking

disjoint union

X

⊔

Y

with obvious group action and by taking

product

X

×

Y

with group action as follows:

(

g

,

(

x

,

y

)

)

⟼

(

g

⁢

x

,

g

⁢

y

)

.

Moreover it is clear that when

X

and

Y

are finite, so are

X

⊔

Y

and

X

×

Y

.

Consider a finite

G

-set

X

. Then there exist a

natural number

n

∈

ℕ

, finite family

{

H

i

}

i

=

1

n

of subgroups of

G

and an

isomorphism

(in

G

-

𝕊

0

category)

X

≃

∐

i

=

1

n

G

/

H

i

.

Therefore (since

G

is finite) family of isomorphism classes of

G

-

𝕊

0

forms a

countable set

.

Denote by

Ω

+

(

G

)

=

{

[

X

]

;

X

∈

G

-

𝕊

0

}

the set of isomorphism classes of category

G

-

𝕊

0

. Then one can turn

Ω

+

⁢

(

G

)

into a

semiring

as follows: for any finite

G

-sets

X

and

Y

define

[

X

]

+

[

Y

]

=

[

X

⊔

Y

]

;

[

X

]

⁢

[

Y

]

=

[

X

×

Y

]

.

Note that here we treat the

empty set

as a

G

-set (with one and unique group action), therefore

Ω

+

⁢

(

G

)

has

zero element

[

∅

]

(the other way is to formally add the zero to

Ω

+

⁢

(

G

)

- this is just technical thing).

Define by

Ω

⁢

(

G

)

=

K

⁢

(

Ω

+

⁢

(

G

)

)

the

Grothendieck group

of

(

Ω

+

⁢

(

G

)

,

+

)

. If

A

is an

abelian semigroup

and

f

:

A

×

A

→

A

is a

bilinear map

, then it can be uniquely extended to a bilinear map

K

⁢

(

f

)

:

K

⁢

(

A

)

×

K

⁢

(

A

)

→

K

⁢

(

A

)

, therefore

Ω

⁢

(

G

)

can be uniquely turned into a ring from

Ω

+

⁢

(

G

)

. This ring is called

the

Burnside ring

of

G

.

Some properties:

(

0

)

each element of

Ω

⁢

(

G

)

can be expressed as a formal diffrence

[

X

]

-

[

Y

]

;

(

1

)

Ω

⁢

(

G

)

is a

commutative

,

unital ring

, where

[

G

/

G

]

is the unity of

Ω

⁢

(

G

)

;

(

2

)

Ω

can be turned into a

contravariant functor

from the category of finite groups to the category of commutative, unital rings;

(

3

)

(

Ω

+

⁢

(

G

)

,

+

)

is a

cancellative semigroup

, therefore it embedds into

Ω

⁢

(

G

)

;

(

4

)

for the

trivial group

E

there is a ring isomorphism

Ω

⁢

(

E

)

≃

ℤ

;

(

5

)

for any group

G

there is a ring monomorphism

φ

:

Ω

⁢

(

G

)

→

⊕

i

=

1

n

ℤ

for some natural number

n

∈

ℕ

; this is called the

characteristic

embedding

;

(

6

)

for any two groups

G

,

H

we have: if

Ω

⁢

(

G

)

and

Ω

⁢

(

H

)

are isomorphic (as a rings), then

|

G

|

=

|

H

|

; generally

G

need not be isomorphic to

H

.

Title

Burnside ring

Canonical name

BurnsideRing

Date of creation

2013-03-22 18:08:02

Last modified on

2013-03-22 18:08:02

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

10

Author

joking (16130)

Entry type

Definition

Classification

msc 16S99

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