calculus of subgroup orders

calculus of subgroup orders

Recall that for any group

G

and any

subgroup

H

of

G

we can define left and/or

right cosets

of

H

in

G

. As the

cardinality

of

left cosets

equals the cardinality of right cosets, the index, denoted

[

G

:

H

]

is well-defined as this cardinality.

The

theorem of Lagrange

is the first of many basic

theorems

on the

calculus

of indices of subgroups and it can be stated as follows:

|

G

|

=

[

G

:

H

]

|

H

|

.

(1)

When

G

is a

finite group

then we can rewrite this in the familiar form (as actually proved by LaGrange)

|

G

|

/

|

H

|

=

[

G

:

H

]

.

(2)

and so conclude the familiar statement: “The order of every subgroup divides the order of the group.”

As the proof of (

2

) can be written with

bijections

instead of specific integer values, the proof of (

1

) is immediate from the usual proof of the Lagrange’s theorem.

The first corollary to the theorem states that given

K

≤

H

≤

G

then

[

G

:

K

]

=

[

G

:

H

]

[

H

:

K

]

.

(3)

So when

G

is finite we have

[

G

:

K

]

[

H

:

K

]

=

[

G

:

H

]

.

(4)

This should be contrasted with the

third isomorphism theorem

which claims if

K

and

H

are normal in

G

then

G

/

K

H

/

K

≅

G

/

H

.

(5)

Remark 1

.

It is preferrable to express the various

equations

relating indices of subgroups with

multiplications

. This is to allow for

infinite

indices, as we can multiply

cardinal numbers

, but we may not always be able to make sense of cardinal number

division

.

When only finite groups are considered so that division is allowed, expressing the theorems as

quotients

is often easier to understand.

Next suppose

H

and

K

are any two subgroups of

G

then we define

H

⁢

K

:=

{

h

⁢

k

:

h

∈

H

,

k

∈

K

}

(sometimes called the

complex

of

H

and

K

.)

Caution:

it is not always true that

H

⁢

K

is a subgroup of

G

. It is true if either

H

or

K

is a

normal subgroup

of

G

and occassionally it is true even without

H

or

K

being normal – for example when

H

⁢

K

=

K

⁢

H

so called

permutable

subgroups.

If

H

and

K

are

finite subgroups

then we can express this as:

|

H

K

|

/

|

K

|

=

[

H

:

H

∩

K

]

(6)

Once again if we have normality, say

K

is normal in

G

, then this is mimicks

second isomorphism theorem

:

H

⁢

K

/

K

≅

H

/

H

∩

K

.

(7)

Notice that if

H

⁢

K

is a subset of the subgroup

⟨

H

,

K

⟩

≤

G

. So it is possible to state (

6

) with all subgroups using

inequalities

such as

[

H

:

H

∩

K

]

≤

[

G

:

K

]

(8)

even when all the groups are infinite. Furthermore, if

H

⁢

K

is a subgroup of

G

then we can write

[

H

:

H

∩

K

]

=

[

H

K

:

K

]

(9)

to apply even for

infinite groups

. This equation is often called the

parallelogram law

for groups because it can be described with with the following picture:

{xy}

<

5

m

m

,

0

m

m

>

:

<

0

m

m

,

10

m

m

>

:

:

(

0

,

5

)

*

+

G

=

"

G

"

;

(

0

,

4

)

*

+

H

K

=

"

H

K

"

;

(

-

2

,

3

)

*

+

H

=

"

H

"

;

(

1

,

2

)

*

+

K

=

"

K

"

;

(

-

1

,

1

)

*

+

H

∩

K

=

"

H

i

K

"

;

(

-

1

,

0

)

*

+

1

=

"

1

"

;

"

G

"

;

"

H

K

"

*

*

@

-

;

"

H

K

"

;

"

H

"

*

*

@

-

;

"

H

K

"

;

"

K

"

*

*

@

-

;

"

H

"

;

"

H

i

K

"

*

*

@

-

;

"

K

"

;

"

H

i

K

"

*

*

@

-

;

"

H

i

K

"

;

"

1

"

*

*

@

-

;

Note the diagram is a Hasse diagram of the

lattice of subgroups

of

G

. We further inforce a policy of drawing edges of the same length if the index of the corresponding subgroups are equal. Thus (

9

) is simply a proof that the picture is accurate: opposite sides of a

parallelogram

are

congruent

.

The most common use of index calculus is for subgroups of finite index in

G

. This allows one to solve for indices from given

assumptions

. It is also quite common to prove certain

configurations

of subgroups are impossible as the indices are relatively prime.

Title

calculus of subgroup orders

Canonical name

CalculusOfSubgroupOrders

Date of creation

2013-03-22 15:48:12

Last modified on

2013-03-22 15:48:12

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

8

Author

Algeboy (12884)

Entry type

Application

Classification

msc 20D99

Synonym

Theorem of Lagrange

Defines

permutable subgroups

Defines

complex of subgroups

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