orbits of a normal subgroup are equal in size when the full group acts transitively

orbits of a normal subgroup are equal in size when the full group acts transitively

The following

theorem

proves that if a group acts transitively on a

finite set

, then any of the orbits of any

normal subgroup

are equal in size and the group acts transitively on them. We also derive an explicit formula for the size of each orbit and the number of orbits.

Theorem 1

.

Let

H

be a normal subgroup of

G

, and assume

G

acts transitively on the finite set

A

. Let

O

1

,

…

,

O

r

be the orbits of

H

on

A

. Then

1.

G

permutes the

𝒪

i

transitively (i.e. for each

g

∈

G

,

1

≤

j

≤

r

, there is

1

≤

k

≤

r

such that

g

⁢

𝒪

j

=

𝒪

k

, and for each

1

≤

j

,

k

≤

r

, there is

g

∈

G

such that

g

⁢

𝒪

j

=

𝒪

k

), and the

𝒪

i

all have the same

cardinality

.

2.

If

a

∈

𝒪

i

, then

|

𝒪

i

|

=

|

H

:

H

∩

G

a

|

and

r

=

|

G

:

H

G

a

|

.

Proof.

Note first that if

g

∈

G

,

a

∈

𝒪

i

, and

g

⋅

a

∈

𝒪

j

, then

g

⁢

𝒪

i

⊂

𝒪

j

. For suppose also

b

∈

𝒪

i

. Then since

a

,

b

are in the same

H

-orbit, we can choose

h

∈

H

such that

b

=

h

⋅

a

. Then

g

⋅

b

=

g

⋅

h

⋅

a

=

g

⋅

h

⋅

g

-

1

⋅

g

⋅

a

=

h

′

⋅

g

⋅

a

∈

h

′

⁢

𝒪

j

⊂

𝒪

j

since

H

is normal in

G

. Thus for each

g

∈

G

,

1

≤

j

≤

r

, there is

1

≤

k

≤

r

such that

g

⁢

𝒪

j

⊂

𝒪

k

.

Given

j

,

k

, choose

a

j

∈

𝒪

j

,

a

k

∈

𝒪

k

. Since

G

is

transitive

on

A

, we may choose

g

∈

G

such that

g

⋅

a

j

=

a

k

. It follows from the above that

g

⁢

𝒪

j

⊂

𝒪

k

.

To prove 1), given

j

,

k

, choose

g

such that

g

⁢

𝒪

j

⊂

𝒪

k

and

g

′

such that

g

′

⁢

𝒪

k

⊂

𝒪

j

. But then

|

𝒪

j

|

≤

|

𝒪

k

|

≤

|

𝒪

j

|

so that

|

𝒪

j

|

=

|

𝒪

k

|

and the subset relationships in the previous two paragraphs are actually set equality.

To prove 2), consider the following diagram:

\xymatrix

⁢

&

⁢

G

⁢

\ar

⁢

@

-

[

d

]

⁢

&

⁢

H

⁢

G

a

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

H

⁢

\ar

⁢

@

-

[

r

⁢

d

]

⁢

&

⁢

&

⁢

G

a

⁢

\ar

⁢

@

-

[

l

⁢

d

]

⁢

&

⁢

H

∩

G

a

Clearly

H

∩

G

a

=

H

a

, and

|

H

:

H

a

|

=

|

𝒪

i

|

by the

orbit-stabilizer theorem

. Using the

second isomorphism theorem

for groups, we then have

|

𝒪

i

|

=

|

H

:

H

a

|

=

|

H

:

H

∩

G

a

|

=

|

H

G

a

:

G

a

|

But

|

G

|

=

r

⁢

|

𝒪

i

|

by the above, so

r

|

𝒪

i

|

=

|

G

|

=

|

G

:

H

G

a

|

⋅

|

H

G

a

:

G

a

|

=

|

G

:

H

G

a

|

⋅

|

𝒪

i

|

and the result follows.
∎

Title

orbits of a normal subgroup are equal in size when the full group acts transitively

Canonical name

OrbitsOfANormalSubgroupAreEqualInSizeWhenTheFullGroupActsTransitively

Date of creation

2013-03-22 17:17:56

Last modified on

2013-03-22 17:17:56

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 20M30

Generated on Fri Feb 9 19:34:35 2018 by

LaTeXML