blocks of permutation groups

blocks of permutation groups

Throughout this article,

A

is a set and

G

is a

permutation group

on

A

.

A

block

is a subset

B

of

A

such that for each

σ

∈

G

, either

σ

⋅

B

=

B

or

(

σ

⋅

B

)

∩

B

=

∅

, where

σ

⋅

B

=

{

σ

⁢

(

b

)

∣

b

∈

B

}

. In other words, if

σ

⋅

B

intersects

B

, then

σ

⋅

B

=

B

.

Note that for any such permutation group, each of

∅

,

A

, and every

element of

A

is a block. These are called

trivial blocks

.

It is obvious that if

H

⊂

G

are permutation groups on

A

, then any block of

G

is also a block of

H

.

Blocks are

closed under

finite

intersection

:

Theorem

.

If

B

1

,

B

2

⊂

A

are blocks of

G

, then

B

=

B

1

∩

B

2

is a block of

G

.

Proof.

Choose

σ

∈

G

.
Note that

σ

⋅

(

B

1

∩

B

2

)

=

(

σ

⋅

B

1

)

∩

(

σ

⋅

B

2

)

. Thus if

(

σ

⋅

B

)

∩

B

≠

∅

, then

(

σ

⋅

B

)

∩

B

=

(

σ

⋅

(

B

1

∩

B

2

)

)

∩

(

B

1

∩

B

2

)

=

(

σ

⋅

B

1

∩

B

1

)

∩

(

σ

⋅

B

2

∩

B

2

)

is nonempty, and thus

σ

⋅

B

i

∩

B

i

≠

∅

for

i

=

1

,

2

. But

B

1

and

B

2

are blocks, so that

σ

⋅

B

i

=

B

i

for

i

=

1

,

2

. Thus

σ

⋅

B

=

σ

⋅

(

B

1

∩

B

2

)

=

(

σ

⋅

B

1

)

∩

(

σ

⋅

B

2

)

=

B

1

∩

B

2

=

B

and

B

is a block.
∎

We show, as a corollary to the following

theorem

, that blocks themselves are permuted by the action of the group.

Theorem

.

If

H

⊂

G

are permutation groups on

A

,

B

⊂

A

is a block of

H

, and

σ

∈

G

, then

σ

⋅

B

is a block of

σ

⁢

H

⁢

σ

-

1

.

Proof.

Choose

τ

∈

H

and assume that

(

(

σ

⁢

τ

⁢

σ

-

1

)

⁢

σ

⋅

B

)

∩

σ

⋅

B

≠

∅

Then, applying

σ

-

1

to this

equation

, we see that

(

τ

⋅

B

)

∩

B

≠

∅

But

B

is a block of

H

, so

τ

⋅

B

=

B

. Multiplying by

σ

, we see that

σ

⋅

(

τ

⋅

B

)

=

σ

⋅

B

and thus

(

σ

⁢

τ

⁢

σ

-

1

)

⁢

σ

⋅

B

=

σ

⋅

B

and the result follows.
∎

Corollary

.

If

B

is a block of

G

,

σ

∈

G

, then

σ

⋅

B

is also a block of

G

.

Proof.

Set

G

=

H

in the above theorem.
∎

Definition

.

If

B

is a block of

G

,

σ

∈

G

, then

B

and

σ

⋅

B

are

conjugate blocks

. The set of all blocks

conjugate

to a given block is a

block system

.

It is clear from the fact that

B

is a block that conjugate blocks are either equal or

disjoint

, so the action of

G

permutes the blocks of

G

. Then if

G

acts transitively on

A

, the union of any nontrivial block and its conjugates is

A

.

Theorem

.

If

G

is finite and

G

acts transitively on

A

, then the size of a nonempty block divides the order of

G

.

Proof.

Since

G

acts transitively,

A

is finite as well. All conjugates of the block have the same size; since the action is

transitive

, the union of the block and all its conjugates is

A

. Thus the size of the block divides the size of

A

. Finally, by the

orbit-stabilizer theorem

, the order of

G

is divisible by the size of

A

.
∎

Title

blocks of permutation groups

Canonical name

BlocksOfPermutationGroups

Date of creation

2013-03-22 17:19:05

Last modified on

2013-03-22 17:19:05

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

15

Author

rm50 (10146)

Entry type

Topic

Classification

msc 20B05

Defines

trivial block

Defines

block

Defines

block system

Defines

conjugate block

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