group actions and homomorphisms

group actions and homomorphisms

Notes on

group actions and homomorphisms

Let

G

be a group,

X

a non-empty set and

S

X

the

symmetric group

of

X

,
i.e. the group of all

bijective

maps on

X

.

⋅

may denote a left

group
action

of

G

on

X

.

1.

For each

g

∈

G

and

x

∈

X

we define

f

g

:

X

→

X

,

x

↦

g

⋅

x

⁢

.

Since

f

g

-

⁢

1

⁢

(

f

g

⁢

(

x

)

)

=

g

-

1

⋅

(

g

⋅

x

)

=

x

for each

x

∈

X

,

f

g

-

⁢

1

is the

inverse

of

f

g

. so

f

g

is bijective and thus element of

S

X

. We define

F

:

G

→

S

X

,

F

⁢

(

g

)

=

f

g

for all

g

∈

G

. This

mapping

is a

group homomorphism

: Let

g

,

h

∈

G

,

x

∈

X

. Then

F

⁢

(

g

⁢

h

)

⁢

(

x

)

=

f

g

⁢

h

⁢

(

x

)

=

(

g

⁢

h

)

⋅

x

=

g

⋅

(

h

⋅

x

)

=

(

f

g

∘

f

h

)

⁢

(

x

)

=

(

F

⁢

(

g

)

∘

F

⁢

(

h

)

)

⁢

(

x

)

for all

x

∈

X

implies

F

⁢

(

g

⁢

h

)

=

F

⁢

(

g

)

∘

F

⁢

(

h

)

. — The same is obviously true for a right
group action.

2.

Now let

F

:

G

→

S

x

be a group homomorphism, and let

f

:

G

×

X

→

X

,

(

g

,

x

)

↦

F

⁢

(

g

)

⁢

(

x

)

satisfy

(a)

f

⁢

(

1

G

,

x

)

=

F

⁢

(

1

g

)

⁢

(

x

)

=

x

for all

x

∈

X

and

(b)

f

(

g

h

,

x

)

=

F

(

g

h

)

(

x

)

=

(

F

(

g

)

∘

F

(

h

)

(

x

)

=

F

(

g

)

(

F

(

h

)

(

x

)

)

=

f

(

g

,

f

(

h

,

x

)

)

,

so

f

is a group action induced by

F

.

Characterization of group actions

Let

G

be a group acting on a set

X

.
Using the same notation as above, we have for each

g

∈

ker

⁡

(

F

)

F

⁢

(

g

)

=

id

x

=

f

g

⇔

g

⋅

x

=

x

,

∀

x

∈

X

⇔

g

∈

∪

x

∈

X

G

x

(1)

and it follows

ker

⁡

(

F

)

=

⋂

x

∈

X

G

x

.

Let

G

act transitively on

X

. Then for any

x

∈

X

,

X

is the
orbit

G

⁢

(

x

)

of

x

. As shown in “

conjugate stabilizer subgroups

’, all

stabilizer

subgroups

of

elements

y

∈

G

⁢

(

x

)

are

conjugate subgroups

to

G

x

in

G

. From
the above it follows that

ker

⁡

(

F

)

=

⋂

g

∈

G

g

⁢

G

x

⁢

g

-

1

.

For a

faithful

operation

of

G

the condition

g

⋅

x

=

x

,

∀

x

∈

X

→

g

=

1

G

is

equivalent

to

ker

⁡

(

F

)

=

{

1

G

}

and therefore

F

:

G

→

S

X

is a

monomorphism

.

For the

trivial

operation of

G

on

X

given by

g

⋅

x

=

x

,

∀

g

∈

G

the stabilizer subgroup

G

x

is

G

for all

x

∈

X

, and thus

ker

⁡

(

F

)

=

G

.

If the operation of

G

on

X

is free, then

G

x

=

{

1

G

}

,

∀

x

∈

X

, thus the kernel of

F

is

{

1

G

}

–like for a faithful operation. But:

Let

X

=

{

1

,

…

,

n

}

and

G

=

S

n

. Then the operation of

G

on

X

given by

π

⋅

i

:=

π

⁢

(

i

)

,

∀

i

∈

X

,

π

∈

S

n

is faithful but not free.

Title

group actions and homomorphisms

Canonical name

GroupActionsAndHomomorphisms

Date of creation

2013-03-22 13:18:48

Last modified on

2013-03-22 13:18:48

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

15

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 20A05

Related topic

GroupHomomorphism

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