properties of group commutators and commutator subgroups

properties of group commutators and commutator subgroups

The purpose of this entry is to collect properties of

http://planetmath.org/node/2812

group
commutators and

commutator subgroups

. Feel free to add more theorems!

Let

G

be a group.

Theorem 1

.

Let

x

,

y

∈

G

, then

[

x

,

y

]

-

1

=

[

y

,

x

]

.

Proof.

Direct computation yields

[

x

,

y

]

-

1

=

(

x

-

1

⁢

y

-

1

⁢

x

⁢

y

)

-

1

=

y

-

1

⁢

x

-

1

⁢

y

⁢

x

=

[

y

,

x

]

.

∎

Theorem 2

.

Let

X

,

Y

be subsets of

G

, then

[

X

,

Y

]

=

[

Y

,

X

]

.

Proof.

By Theorem

1

, the elements from

[

X

,

Y

]

or

[

Y

,

X

]

are products of commutators of the form

[

x

,

y

]

or

[

y

,

x

]

with

x

∈

X

and

y

∈

Y

.
∎

Theorem 3

(Hall–Witt identity).

Let

x

,

y

,

z

∈

G

, then

y

-

1

⁢

[

x

,

y

-

1

,

z

]

⁢

y

⁢

z

-

1

⁢

[

y

,

z

-

1

,

x

]

⁢

z

⁢

x

-

1

⁢

[

z

,

x

-

1

,

y

]

⁢

x

=

1

.

Proof.

This is mainly a brute-force calculation. We can easily calculate the
first factor

y

-

1

⁢

[

x

,

y

-

1

,

z

]

⁢

y

explicitly using
theorem

1

:

y

-

1

⁢

[

x

,

y

-

1

,

z

]

⁢

y

=

y

-

1

⁢

[

y

-

1

,

x

]

⁢

z

-

1

⁢

[

x

,

y

-

1

]

⁢

z

⁢

y

=

y

-

1

⁢

y

⁢

x

-

1

⁢

y

-

1

⁢

x

⁢

z

-

1

⁢

x

-

1

⁢

y

⁢

x

⁢

y

-

1

⁢

z

⁢

y

=

x

-

1

⁢

y

-

1

⁢

x

⁢

z

-

1

⁢

x

-

1

⁢

y

⁢

x

⁢

y

-

1

⁢

z

⁢

y

.

Let

h

1

:=

x

-

1

⁢

y

-

1

⁢

x

⁢

z

-

1

⁢

x

-

1

, the “first half” of

y

-

1

⁢

[

x

,

y

-

1

,

z

]

⁢

y

. Let

h

2

be the element obtained from

h

1

by
the cyclic shift

S

:

x

↦

y

↦

z

↦

x

, and

h

3

be
the element obtained from

h

2

by

S

. We have

h

2

-

1

=

(

y

-

1

⁢

z

-

1

⁢

y

⁢

x

-

1

⁢

y

-

1

)

-

1

=

y

⁢

x

⁢

y

-

1

⁢

z

⁢

y

which gives us

y

-

1

⁢

[

x

,

y

-

1

,

z

]

⁢

y

=

h

1

⁢

h

2

-

1

,

and, by applying

S

twice

z

-

1

⁢

[

y

,

z

-

1

,

x

]

⁢

z

=

h

2

⁢

h

3

-

1

,

x

-

1

⁢

[

z

,

x

-

1

,

y

]

⁢

x

=

h

3

⁢

h

1

-

1

.

In total, we have

y

-

1

⁢

[

x

,

y

-

1

,

z

]

⁢

y

⁢

z

-

1

⁢

[

y

,

z

-

1

,

x

]

⁢

z

⁢

x

-

1

⁢

[

z

,

x

-

1

,

y

]

⁢

x

=

h

1

⁢

h

2

-

1

⁢

h

2

⁢

h

3

-

1

⁢

h

3

⁢

h

1

-

1

=

1

.

∎

Theorem 4

(Three subgroup lemma).

Let

N

be a

normal subgroup

of

G

. Furthermore, let

X

,

Y

and

Z

be

subgroups

of

G

, such that

[

X

,

Y

,

Z

]

and

[

Y

,

Z

,

X

]

are contained
in

N

. Then

[

Z

,

X

,

Y

]

is contained in

N

as well.

Proof.

The group

[

Z

,

X

,

Y

]

is generated by all elements of the form

[

z

,

x

-

1

,

y

]

with

x

∈

X

,

y

∈

Y

and

z

∈

Z

. Since

N

is
normal,

y

-

1

⁢

[

x

,

y

-

1

,

z

]

⁢

y

and

x

-

1

⁢

[

z

,

x

-

1

,

y

]

⁢

x

are elements
of

N

. The Hall–Witt

identity

then implies that

x

-

1

⁢

[

z

,

x

-

1

,

y

]

⁢

x

is an element of

N

as well. Again, since

N

is normal,

[

z

,

x

-

1

,

y

]

∈

N

which concludes the proof.
∎

Theorem 5

.

For any

x

,

y

,

z

∈

G

we have

[

x

⁢

y

,

z

]

=

[

x

,

z

]

y

⁢

[

y

,

z

]

[

x

,

y

⁢

z

]

=

[

x

,

z

]

⁢

[

x

,

y

]

z

[

x

,

y

]

z

=

[

x

z

,

y

z

]

[

x

z

,

y

]

=

[

x

,

y

z

-

1

]

where

a

b

denotes

b

-

1

⁢

a

⁢

b

Proof.

By expanding:

[

x

⁢

y

,

z

]

=

y

-

1

⁢

x

-

1

⁢

z

-

1

⁢

x

⁢

y

⁢

z

=

y

-

1

⁢

x

-

1

⁢

z

-

1

⋅

x

⁢

z

⋅

z

-

1

⁢

x

-

1

⋅

x

⁢

y

⁢

z

=

y

-

1

⁢

[

x

,

z

]

⋅

y

⋅

y

-

1

⋅

z

-

1

⁢

x

-

1

⋅

x

⁢

y

⁢

z

=

[

x

,

z

]

y

⋅

y

-

1

⁢

z

-

1

⁢

y

⁢

z

=

[

x

,

z

]

y

⁢

[

y

,

z

]

The other identities are proved similarly.
∎

Title

properties of group commutators and commutator subgroups

Canonical name

PropertiesOfGroupCommutatorsAndCommutatorSubgroups

Date of creation

2013-03-22 15:30:50

Last modified on

2013-03-22 15:30:50

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

11

Author

GrafZahl (9234)

Entry type

Theorem

Classification

msc 20F12

Related topic

NormalSubgroup

Defines

Hall-Witt identity

Defines

three subgroup lemma

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