central product of groups

central product of groups

1

Definitions

A

central decomposition

is a set

ℋ

of

subgroups

of a group

G

where

1.

for

𝒥

⊆

ℋ

,

G

=

⟨

𝒥

⟩

if, and only if,

𝒥

=

ℋ

, and

2.

[

H

,

⟨

ℋ

-

{

H

}

⟩

]

=

1

for all

H

∈

ℋ

.

A group

G

is

centrally

indecomposable

if its only central decomposition is

{

G

}

.
A central decomposition is

fully refined

if its members are centrally indecomposable.

Remark 1

.

Condition 1 is often relaxed to

G

=

⟨

H

⟩

but this
has the negative affect of allowing, for example,

R

2

to have the central decomposition
such sets as

{

⟨

(

1

,

0

)

⟩

,

⟨

(

0

,

1

)

⟩

,

⟨

(

1

,

1

)

⟩

,

0

,

R

2

}

and
in general a decomposition of any possible size. By impossing 1, we then restrict
the central decompositions of

R

2

to direct decompositions. Furthermore, with condition 1,
the meaning of indecomposable is easily had.

A

central product

is a group

G

=

(

∏

H

∈

ℋ

H

)

/

N

where

N

is a

normal subgroup

of

∏

H

∈

ℋ

H

and

H

∩

N

=

1

for all

H

∈

ℋ

.

Proposition 2

.

Every finite central decomposition

H

is a central product of the members in

H

.

Proof.

Suppose that

ℋ

is a a finite central decomposition of

G

. Then
define

π

:

∏

H

∈

ℋ

H

→

G

by

(

x

H

:

H

∈

ℋ

)

↦

∏

H

∈

ℋ

x

H

.
Then

G

=

(

∏

H

∈

ℋ

H

)

/

ker

⁡

π

. Furthermore,

H

∩

ker

⁡

π

=

1

for
each

direct factor

H

of

∏

K

∈

ℋ

K

. Thus,

G

is a central product of

ℋ

.
∎

2

Examples

1.

Every

direct product

is also a central product and so also every direct decomposition is a
central decomposition. The converse is generally false.

2.

Let

E

=

{

[

1

α

γ

0

1

β

0

0

1

]

:

α

,

β

,

γ

∈

K

}

, for a field

K

. Then

G

is a centrally indecomposable group.

3.

If

F

=

{

[

1

α

1

…

α

n

γ

0

1

0

β

1

⋮

⋱

0

⋮

1

β

n

1

]

:

α

i

,

β

i

,

γ

∈

K

,

1

≤

i

≤

n

}

and

H

i

=

{

A

∈

F

:

α

j

=

0

=

β

j

⁢

∀

j

≠

i

,

1

≤

j

≤

n

}

then

{

H

1

,

…

,

H

n

}

is a central decomposition of

G

. Furthermore, each

H

i

is

isomorphic

to

E

and so

ℋ

is a fully refined central decomposition.

4.

If

D

8

=

⟨

a

,

b

|

a

4

,

b

2

,

(

a

⁢

b

)

2

⟩

– the

dihedral group

of order 8,
and

Q

8

=

⟨

i

,

j

|

i

4

,

i

2

=

j

2

,

i

j

=

i

-

1

⟩

– the

quaternion group

of order

8

, then

D

8

∘

D

8

=

D

8

×

D

8

/

⟨

(

a

2

,

a

2

)

⟩

is isomorphic to

Q

8

∘

Q

8

=

(

Q

8

×

Q

8

)

/

⟨

(

i

2

,

i

2

)

⟩

; yet,

D

8

and

Q

8

are nonisomorphic and
centrally indecomposable. In particular, central decompositions are not unique even up to

automorphisms

.
This is in contrast the well-known

Krull-Remak-Schmidt theorem

for direct products of groups.

3

History

The name

central product

appears to have been coined by Philip Hall

[

1

,

Section

3.2]

though the principal

concept

of such a

product

appears in earlier work (e.g.

[

2

, Theorem II]

).
Hall describes central products as “…the group obtained from the direct product by identifying
the centres of the direct factors…”. The modern definition clearly out grows this original version
as now centers may be only partially identified.

References

1

P. Hall,

Finite-by-nilpotent groups

, Proc. Camb. Phil. Soc., 52 (1956), 611-616.

2

B. H. Neumann, and H. Neumann,

A remark on generalized

free products

, J. London Math. Soc. 25 (1950), 202-204.

Title

central product of groups

Canonical name

CentralProductOfGroups

Date of creation

2013-03-22 18:49:45

Last modified on

2013-03-22 18:49:45

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

6

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 20E34

Synonym

central decomposition

Defines

central decomposition

Defines

central product

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