FC-group

FC-group

\PMlinkescapephrase

locally finite

An

FC-group

is a group in which every

element

has only finitely many

conjugates

. Equivalently, a group

G

is an FC-group if and only if the

centralizer

C

G

⁢

(

x

)

is of finite index in

G

for each

x

∈

G

.

All

finite groups

and all

abelian groups

are obviously FC-groups.
Further examples of FC-groups can be obtained by taking

restricted direct products

of such groups.

The term

FC-group

was introduced by Baer

[

1

]

;
the

FC

is simply a mnemonic for the

definition

involving finite

conjugacy classes

.

1

Some theorems

Theorem 1

.

Every

subgroup

(

http://planetmath.org/Subgroup

) of an FC-group is an FC-group.

Theorem 2

.

Every

homomorphic image

of an FC-group is an FC-group.

Theorem 3

.

Every restricted direct product of FC-groups is an FC-group.

Theorem 4

.

Every

periodic

FC-group is locally finite (

http://planetmath.org/LocallyFiniteGroup

).

Theorem 5

.

Let

G

be an FC-group.
The elements of

finite order

in

G

form a subgroup,
which will be denoted by

Tor

⁡

(

G

)

.
The subgroup

Tor

⁡

(

G

)

is a periodic FC-group,
and the

quotient

(

http://planetmath.org/QuotientGroup

)

G

/

Tor

⁡

(

G

)

is a torsion-free abelian group.

Corollary 1

.

Every

torsion-free

FC-group is

abelian

.

Theorem 6

.

If

G

is a

finitely generated

FC-group,
then

G

/

Z

⁢

(

G

)

and

Tor

⁡

(

G

)

are both finite.

Theorem 7

.

Every FC-group is a

subdirect product

of a periodic FC-group
and a torsion-free abelian group.

From

Theorem

4 above it follows that a group

G

is a periodic FC-group
if and only if every finite subset of

G

has a finite

normal closure

.
For this reason, periodic FC-groups are sometimes called

locally normal

(or

locally finite and normal

) groups.

Stronger properties

The following two

properties

are sometimes encountered,
both of which are somewhat stronger than being an FC-group.
For finitely generated groups they are in fact

equivalent

to being an FC-group,
by Theorem 6 above.

A

BFC-group

is a group

G

such that every conjugacy class of elements of

G

has at most

n

elements, for some fixed

integer

n

.
B. H. Neumann showed

[

2

]

that

G

is a BFC-group if and only if its

commutator subgroup

[

G

,

G

]

is finite
(which in turn is easily shown to be equivalent to

G

being

finite-by-abelian

, that is,
having a finite

normal subgroup

N

such that

G

/

N

is abelian).

A

centre-by-finite

(or

central-by-finite

) group
is a group

G

such that the

central quotient

G

/

Z

⁢

(

G

)

is finite.
A centre-by-finite group is necessarily a BFC-group,
because the centralizer of any element contains the centre.

References

1

R. Baer,

Finiteness properties of groups

,
Duke Math. J. 15 (1948), 1021–1032.

2

B. H. Neumann,

Groups covered by

permutable

subsets

,
J. London Math. Soc. 29 (1954), 236–248.

Title

FC-group

Canonical

name

FCgroup

Date of creation

2013-03-22 14:52:28

Last modified on

2013-03-22 14:52:28

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

22

Author

yark (2760)

Entry type

Definition

Classification

msc 20F24

Synonym

FC group

Defines

FC

Defines

locally normal

Defines

locally normal group

Defines

locally finite and normal

Defines

locally finite and normal group

Defines

BFC-group

Defines

BFC group

Defines

BFC

Defines

finite-by-abelian

Defines

finite-by-abelian group

Defines

centre-by-finite group

Defines

center-by-finite group

Defines

central-by-finite group

Defines

centre-by

Generated on Fri Feb 9 19:29:20 2018 by

LaTeXML