1 Some theorems
Every subgroup (http://planetmath.org/Subgroup) of an FC-group is an FC-group.
Every homomorphic image of an FC-group is an FC-group.
Every restricted direct product of FC-groups is an FC-group.
Every periodic FC-group is locally finite (http://planetmath.org/LocallyFiniteGroup).
If is a finitely generated FC-group, then and are both finite.
Every FC-group is a subdirect product of a periodic FC-group and a torsion-free abelian group.
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 such that every conjugacy class of elements of has at most elements, for some fixed integer . B. H. Neumann showed that is a BFC-group if and only if its commutator subgroup is finite (which in turn is easily shown to be equivalent to being finite-by-abelian, that is, having a finite normal subgroup such that is abelian).
A centre-by-finite (or central-by-finite) group is a group such that the central quotient is finite. A centre-by-finite group is necessarily a BFC-group, because the centralizer of any element contains the centre.
- 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.
|Date of creation||2013-03-22 14:52:28|
|Last modified on||2013-03-22 14:52:28|
|Last modified by||yark (2760)|
|Defines||locally normal group|
|Defines||locally finite and normal|
|Defines||locally finite and normal group|