# FC-group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

*This property makes sense for infinite groups. For finite groups, it is always true*

This is a variation of Abelianness|Find other variations of Abelianness |

## Contents

## Definition

### Symbol-free definition

A group is said to be a **FC-group** if it satisfies the following equivalent conditions:

- Every conjugacy class in it is finite
- The centralizer of any element is a subgroup of finite index

### Definition with symbols

A group is said to be an **FC-group** if for any element in , the following equivalent conditions are satisfied:

- There are only finitely many elements in its conjugacy class, that is, every element has only finitely many conjugates.
- The centralizer has finite index in , viz is finite.

## Relation with other properties

### Stronger properties

### Weaker properties

## Metaproperties

### Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property

View a complete list of subgroup-closed group properties

Any subgroup of a FC-group is a FC-group. This follows from the fact that if are groups, and is a conjugacy class in , then all elements of are conjugate in , and hence is contained inside a conjugacy class in .

### Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property

View other direct product-closed group properties

A direct product of FC-groups is an FC-group. This follows from the fact that the equivalence relation of being conjugate is closed under direct products.

## Study of this notion

### Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F24

The subject classification 20F24 is used for FC-groups, and their generalizations.