# FC-group

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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 properties
VIEW 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 |

## Definition

### Symbol-free definition

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

### Definition with symbols

A group $G$ is said to be an FC-group if for any element $x$ in $G$, 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 $C_G(x)$ has finite index in $G$, viz $[G:C_G(x)]$ is finite.

## 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 $H \le G$ are groups, and $C$ is a conjugacy class in $H$, then all elements of $C$ are conjugate in $G$, and hence $C$ is contained inside a conjugacy class in $G$.

### 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.