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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 |


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.

Relation with other properties

Stronger properties

Weaker properties



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.

External links

Definition links