This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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 is a variation of Abelianness|Find other variations of Abelianness |
A group is said to be a FC-group if it satisfies the following equivalent conditions:
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
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 .
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.