FC-group

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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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:

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

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle [G:C_{G}(x)]} 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 $H\leq G$ are groups, and $C$ is a conjugacy class in $H$ , then all elements of $C$ are conjugate in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} , and hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is contained inside a conjugacy class in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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