# FC-center

From Groupprops

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to FC-center, all facts related to FC-center) |Survey articles about this | Survey articles about definitions built on this

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## Definition

### Symbol-free definition

The **FC-center** of a group is the set of elements in the group whose conjugacy class is finite in size. Since the size of the conjugacy class of a product is bounded from above by the product of the sizes of the conjugacy classes, the FC-center is a subgroup.

## Property theory

### Fixed-point operator

A group is a fixed-point under the FC-center operator if and only if it is a FC-group.