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.
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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.