# FC not implies BFC

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., FC-group) neednotsatisfy the second group property (i.e., BFC-group)

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## Statement

It is possible to have a FC-group (i.e., every conjugacy class in is finite) that is not a BFC-group (i.e., there is no finite upper bound on the sizes of conjugacy classes).

## Proof

## Proof

Suppose is a finite group that is not abelian, so the derived subgroup of is nontrivial. Let be the restricted external direct product of a countably infinite number of copies of . We can verify that:

- is a FC-group: any element of has all but finitely many coordinates equal to the identity element of , and therefore all its conjugates all have those coordinates as the identity element, so there can be only finitely many conjugates.
- There is no finite upper bound on the sizes of conjugacy classes in : Let be a non-central element, with a conjugacy class of size . Then, an element with coordinates that are copies of , and the remaining coordinates the identity element, has conjugates. As can be arbitrarily large, this allows for arbitrarily large conjugacy class sizes.