BFC-group

Symbol-free definition
A group is termed a BFC-group if there is a finite constant $$d$$ such that no element in the group has more than $$d$$ distinct conjugates. Such a group is also termed a $$d$$-BFC-group.

Any Abelian group is a 1-BFC-group.

Stronger properties

 * Finite group
 * Abelian group
 * FZ-group

Weaker properties

 * FC-group

Metaproperties
Any subgroup of a BFC-group is a BFC-group. In fact, any subgroup of a $$d$$-BFC group is a $$d$$-BFC group. This follows from the fact that if two elements in the subgroup are conjugate in the subgroup, they are conjugate in the whole group. Hence, any conjugacy class in the subgroup is contained in a conjugacy class in the whole group.

A direct product of BFC-groups is a BFC-group. In fact, a direct product of a $$d_1$$-BFC-group and a $$d_2$$-BFC-group is a $$d_1d_2$$-BFC-group. This follows from the fact that if two elements in the direct product of $$G_1$$ and $$G_2$$ are conjugate, then their $$G_1$$-coordinates are conjugate and their $$G_2$$-coordinates are conjugate. Hence, conjugacy classes in $$G_1 \times G_2$$ are simply the pairwise direct products of conjugacy classes in $$G_1$$ and $$G_2$$.