Groups with boundedly finite classes of conjugate elements

An n-BFC group is one in which no element has more than n conjugates, and some element has exactly n conjugates. A bound in terms of n for the order of the derived group of an n-BFC group is determined, and it is shown that the derived group of a p-BFC group is cyclic of order p, where p is prime. The case of 4-BFC groups is also considered.