A stronger form of Neumann's BFC-theorem

摘要

Given a group G, we write x(G) for the conjugacy class of G containing the element x. A famous theorem of B. H. Neumann states that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group G ' is finite. We establish the following result. Let n be a positive integer and K a subgroup of a group G such that divide x(G) divide <= n for each x is an element of K. Let H = < K-G & rang; be the normal closure of K. Then the order of the derived group H ' is finite and n-bounded. Some corollaries of this result are also discussed.