Groups in which squares have boundedly many conjugates

Abstract

Abstract Given a group G, we write {x^{G}} for the conjugacy class of G containing the element x. A famous result 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^{\prime}} is finite. Recently we showed that if {|x^{G}|\leq n} for any commutator x, then {|G^{\prime\prime}|} is finite and n-bounded. If {|x^{G^{\prime}}|\leq n} for any commutator x, then {|\gamma_{3}(G^{\prime})|} is finite and n-bounded. The present article deals with groups in which the conjugacy classes containing squares are finite with bounded size. The following theorem is proved. Let n be a positive integer, G a group and H the subgroup generated by all squares in G. If {|x^{H}|\leq n} for any square {x\in G} , then the order of {\gamma_{3}(H)} is finite and n-bounded.