Free pairs of symmetric elements in normal subgroups of division rings

Abstract

Let D be a division ring with central subfield k of characteristic ≠2, let ⁎ be a k-involution of D, and let N⊲D• be a normal subgroup of the multiplicative group D• of D. We show that if G is a ⁎-stable nonabelian subgroup of N that is either torsion-free polycyclic-by-finite but not abelian-by-finite, or finite of odd order, then N contains a pair (u,v) of elements such that u⁎=u, v⁎=v and such that 〈u,v〉 is a noncyclic free group.One aspect of the above proof requires that we extend a theorem of Bergman on invariant ideals in commutative group algebras. This new result is surely of interest in its own right. It appears in the Appendix and can be read independently of the remainder of the paper.