Normal subgroups generated by a single pure element in quaternion algebras

Abstract

Let D be a quaternion division algebra whose center is an arbitrary infinite field K of characteristic ≠2, and let e ∈ D be a pure quaternion. Hence, by definition, e ∈ D ∖ K and e 2 ∈ K . We show that if the characteristic of K is >2, then D × / 〈 e D × 〉 is abelian-by-nilpotent-by-abelian. Note that by [A.S. Rapinchuk, L. Rowen, Y. Segev, Nonabelian free subgroups in homomorphic images of valued quaternion division algebras, Proc. Amer. Math. Soc., in press] this result is false in characteristic zero. As a consequence we show that the Whitehead group W ( G , k ) , where G is an absolutely simple simply connected algebraic group of type D 4 3 , 6 defined over a field k of odd characteristic and of k-rank 1, is abelian-by-nilpotent-by-abelian.