TY - JOUR

T1 - Normal subgroups generated by a single pure element in quaternion algebras

AU - Rowen, Louis

AU - Segev, Yoav

N1 - Funding Information:
* Corresponding author. E-mail addresses: rowen@macs.biu.ac.il (L. Rowen), yoavs@math.bgu.ac.il (Y. Segev). 1 Partially support by BSF grant no. 2004-083 and by the Israel Science Foundation Center of Excellence. 2 Partially supported by BSF grant no. 2004-083.

PY - 2006/11/1

Y1 - 2006/11/1

N2 - 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 {set minus} K and e2 ∈ K. We show that if the characteristic of K is >2, then D× / 〈 eD× 〉 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 3, 6D4 defined over a field k of odd characteristic and of k-rank 1, is abelian-by-nilpotent-by-abelian.

AB - 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 {set minus} K and e2 ∈ K. We show that if the characteristic of K is >2, then D× / 〈 eD× 〉 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 3, 6D4 defined over a field k of odd characteristic and of k-rank 1, is abelian-by-nilpotent-by-abelian.

KW - Pure quaternion

KW - Quaternion division algebra

KW - Whitehead group

UR - http://www.scopus.com/inward/record.url?scp=33748757158&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2005.11.015

DO - 10.1016/j.jalgebra.2005.11.015

M3 - Article

AN - SCOPUS:33748757158

VL - 305

SP - 130

EP - 145

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 1

ER -