## 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 {set minus} 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 ^{3, 6}D_{4} defined over a field k of odd characteristic and of k-rank 1, is abelian-by-nilpotent-by-abelian.

Original language | English |
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Pages (from-to) | 130-145 |

Number of pages | 16 |

Journal | Journal of Algebra |

Volume | 305 |

Issue number | 1 |

DOIs | |

State | Published - 1 Nov 2006 |

### Bibliographical note

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.

## Keywords

- Pure quaternion
- Quaternion division algebra
- Whitehead group