Some Completely Reducible Linear Groups over a Quaternion Division Ring Containing a Root Subgroup

Abstract

Let F be a field of characteristic ≠2, D be a quaternion division algebra over F, and Q be a subgroup of the additive group of D which satisfies the following two conditions:

1.	Q contains a subfield k of F such that D is algebraic over k.
2.	Q ⊈ F.

Let n be an integer, n ≥ 2. We study the completely reducible subgroups of GL _n (D) that comprise a conjugate in GL _n (D) of the group of all matrices diag(, 1,…, 1) ∈ GL _n (D), a ∈ Q.

Key Words:

• Quaternion division ring
• Linear group

Acknowledgments