Quadratic forms, rigid elements and nonreal preorders

Abstract:

A nonreal preorder of a quaternionic structure $q:G \times G \to B$ is a subgroup $T \subseteq G$ such that $- 1 \in T$ and $- 1 \ne t \in T$ implies $D\left \langle {1,t} \right \rangle \subseteq T$. The basic part of $q$ is defined to be the set $B = \left \{ { \pm 1} \right \} \cup \left \{ {a \in G|a\;{\text {is}}\;{\text {not}}\;2 - {\text {sided}}\;{\text {rigid}}} \right \}$. A. Carson and M. Marshall have shown that if $\left | G \right | < \infty$ then every nontrivial nonreal preorder $T$ must contain $B$. The main purpose of this note is to extend this result by replacing $\left | G \right | < \infty$ with $[G:T] < \infty$.