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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Quadratic forms, rigid elements and nonreal preorders


Authors: Kazimierz Szymiczek and Joseph Yucas
Journal: Proc. Amer. Math. Soc. 88 (1983), 201-204
MSC: Primary 10C05
MathSciNet review: 695240
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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\vert a\;{\text{is}}\;{\text{not}}\;2 - {\text{sided}}\;{\text{rigid}}} \right\}$. A. Carson and M. Marshall have shown that if $ \left\vert G \right\vert < \infty $ then every nontrivial nonreal preorder $ T$ must contain $ B$. The main purpose of this note is to extend this result by replacing $ \left\vert G \right\vert < \infty $ with $ [G:T] < \infty $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0695240-0
PII: S 0002-9939(1983)0695240-0
Article copyright: © Copyright 1983 American Mathematical Society