Quadratic Forms, Rigid Elements and Nonreal Preorders

A nonreal preorder of a quaternionic structure q: G × G → B is a subgroup$T \subseteq G$such that -1 ∈ T and -1 ≠ t ∈ T implies$D\langle 1, t \rangle \subseteq T$. The basic part of q is defined to be the set$B = \{\pm1 \} \cup \{a \in G\mid a\quad\text{is not 2-sided rigid}\quad\}$. A. Carson and M...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 1983-06, Vol.88 (2), p.201-204
Main Authors: Szymiczek, Kazimierz, Yucas, Joseph
Format: Article
Language:English
Subjects:
Cassava
Mathematical induction
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Summary:A nonreal preorder of a quaternionic structure q: G × G → B is a subgroup$T \subseteq G$such that -1 ∈ T and -1 ≠ t ∈ T implies$D\langle 1, t \rangle \subseteq T$. The basic part of q is defined to be the set$B = \{\pm1 \} \cup \{a \in G\mid a\quad\text{is not 2-sided rigid}\quad\}$. A. Carson and M. Marshall have shown that if$|G| < \infty$then every nontrivial nonreal preorder T must contain B. The main purpose of this note is to extend this result by replacing$|G| < \infty$with$\lbrack G: T \rbrack < \infty$.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-1983-0695240-0