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

 

Relations in hyperreflection groups


Author: Erich W. Ellers
Journal: Proc. Amer. Math. Soc. 81 (1981), 167-171
MSC: Primary 20H15; Secondary 20F05, 20H20, 51F15
MathSciNet review: 593448
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Abstract: Hyperreflection groups $ {G_m}$ are generalizations of groups generated by reflections. A hyperreflection group is generated by hyperreflections if $ \dim V$ is finite. A hyperreflection is a simple mapping $ \sigma $ such that $ \det \sigma = \gamma $, where $ {\gamma ^m} = 1$. If the field of scalars is commutative, the order of $ \sigma $ is $ m$. Our main result states that every relation between hyperreflections and their inverses is a consequence of relations of lengths 2, 4, and $ m$. The most interesting special case occurs for $ m = 2$. Then our result refers to relations between reflections.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0593448-4
PII: S 0002-9939(1981)0593448-4
Article copyright: © Copyright 1981 American Mathematical Society