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

 

Multiplicative subgroups of finite index in a division ring


Author: Gerhard Turnwald
Journal: Proc. Amer. Math. Soc. 120 (1994), 377-381
MSC: Primary 12E99; Secondary 05D10, 11T99, 12E15
MathSciNet review: 1215206
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Abstract: If $ G$ is a subgroup of finite index $ n$ in the multiplicative group of a division ring $ F$ then $ G - G = F$ or $ \vert F\vert < {(n - 1)^4} + 4n$. For infinite $ F$ this is derived from the Hales-Jewett theorem. If $ \vert F\vert > {(n - 1)^2}$ and $ - 1$ is a sum of elements of $ G$ then every element of $ F$ has this property; the bound $ {(n - 1)^2}$ is optimal for infinitely many $ n$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1215206-9
PII: S 0002-9939(1994)1215206-9
Keywords: Multiplicative subgroup, field, division ring, Hales-Jewett theorem
Article copyright: © Copyright 1994 American Mathematical Society