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The group of rotations in a plane over GF(2n)

Published online by Cambridge University Press:  26 February 2010

D. W. Crowe
Affiliation:
University College, Ibadan, Nigeria.
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Extract

Archbold [1] has shown how a “distance” can be denned in an affine plane over the field GF(2n) of 2n elements. In terms of this distance, he has shown how to define a group, R(2, 2n), of 2×2 “rotational” matrices which have certain properties of ordinary orthogonal matrices. In the present note we find a standard form for such matrices. Using this standard form, we show that the order of R(2, 2n) is 2n+1+2 and that it has a “proper rotational” subgroup, R+(2, 2n), of index 2. The multiples of R+(2, 2n) by elements of GF(2n) are shown to form a field, which is necessarily isomorphic to GF(22n). The groups R+(2, 2n) and R(2, 2n) are then shown to be cyclic and dihedral groups respectively.

Type
Research Article
Copyright
Copyright © University College London 1962

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References

1. Archbold, J. W., “A metric for plane affine geometry over GF(2n)”, Mathematika, 7 (1960), 145148.CrossRefGoogle Scholar
2. Segre, B., Lectures on modern geometry (Rome, 1961).Google Scholar