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

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.