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.