Determination of the Finite Quaternion Groups

If we apply to a set of quaternions the definitioni of a group as given in the Theory of Substitutions, then a quaternion group of the mth order means a set of m quaternions (scalar unity always included) whose products and powers are also quaternions of the same set or group. The object of the present paper is to determine all the possible finite quaternion groups. These groups, or rather their analogues in the ordinary Theory of Functions, have usually been interpreted geometrically as the linear transformations of the plane of complex variables in itself-automorph1dc traisformnations. The forimulae for these linear transform-nations were first given by Professor G0ordan. in his paper "Ueber endliche Gruppen linearer Traiisforinationen einer Veriinderlichen." * The quaternion formulae, although they have their exact correspondents in Gordan's algebraic formulae, have, when interpreted geometrically, a more general character, in that they represent certain automorphic linear transtbrmations of a three-dimensional infinite homoloidal space, or what is the same thing, of a three-fold extended sphere. The determination of these formulae might be made to depend upon Gordan's solution, but the following is a shorter and simpler solution than that of Gordan, and includes it as a special case.