This paper studies a family of groups of linear transformations, G(c), of the plane onto itself. If the parameter, c, is zero or infinity, the group reduces to the Galilean group in one space and one time dimension. If c is real, not zero, G is the corresponding Lorentz group, with light velocity c. If c is imaginary, the group is SO(2). The approach of this paper is to identify the plane with an algebra, ''dual numbers'', which is actually a subalgebra of the Pauli algebra. The author discusses the physical interpretation of his formalism and generalizes it to the full four dimensions, essentially by extending the subalgebra to the full Pauli algebra. Appropriate comparisons are made to quaternions, Clifford algebras and the SL(2,\({\mathbb{C}})\) covering group of the Lorentz group.