How may we characterise the intrinsic structure of physical quantities such as mass, length, or electric charge? This paper shows that group-theoretic methods—specifically, the notion of a free and transitive group action—provide an elegant way of characterising the structure of scalar quantities, and uses this to give an intrinsic treatment of vector quantities. It also gives a general account of how different scalar or vector quantities may be algebraically combined with one another. Finally, it uses this apparatus to give a simple intrinsic treatment of Newtonian gravitation.