ABSTRACT
This chapter explores the continuous symmetries of space and time. It introduce these symmetries in their defining representation, acting on the base space of spacetime coordinates. Rotations and translations are the symmetries of nonrelativistic space. The rotation group is, in a sense, the drosophila within linear groups and provides enough structure to illustrate the central notions of Lie group theory. The Euclidean group is the symmetry group of classical affine space. The rotation group does not need to act necessarily on 3-vectors. Euclidean transformations can be made linear, when people view them in a 4-dimensional space with the vector representation. It is interesting to note that the commutator of a rotation and a translation corresponds to a pure translation. This corresponds to the fact that the generator of translations behaves as a vector under rotations.
