Abstract: This talk will start with some facts about symmetries in general relativity theory, for example, the idea of a Killing vector field and the concepts of orbit and isotropy in symmetry theory. Some results will be proved regarding the relationships between the dimensions of the Killing algebra and the corresponding orbits and isotropies. Next the general physical idea behind cosmological symmetry will be discussed and the above result will be applied to obtain, in a convenient way, the usual FRWL models used in cosmology. Next, the general symmetries associated with wave motion and, in particular, with plane waves in general relativity theory, will be described and which include the idea of a wave surface. Then the general results given at the beginning of the lecture will be used to give a very short convenient definition of a plane wave. Finally, the idea of homothetic symmetry will be introduced and it will be shown how one simple feature of it leads to an alternative definition of a plane wave.