groups are frequently studied through "group representations". in their most general form, these consist of a choice of group, a set, and an "action" of the group on the set, that is, an operation which takes an element of the group and an element of the set and returns an element of the set. most often, the set is a vector space, and the group represents symmetries of the vector space. for example, there is a group which represents the rigid rotations of space. this is a type of symmetry of space, because space itself does not change when it is rotated even though the positions of objects in it do. noether used these sorts of symmetries in her work on invariants in physics.