or as a more general function on the object.) The group of isometries of space induces a group action on objects in it, and the symmetry group Sym(X) consists of those isometries which map X to itself (as well as mapping any further pattern to itself). We say X is invariant under such a mapping, and the mapping is a symmetry of X.

The above is sometimes called the full symmetry group of X to emphasize that it includes orientation-reversing isometries (reflections, glide reflections