a transformation group

Think of all the ways you can pick up a flat shape and set it down again without distorting it: slide it, spin it, flip it over. Each such move is a transformation — a rule that takes every point of the plane to a new point. A transformation group is a collection of such moves that is self-contained in a precise way: it is the toolbox of transformations a particular geometry is allowed to use.

To qualify as a group, the collection must satisfy three conditions, and they are exactly the conditions that make 'undoing and combining moves' work cleanly. First, doing nothing — the identity transformation that leaves every point where it is — must be in the collection. Second, every transformation in the collection must have its reverse also in the collection, so any move can be undone (a rotation by 30 degrees is undone by a rotation by minus 30 degrees). Third, performing one transformation and then another must give a transformation that is again in the collection (a translation followed by a translation is a translation). This third property is called closure under composition. Because composing functions is automatically associative, those three checks are all you need. The group of rigid motions of the plane, the group of similarities, the group of affine maps, and the group of projective transformations are all transformation groups, and they sit one inside another.

The reason geometry insists on a group rather than just any set of transformations is that the group laws are precisely what make 'invariant property' a sensible idea. If you could apply a transformation but not undo it, 'same figure' would not be a symmetric relation; if combining two allowed moves could fall outside the toolbox, the relation would not be transitive. The group axioms guarantee that 'A is equivalent to B' behaves like a genuine notion of sameness — reflexive, symmetric, transitive — which is exactly what a geometry needs to classify figures.