Definition. A set G of transformations of a set X is called a transformation group if the following conditions are satisfied.
G contains the identity transformation.
If g_{1} and g_{2} are transformations in G, then the composition g_{2} \circ g_{1} is also in G.
If g is a transformation in G, then the inverse transformation, usually denoted g^{-1}, is also in G.
Definition. A subgroup of a transformation group is a subset of the group that is a transformation group by itself.
Definition. A transformation group is commutative if one has g_{1} \circ g_{2} = g_{2} \circ g_{1} for any pair g_{1}, g_{2} of its elements.
Definition. A subgroup H of a transformation group G is said to be a normal subgroup of G if every conjugate of an element of H by an element of G is an element of H, i.e., if g \circ h \circ g^{-1} is in H whenever h is in H and g is in G.
Examples of Plane Transformation Groups The following are transformation groups that are subgroups of the group of all affine transformations of R^{n}:
The group of all invertible linear transformations.
The group of all translations.
The group of all isometries.
The group of all orientation-preserving isometries.
The group of all isometries that fix a given point.
The group of all isometries that stabilize a given line
The group of all translations by vectors with integer coordinates.
Find examples of commutative plane transformation groups of orders 2, 4, 6, and 8.
Find plane transformation groups of orders 6 and 8 that are not commutative
Show that the group of all orientation-preserving affine transformations of R^{n} is a normal subgroup of the group of all affine transformations of R^{n}.
Show that the group of all rotations about a point in the plane is commutative.
Show that the group of all plane isometries that fix the origin is not commutative.
Show that the group of plane isometries is not a normal subgroup of the group of all affine transformations of the plane.
What is the smallest subgroup of the group of plane isometries that contains the three reflections in the sides of a given equilateral triangle?
Show that a reflection in the plane always commutes with a translation of the plane that is parallel to the axis of that reflection. Must a reflection commute with any translation? Must a reflection commute with any half-turn?
Show that the group of translations is a normal subgroup of the group of all affine transformations of R^{n}.