Definition. If f and g are transformations of a set X, the commutator of f and g is the transformation f g f^{-1} g^{-1}. Sometimes the commutator of f and g is denoted by [f, g].
A group of transformations is commutative if and only if all commutators of transformations in that group are equal to the identity.
While [f, g]^{-1} = [g, f] and, therefore, the inverse of any commutator is always a commutator, it is not necessarily true that the product of two commutators is a commutator.
Definition. A transformation f of R^{n} is called a similarity if there is a positive scalar r such that for all points x, y in R^{n} one has the distance relation
The number r is called the scaling factor of f.
Proposition. The similarities of R^{n} form a group. When two similarities are composed, the scaling factor of the composite similarity is the product of the scaling factors of the two original similarities.
Theorem. A transformation f of R^{n} is a similarity if and only if in an affine coordinate system it is given by the formula
where r is a positive scalar, U is an n \times n orthogonal matrix, and b is a point of R^{n}.
Given three lines a, b, and c in the plane that intersect in pairs so as to delimit a triangle, construct a point p and a line m so that the product of the reflections in the three given lines satisfies the equation
where sigma_{m} is reflection in the line m and h_{p} is the half turn about the point p.
Show that the commutator of two affine transformations of R^{n} must always be orientation-preserving and volume-preserving but need not be an isometry.
Show that the commutator of two plane reflections in parallel lines is always a translation and that every translation may be obtained this way.
Show that the commutator of two plane reflections in intersecting lines is always a rotation and that every rotation may be obtained in this way.
Show that the commutator of any two similarities of R^{n} must be an isometry.
How many isometries of R^{2} permute the vertices of a given regular hexagon? Note that they form a group. Which of these are commutators in that group?
Show that for r > 0 and c a given point of R^{n} the formula
defines a similarity f with scaling factor r that has c as a fixed point and that commutes with every affine transformation of R^{n} that has c as a fixed point. (Note that inasmuch as this definition of f involves a barycentric combination of c and x, the transformation f thereby defined does not depend on the choice of an affine coordinate system. Use this observation to simplify your argument.)