Definition. If G is a group of transformations of a set X, then the set of all g in G that fix a given point x in X is itself a group called the isotropy group of G at the point X.
Theorem. If L is a vector lattice in a finite-dimensional vector space V, and if x_{1}, x_{2}, ... x_{n} denote coordinates with respect to a basis of the lattice, i.e., if the lattice consists of all points for which the coordinates x_{1}, x_{2}, ..., x_{n} are all integers, then the affine transformations of V that leave L invariant are precisely the transformations represented in the given L-compatible coordinate system by x -> M x + c where c is an integer vector and M is an integer matrix with determinant {+/-} 1. Note. It is not assumed here that the coordinate axes are mutually perpendicular lines in a Euclidean vector space. Thus, it is not geometrically meaningful to ask if the matrix M is orthogonal.
Proof. Let G_{L} denote the group of affine transformations that leave L invariant. Certainly all transformations of the form x -> x + c with integer coordinate vector c leave the lattice invariant. Likewise, all of the transformations x -> M x are included when M is an integer matrix with determinant {+/-} 1. Since G_{L} is a group, it follows that all of the x -> M x + c are in G_{L}, and, thus, we are faced with the converse question of whether every element of G_{L} has this explicit form. It is clear that the set of transformations of the explicit form x -> M x + c is itself a group, which will be denoted by H, that is a subgroup of G_{L}. Let f be given in G_{L}. Then certainly f(0) is in L, and f followed by the translation x -> x - f(0) (1) is in G_{L} and (2) is in H if f is in H. That is, we are led to the observation, if the theorem is true, that the isotropy group of G_{L} at the origin must be equal to the isotropy group of H at the origin, which is the set of transformations of the form x -> M x with M as above, and that, conversely, if every element of G_{L} that fixes the origin has this form, then the theorem is true. So we suppose that f fixes the origin. Then f must be linear, and, therefore, f(x) = M x for some invertible real matrix M. Since f carries integer vectors to integer vectors, the columns of the identity matrix must be carried to integer vectors, which means that M must be an integer matrix. Since f(L) is not just contained in L but is equal to L (by the hypothesis of invariance), the inverse transformation must have the same property as f, and, therefore, M^{-1} is also an integer matrix. Since the product of the two integers can only be 1 if the two integers are {+/-} 1, we see that the determinants of M and M^{-1} must be {+/-} 1. (Cramer's rule shows that the inverse of an integer matrix with determinant {+/-} 1 must be an integer matrix; this needed to see that H is a group.)
Prove that the group of all orientation-preserving isometries of R^{n} is the smallest subgroup of itself that contains:
all translations of R^{n}.
all rotations about the origin of R^{n}.
Prove that any subgroup of the group of all transformations of R^{n} that contains
all translations of R^{n},
all rotations about the origin of R^{n}, and
a single reflection
Given a square show that the group of its symmetries, i.e, the group of plane isometries leaving invariant the set of its four vertices (which we know to ``to be'' the dihedral group D_{4}) is the smallest group of isometries of the plane containing the following four reflections: the reflections in its two diagonals and the reflections in the perpendicular bisectors of its opposite sides.
Given a square what is the smallest group of isometries that contains the reflection in one of its diagonals and the reflection in one of the perpendicular bisectors of its sides?
Given a square identify a plane lattice with the property that the group of isometries of the plane leaving the lattice invariant is the smallest subgroup of the group of isometries of the plane that contains the following eight reflections: the four reflections in the sides of the square, the reflections in its two diagonals, and the reflections in the two perpendicular bisectors of its opposite sides.
What subgroup of the group leaving invariant the lattice of the previous exercise is the smallest group containing the four reflections in the sides of the given square?