Math 331 - April 12, 1999

Homomorphisms and Isomorphisms of Groups

• Definition. A homomorphism from a group G to a group G' is a map h : G -----> G' that obeys the rule that

  h(alpha \circ beta) = h(alpha) \circ h(beta)

  for all alpha, beta in G.

• Proposition & Definition. If a homomorphism from a group G to a group G' is an invertible map, then its inverse is a homomorphism from G' to G. An invertible homomorphism is called an isomorphism.

• Examples.

  1. If theta is a real number, and if rho(theta) denotes the rotation of the plane R^{2} about its origin through the angle of radian measure theta, then a homomorphism from the group of translations of the line R to the group of isometries of the plane is given by

     {translation by theta } ---> rho(theta) .

  2. If S and T are sets and phi : S --> T is a bijection, then an isomorphism from the group of permutations of S to the group of permutations of T is furnished by sending a permutation sigma of S to the permutation phi\circ sigma\circ phi^{-1}.

  3. An isomorphism from the group S_{3} of all permutations of three objects to the dihedral group D_{3} (consisting of all isometries of the plane that permute the vertices of the equilateral triangle inscribed in the unit circle with a vertex at the point (1, 0)) is provided by sending a permutation to the unique affine transformation of the plane that effects that permutation of the vertices of the triangle.

  4. If f is an affine transformation of R^{2}, say f(x) = M x + b, let I(f) to be the linear transformation of R^{3} given by

     (x, t) ---> (M x + t b, t) .

     Then I, i.e., the map f -> I(f), is an injective homomorphism from the group of all affine transformations of R^{2} to the group of affine transformations of R^{3} that fix the origin. (What is the matrix of I(f) ?)

• There is a notion of abstract group that generalizes the notion of a group of transformations. In this generalization the elements of a group G may be members of any set, rather than members of the set of transformations of a set, and there is an undefined operation, perhaps denoted by \circ or simply by juxtaposition. One assumes that the operation is associative, that G contains an element 1 = 1_{G} satisfying g\circ 1 = 1\circ g = g for all g and that for each element g in G there is an inverse i(g) satisfying g\circ i(g) = i(g)\circ g = 1_{G}. The notions of homomorphism and isomorphism, as above, make sense in this broader context. One picks up obvious examples based on arithmetic.

Assignment for Wednesday, April 14

1. Let L be the plane lattice consisting of all points (x, y) where x is an integer and y is an even integer. Find the group of all isometries of the plane that leave L invariant.

2. Show that matrix multiplication regarded as an abstract group operation on the set of all invertible n \times n matrices yields an abstract group GL_{n}(R) that is isomorphic to the group of all affine transformations of R^{n} that fix the origin.

3. Bearing in mind that the set of translations of n-dimensional affine space A^{n} is a vector space, show that if f is any affine transformation of A^{n}, the map lambda_{f} from the vector space of translations to itself given by conjugating a translation tau by f, i.e.,

   lambda_{f}(tau) = f \circ tau \circ f^{-1} ,

   is an invertible linear map.

4. (Continuation.) Show that the map

   lambda : f ---> lambda_{f}

   is a homomorphism from the group of affine transformations of the affine space A^{n} to the group of invertible linear maps of its vector space of translations.