Homomorphism

f(x \cdot y) = f(x) \cdot f(y)

A homomorphism is a transformation of a first set into a second set that preserves in the second set the relations between elements of the first set. We define four symmetry operations below, with the rotation being a 180 degree rotation.

Rotation = R_z = \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & +1 \end{bmatrix}

Reflection (xz) = \sigma_{xz} = \begin{bmatrix} +1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & +1 \end{bmatrix}

Reflection(yz) = \sigma_{yz} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & +1 & 0 \\ 0 & 0 & +1 \end{bmatrix}

Identity = E = \begin{bmatrix} +1 & 0 & 0 \\ 0 & +1 & 0 \\ 0 & 0 & +1 \end{bmatrix}

  • The first set is the four symmetry operations, represented by symbols
  • The second set is the three matrices

We will use the these in an example. If we take a point and a 180 degree rotation around the z-axis and then we do a reflection across the xz plane, we get the same result we would get if we had done a a reflection across the yz plane.

\sigma_{yz} = \sigma_{xz} R_z

Like wise, if we multiply the matrix for a 180 degree z-axis rotation by the matrix for a reflection across the xz plane, the result is the matrix for the a reflection across the yz plane.

\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

Now we use the above math to provide an example for the equation given at the beginning:

f(\sigma_{xz} \cdot R_z) = f(\sigma_{xz}) f(\cdot R_z)

f(\sigma_{yz}) = f(\sigma_{xz}) \cdot f(R_z)

\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

\begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

A homomorphism that works both ways is an Isomorphism. A good homework assignment would be to do this starting with the matrices and working to the symbols.