Affine isomorphism

Each symmetry operation of crystallographic group in E³ may be represented by a 3×3 matrix W (the linear part) and a vector w. Two crystallographic groups G₁ = {(W_1i,w_1i)} and G₂ = {(W_2i,w_2i)} are called affine isomorphic is there exists a non-singular 3×3 matrix A and a vector a such that:

G₂ = {(A,a)(W_1i,w_1i)(A,a)^-1}

Two crystallographic groups are affine isomorphic if and only if their arrangement of symmetry elements may be mapped onto each other by an affine mapping of E³. Two affine isomorphic groups are always isomorphic.

See also

Group isomorphism