# Affine isomorphism

### From Online Dictionary of Crystallography

Each symmetry operation of a crystallographic group in *E*^{3} may be represented by a 3×3 matrix **W** (the *linear part*) and a vector **w**. Two crystallographic groups *G*_{1} = {(**W**_{1i},**w**_{1i})} and *G*_{2} = {(**W**_{2i},**w**_{2i})} are called **affine isomorphic** if there exists a non-singular 3×3 matrix **A** and a vector **a** such that:

*G*_{2} = {(**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 *E*^{3}. Two affine isomorphic groups are always isomorphic.