Translation subgroup
= Translation subgroup
The translation subgroup of a group of <Euclidean isometries> consists of its <translations>. It is a <normal subgroup>, because conjugation by $x\mapsto Ax+b$ takes translation by $v$ to translation by $Av$. Its vectors form an additive subgroup of Euclidean space; for a <plane crystallographic group> they form a rank-two <translation lattice>.