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.

A translation lattice is the additive group generated by linearly independent vectors $v_1,\dots ,v_r$ in a Euclidean space. It defines a discrete subgroup of translations. A full-rank translation lattice has a compact parallelepiped as a fundamental region and therefore a cocompact group action. This geometric usage is distinct from an order-theoretic lattice.