The translation group is the set of all transformations that move or displace the orgin of a coordinate frame to a new location, forming a new coordinate fram . This can be visualized with the following graph:
from which we see that if we displace by the arbitrary vector , then:
[Aside: This can be written as a matrix to form a continous group using matrix multipication as the group composition, but doing so is tricky (it requires extending the dimension of by one) and we will leave it in this easily understood form, where it is hopefully obvious that the set of all such transformations (indeed, vector addition itself) form a group.]
One can easily prove that the transformations of this group leave displacement vectors unchanged. Newtonian mechanics are invariant under the action of this group provided that is either a constant or a linear function of time (inertial frame transformations) because in this case the group leaves acceleration unchanged.