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:

shift-coordinate-frame.eps

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.