The following is based on Lévy-LeBlond’s Galilei Group and Galilean Invariance, §2 (Nuovo Cimento, Jan. 1973).
Let Ω be the complete Newtonian space, the points (events) of which we label by their coordinates in some complete Galilean frame, using the notation
y = (x(t), z(s)). (1)
The complete proper Galilei group G (or Galilei group for short) is a group of linear endomorphisms of Ω, of the form
y′ = (x′(t), z′(s))
which are
t′ = t + g, (2a)
s′ = s + h, (2b)
x′ = Rx + vt + a, (2c)
z′ = Pz + ws + b, (2d)
where R and P are 3 × 3 orthogonal matrices. The physical interpretation is straight-forward: The point y′ is obtained from y by a time translation g, a freelength translation h, a length-space translation a, a duration-space translation b, a length-space Galilean transformation v, a duration-space Galilean transformation w, a length-space rotation R and a duration-space rotation P.
The length-space Galilean transformation consists in a uniform motion with velocity v. The duration-space Galilean transformation consists in a uniform motion with lenticity w. It is important to notice that a length-space Galilean transformation is effected at a very definite instant of time, t = 0, and a duration-space Galilean transformation is effected at a very definite point of freelength, s = 0, in the parameterization of (2).
Instead of this active point of view, one can adopt the equivalent passive point of view; (2) would then give the relation between the coordinates of the same event as described in two different complete Galilean frames.
The complete Galilei group thus appears as a subgroup of the affine group. Under such a general Galilean transformation, the time interval between any two events is invariant,
t2 − t1 = const, (3)
and the freelength interval between any two events is invariant,
s2 − s1 = const. (4)
This also applies to the length-space distance between simultaneous events,
|x2 − x1| = const if t2 = t1. (5)
and the length-space distance between simulindistant events,
|z2 − z1| = const if s2 = s1. (6)
Conversely, the complete Galilei group is the most general linear group of transformations in Ω, such that (3) to (6) hold true. The restriction to linear transformations is crucial since otherwise any “rigid transformation”
t′ = t + g, Rx + f(t), (7)
where f is an arbitrary function of time, obeys (3) and (5). Similarly, without the restriction to linear transformations, any “rigid transformation”
s′ = s + h, Pz + k(s), (8)
where k is an arbitrary function of freelength, obeys (4) and (6).
The complete Galilei group G also has as a subgroup the Euclidean group E of length-space displacements and duration-space dischronments.