I think this is a quite general topic and not only interesting in the context of quantum theory. In quantum theory, however, this approach becomes very important to define observables.
The basic idea behind the group theoretical approach is to start from a symmetry principle. Most fundamental are the symmetries of the space-time description. In the case of classical Newtonian physics the space-time structure is determined by the proper orthochronous Galilei group, which is a semidirect product of translations in space and time, rotations in space, and boosts, making up the 10-dimensional Lie group of space-time symmetries.
Each one-parameter subgroup leads, according to Noether's theorems, to a conserved quantity for closed systems. Thus we have 10 conserved quantities for any closed system, i.e., energy (time translations), 3 momentum components (spatial translations), 3 angular momentum components (spatial rotations) and the three coordinates of the center of mass (boosts).
In analyzing the properties of the Galilei group in terms of its Lie algebra, you find a little subtlety: There is a non-trivial central charge, the total mass of the system, which then can be interpreted as an 11th conservation law for mass. This is important for quantum theory, since there only the corresponding central extension of the Galilei group's covering group (substituting the rotation group SO(3) by its covering group, SU(2)) gives a physically meaningful dynamics of quantum systems.
All this is pretty nicely written up in
Ballentine, Quantum Mechanics, World Scientific 1998.