  By means of a digit function that has been introduced in a recent formulation
of classical and quantum mechanics, we provide a new construction of some
infinite families of finite groups, both abelian and nonabelian, of importance
for theoretical, atomic and molecular physics. Our construction is not based on
algebraic relationships satisfied by generators, but in establishing the
appropriate law of composition that induces the group structure on a finite set
of nonnegative integers (the cardinal of the set being equal to the order of
the group) thus making computations with finite groups quite straightforward.
We establish the abstract laws of composition for infinite families of finite
groups including all cyclic groups (and any direct sums of them), dihedral,
dicyclic and other metacyclic groups, the symmetric groups of permutations of
$p$ symbols and the alternating groups of even permutations. Specific examples
are given to illustrate the expressions for the law of composition obtained in
each case.
