Digital calculus and finite groups in quantum mechanics

TL;DR

This paper introduces a novel method using a digit function to construct and compute finite groups relevant to quantum mechanics, simplifying group operations without relying on traditional algebraic generator relations.

Contribution

It presents a new approach to constructing finite groups via a digit function, enabling straightforward computation of group laws for various important groups in physics.

Findings

Constructed infinite families of finite groups including cyclic, dihedral, and symmetric groups.

Provided explicit laws of composition for each group family.

Demonstrated the method's applicability to groups relevant in quantum mechanics.

Abstract

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.