The integer group determinants for the symmetric group of degree four

TL;DR

This paper determines all possible integer values of the group determinant for the symmetric group S4 when matrix entries are integers, providing a complete characterization of these determinants.

Contribution

It explicitly computes all integer group determinants for S4, a previously unresolved problem in algebraic combinatorics.

Findings

All integer values of the group determinant for S4 are identified.

The set of these determinants is explicitly characterized.

The results extend understanding of group determinants for non-abelian groups.

Abstract

For the symmetric group $S_4$ we determine all the integer values taken by its group determinant when the matrix entries are integers.