Generalized group determinant gives a necessary and sufficient condition for a subset of a finite group to be a subgroup

TL;DR

This paper generalizes the group determinant concept to establish a necessary and sufficient condition for a subset of a finite group to be a subgroup, extending previous work on group isomorphisms.

Contribution

It introduces a new criterion based on the generalized group determinant to identify subgroups within finite groups, expanding the theoretical framework of group theory.

Findings

Derived a necessary and sufficient condition for subgroup identification

Extended the group determinant characterization to subsets

Generalized previous isomorphism results based on group determinants

Abstract

We generalize the concept of the group determinant and prove a necessary and sufficient novel condition for a subset to be a subgroup. This development is based on the group determinant work by Edward Formanek, David Sibley, and Richard Mansfield, where they show that two groups with the same group determinant are isomorphic. The derived condition leads to a generalization of this result.