Realizing a finite group as a subgroup of a product of two groups of permutation matrices

TL;DR

This paper demonstrates that any finite group can be represented as a subgroup of a product of two permutation matrix groups, using a specific matrix equation and row permutation techniques.

Contribution

It introduces a novel method to realize any finite group as a subgroup of permutation matrices via a matrix equation involving permutation matrices.

Findings

Any finite group of order n can be embedded as solutions to a matrix equation.

The group is isomorphic to a subgroup of permutation matrices obtained by row permutations.

The approach generalizes Cayley's theorem using matrix and permutation techniques.

Abstract

In this paper we prove that any finite group of order $n$ can be viewed as the group of the solutions of a certain matrix equation $XB=BY$, where the unknowns $X,Y$ are two permutation matrices of order $n$ and $(1+k)n+2 $ respectively and where $k\in \Bbb N$ is given by Cayley's theorem. Moreover, we show that $G$ is isomorphic to a certain subgroup formed by permutation matrices of order $(1+k)n$ obtained by permuting all the rows of the identity matrix $I_{(1+k)n}$.