Groups as Graphs

TL;DR

This paper introduces a novel way to represent finite groups as graphs called identity graphs, enabling visual analysis of group properties and substructures.

Contribution

It is the first work to represent every finite group as a graph, linking algebraic properties to graphical features for the first time.

Findings

Graph representation reveals the number of self-inversed elements.

Subgroups and normal subgroups are studied via the identity graph.

Conjugate elements and p-Sylow subgroups are analyzed using the graph.

Abstract

For the first time we represent every finite group in the form of a graph in this book. The authors choose to call these graphs as identity graph, since the main role in obtaining the graph is played by the identity element of the group. This study is innovative because through this description one can immediately look at the graph and say the number of elements in the group G which are self-inversed. Also study of different properties, like the subgroups of a group, normal subgroups of a group, p-sylow subgroups of a group and conjugate elements of a group are carried out using the identity graph of the group in this book. This book has four chapters. The first chapter is introductory. The second chapter represents groups as graphs. In the third chapter, we have defined similar types of graphs for algebraic structures like commutative semigroups, loops, commutative groupoids and commutative rings. The final chapter poses 52 problems.