On the critical group of the missing Moore graph

TL;DR

This paper investigates the structure of the critical group of a hypothetical Moore graph with specific parameters, providing constraints on its Sylow p-subgroups and linking the 5-rank of the Laplacian to the group's form.

Contribution

It establishes the Sylow p-subgroup structure for the critical group and relates the 5-rank of the Laplacian to the group's possible configurations.

Findings

All Sylow p-subgroups are elementary abelian except for p=5.

The 5-rank of the Laplacian determines the critical group up to two options.

Constraints on the critical group's structure for the hypothetical Moore graph.

Abstract

We consider the critical group of a hypothetical Moore graph of diameter $2$ and valency $57$. Determining this group is equivalent to finding the Smith normal form of the Laplacian matrix of such a graph. We show that all of the Sylow $p$-subgroups of the critical group must be elementary abelian with the exception of $p = 5$. We prove that the $5$-rank of the Laplacian matrix determines the critical group up to two possibilities.