On the automorphism group of the Aschbacher graph

Abstract

A Moore graph is a regular graph of degree k and diameter d with v vertices such that v ≤ 1 + k + k(k − 1) + ... + k(k − 1)d−1. It is known that a Moore graph of degree k ≥ 3 has diameter 2; i.e., it is strongly regular with parameters λ = 0, µ = 1, and v = k 2 + 1, where the degree k is equal to 3, 7, or 57. It is unknown whether there exists a Moore graph of degree k = 57. Aschbacher showed that a Moore graph with k = 57 is not a graph of rank 3. In this connection, we call a Moore graph with k = 57 the Aschbacher graph and investigate its automorphism group G without additional assumptions (earlier, it was assumed that G contains an involution).