On the prime spectrum of an automorphism group of an $AT4(p,p+2,r)$-graph

Abstract

This paper is devoted to the problem of classification of $AT4(p,p+2,r)$-graphs. There is a unique $AT4(p,p+2,r)$-graph with $p=2$, namely, the distance-transitive Soicher graph with intersection array $\{56,45,16,1;1,8,45,56\}$, whose local graphs are isomorphic to the Gewirtz graph. It is still unknown whether an $AT4(p,p+2,r)$-graph with $p>2$ exists. The local graphs of each $AT4(p,p+2,r)$-graph are strongly regular with parameters $((p+2)(p^2+4p+2),p(p+3),p-2,p)$. In the present paper, we find an upper bound for the prime spectrum of an automorphism group of a strongly regular graph with such parameters, and we also obtain some restrictions for the prime spectrum and the structure of an automorphism group of an $AT4(p,p+2,r)$-graph in case when $p$ is a prime power. As a corollary, we show that there are no arc-transitive $AT4(p,p+2,r)$-graphs with $p\in \{11,17,27\}$.

References in corpus (1)

• Finite simple groups with narrow prime spectrum