Automorphisms of Strongly Regular Krein Graphs without Triangles

A strongly regular graph is called a Krein graph if, in one of the Krein conditions, an equality obtains for it. A strongly regular Krein graph $Kre(r)$ without triangles has parameters $((r^2+3r)^2,r^3+3r^2+r,0,r^2+r)$. It is known that $Kre(1)$ is a Klebsh graph, $Kre(2)$ is a Higman –Sims graph, and that a graph of type $Kre(3)$ does not exist. Let $G$ be the automorphism group of a hypothetical graph $\Gamma=Kre(5)$, $g$ be an element of odd prime order $p$ in $G$, and $\Omega=\operatorname{Fix}(g)$. It is proved that either $\Omega$ is the empty graph and $p=5$, or $\Omega$ is a one-vertex graph and $p=41$, or $\Omega$ is a $2$-clique and $p=17$, or $\Omega$ is the complete bipartite graph $K_{8,8}$, from which the maximal matching is removed, and $p=3$.