On the automorphism group of the m-coloured random graph

Abstract

Let $R_m$ be the (unique) universal homogeneous $m$-edge-coloured countable complete graph ($m\ge2$), and $G_m$ its group of colour-preserving automorphisms. The group $G_m$ was shown to be simple by John Truss. We examine the automorphism group of $G_m$, and show that it is the group of permutations of $R_m$ which induce permutations on the colours, and hence an extension of $G_m$ by the symmetric group of degree $m$. We show further that the extension splits if and only if $m$ is odd, and in the case where $m$ is even and not divisible by~$8$ we find the smallest supplement for $G_m$ in its automorphism group. (This unpublished paper from 2007 is placed here because of renewed interest in the topic.)