On the automorphism group of the m-coloured random graph

TL;DR

This paper investigates the automorphism group of the m-coloured random graph, revealing its structure as an extension of the automorphism group of the graph by the symmetric group, with splitting properties depending on the parity of m.

Contribution

It characterizes the automorphism group of the automorphism group of the m-coloured random graph and determines when the extension splits based on the parity of m.

Findings

The automorphism group of G_m is the group of permutations inducing permutations on colours.

The extension of G_m by the symmetric group splits if and only if m is odd.

For even m not divisible by 8, the smallest supplement of G_m in its automorphism group is identified.

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 (This unpublished paper from 2007 is placed here because of renewed interest in the topic.)