automorphism group of orbital graphs - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T16:28:33Z http://mathoverflow.net/feeds/question/117774 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/117774/automorphism-group-of-orbital-graphs automorphism group of orbital graphs majid arezoomand 2013-01-01T09:50:54Z 2013-01-02T17:24:49Z <p>Let $G$ be a transitive group on $\Omega$. Every orbits of $G$ on its natural action on $\Omega\times\Omega$ is called an orbital of $G$ on $\Omega$. For each orbital $\Delta$ of $G$ on $\Omega$, the orbital digraph $Graph(\Delta)$ is a digraph with vertex set $\Omega$ and edge set $\Delta$. Clearly, $G$ is a subgroup of automorphism group of $Graph(\Delta)$. Is there any paper or book that determine for which groups $G$, there is an orbital $\Delta$ that $G$ is equal to the automorphism group of $Graph(\Delta)$? Clearly, the action of $S_5$ on 2-subets of ${1,\ldots,5}$ is an example.</p> http://mathoverflow.net/questions/117774/automorphism-group-of-orbital-graphs/117788#117788 Answer by Dima Pasechnik for automorphism group of orbital graphs Dima Pasechnik 2013-01-01T17:01:15Z 2013-01-02T17:24:49Z <p>The property of $G$ you are looking at is called <em>2-closure</em>, i.e., you ask for a classification of 2-closed permutation groups. See the paper by Liebeck, Praeger, Saxl, On the 2-closures of finite permutation groups. <em>J. London Math. Soc.</em> (2) <strong>37</strong> (1988), no. 2, 241–252, where this question is investigated for a particular class of primitive permutation groups.</p> <p>More precisely, this is not 100% equivalent to your question (it would be, if you allowed graphs have coloured arcs), as there exist 2-closed permutation groups $G$ such that for each orbital the automorphism group of the underlying (di)graph is strictly bigger than $G$.</p> <p>In general such a classification is not known, and the problem is hopelessly hard, I think.</p>