Automorphism groups of self-complementary vertex-transitive graphs (Q2800094)

A self-complementary graph is a graph admitting a bijective involution on the vertices that maps edges to non-edges. In this paper, the authors study the automorphism groups of self-complementary vertex-transitive graphs. In particular, they consider what groups can appear as sections of these automorphism groups. A section of a group \(G\) is a group \(H/N\), where \(N \triangleleft H \leq G\).NEWLINENEWLINEFor a self-complementary graph \(\Gamma\) of order \(n\), they show that if \(n\) is not divisible by a fourth power and \(\mathrm{Aut}(\Gamma)\) admits a nonabelian simple group \(T\) as a section, then \(T \cong A_5\), \(T \cong A_6\), or \(T \cong\mathrm{PSL}(2,7)\); if \(n\) is not divisible by a cube and \(\mathrm{Aut}(\Gamma)\) admits a nonabelian simple group \(T\) as a section, then \(T \cong A_5\); and if \(n\) is squarefree then \(\mathrm{Aut}(\Gamma)\) is solvable. The second of these points is closely related to a result of \textit{C. H. Li} et al. [J. Algebr. Comb. 40, No. 4, 1135--1144 (2014; Zbl 1304.05071)] in which it was proved that the only possible non-solvable composition factor of the automorphism group a self-complementary metacirculant is \(A_5\).NEWLINENEWLINEHowever, they also show that for any collection \(T_1, \dots, T_n\) of simple groups, there are infinitely many self-complementary vertex-transitive graphs such that \(T_1\times \cdots \times T_n\) is a subgroup of each automorphism group. In particular, every nonabelian simple group is a section and in fact a subgroup of the automorphism group of infinitely many self-complementary vertex-transitive graphs.