On Sylow subgraphs of vertex-transitive self-complementary graphs

One of the basic facts of group theory is that each finite group contains a Sylow p-subgroup for each prime p which divides the order of the group. In this note we show that each vertex-transitive self-complementary graph has an analogous property. As a consequence of this fact, we obtain that each prime divisor p of the order of a vertex-transitive self-complementary graph satisfies the congruence p^m ≡ 1(mod 4), where p^m is the highest power of p which divides the order of the graph.