On automorphism groups of circulant digraphs of square-free order

We show that the full automorphism group of a circulant digraph of square-free order is either the intersection of two 2-closed groups, each of which is the wreath product of 2-closed groups of smaller degree, or contains a transitive normal subgroup which is the direct product of two 2-closed groups of smaller degree. The work in this paper makes contributions to the solutions of two problems in graph theory. The most general, known as the König problem, asks for a concrete characterization of all automorphism groups of graphs. While it is known that every group is isomorphic to the automorphism group of a graph [12], determining the concrete characterization seems intractable. Thus, the natural approach is to consider either certain classes of graphs, or certain classes of groups. The second problem considered in this paper was posed by Elspas and Turner [11], when they asked for a polynomial time algorithm to calculate the full automorphism group of a circulant graph. (Note that it is unclear if a polynomial time algorithm exists.) That is, they essentially asked for an efficient solution to the König problem restricted to the class of graphs consisting of circulant graphs. In this paper, we will consider the class of circulant graphs of square-free order. We will show that the full automorphism group of a circulant digraph of square-free order is either the intersection of two 2-closed groups, each of which is the wreath product of 2-closed groups of smaller degree, or contains a transitive normal subgroup which is the direct Email addresses: [email protected] (Edward Dobson), [email protected] (Joy Morris). 1 This author gratefully acknowledges support from NSERC grant # 40188 Preprint submitted to Elsevier Science 2 September 2003 product of two 2-closed groups of smaller degree. Several remarks are now in order. First, in the latter case, the possible over groups of the direct product of the 2-closed groups of smaller degree are found in this paper. Second, although this result in and of itself will not solve Elspas and Turner’s original problem for circulant graphs of square-free order, we will show in a subsequent paper [22] that a polynomial time algorithm to calculate the full automorphism group of a circulant digraph of square-free order can be derived using this result. This algorithm is only polynomial time provided that the prime power decomposition on the order of the graph is known. Finally, several results have been previously obtained on Elspas and Turner’s problem. The full automorphism groups of circulant digraphs of prime order [2] and [1], prime-squared order [18] (see [10] for another proof of this result), odd prime power order [19], and of a product of two distinct primes [18] have been obtained, and all of these results lead to polynomial time algorithms. The proof of these results are presented in the four sections that follow. The first section includes preliminaries: primarily results from other sources that are used in this paper, and definitions. The second section looks at the structure of actions on blocks. More specifically, it uses results from the Classification of Finite Simple Groups and the structure of specific groups to prove Lemma 16, showing that faithful doubly-transitive nonsolvable actions on blocks must be equivalent. In the third section, under the hypothesis that a certain kind of block system exists, we prove results about the structure of blocks that are minimal with respect to the partial order defined in the preliminaries. Finally, we use the results of sections 2 and 3 to establish the main results described above.