Abstract
Let Γ = Cay(G, S) be a Cayley digraph on a group G and let A = Aut(Γ). The Cayley index of Γ is |A : G|. It has previously been shown that, if p is a prime, G is a cyclic p-group and A contains a noncyclic regular subgroup, then the Cayley index of Γ is superexponential in p. We present evidence suggesting that cyclic groups are exceptional in this respect. Specifically, we establish the contrasting result that, if p is an odd prime and G is abelian but not cyclic, and has order a power of p at least p3, then there is a Cayley digraph Γ on G whose Cayley index is just p, and whose automorphism group contains a nonabelian regular subgroup.
Highlights
Every digraph and group in this paper is finite
A digraph Γ consists of a set of vertices V(Γ) and a set of arcs A(Γ), each arc being an ordered pair of distinct vertices. (Our digraphs do not have loops.) We say that Γ is a graph if, for every arc (u, v) of Γ, (v, u) is an arc
The index of G in Aut(Γ) is called the Cayley index of Γ. It is well-known that a digraph is a Cayley digraph on G if and only if its automorphism group contains the right regular representation of G
Summary
Every digraph and group in this paper is finite. A digraph Γ consists of a set of vertices V(Γ) and a set of arcs A(Γ), each arc being an ordered pair of distinct vertices. (Our digraphs do not have loops.) We say that Γ is a graph if, for every arc (u, v) of Γ, (v, u) is an arc. The index of G in Aut(Γ) is called the Cayley index of Γ It is well-known that a digraph is a Cayley digraph on G if and only if its automorphism group contains the right regular representation of G. If G has order at least p3 and is not cyclic, there exists a proper Cayley digraph on G with Cayley index p and whose automorphism group contains a nonabelian regular subgroup. We expect that “most” groups admit a Cayley digraph of “small” Cayley index such that the automorphism group of the digraph contains another (or even a nonisomorphic) regular subgroup. (Lemma 3.2 shows that the smallest index of a proper subgroup of either of the regular subgroups is a lower bound – and that the Cayley index of p in Theorem 1.1 is best possible.) As an example, we prove the following.
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