Semiregular automorphisms in vertex-transitive graphs of order 3 p 2

It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally 2-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. It is the purpose of this paper to prove that vertex-transitive graphs of order 3p2, where p is a prime, contain semiregular automorphisms. Mathematics Subject Classifications: 20B25, 05C25


Introduction
It is known that every finite transitive permutation group contains a fixed-point-free element of prime power order (see [5,Theorem 1]), but not necessarily a fixed-point-free element of prime order (which is equivalent to existence of a semiregular element) [3,5]. In 1981 it was asked if every vertex-transitive digraph admits a semiregular automorphism (see [17,Problem 2.4]). The existence of such automorphisms plays an important role in solutions to many important open problems in algebraic graph theory, such as, for example, in the classifications of graphs satisfying certain prescribed symmetry conditions (see [14,15,21,23,26]). Semiregular automorphisms have also proved useful in a long standing hamiltonicity problem for connected vertex-transitive graphs and in a recently explored dichotomy of even/odd automorphisms (see [1,12,16]).
In 1997 Klin generalized the semiregularity problem conjecturing that every transitive 2-closed permutation group contains a semiregular element (see [2]) -the term polycirculant conjecture is sometimes used for the semiregularity problem in this wider context. (Recall that for a finite permutation group G on a set V the 2-closure G (2) of G is the largest subgroup of the symmetric group Sym(V ) containing G and having the same orbits as G in the induced action on V × V .) The problem has spurred a lot of interest in the mathematical community producing several partial results. In particular, Giudici [9] settled the question for quasiprimitive group actions, leaving as one of the main open cases graphs admitting solvable group actions (see [19]). Furthermore, there have also been a number of papers dealing with semiregularity problem for vertex-transitive graphs satisfying certain valency and order restrictions (see, for instance, [3,4,5,6,7,8,9,10,11,13,20,22,24,25]). For example, it is known that every 2-closed group of square-free degree admits semiregular elements (see [7]). As for composite nonsquare orders the only positive result is, if we disregard prime power orders, that every vertex-transitive graph of order 2p 2 , p a prime, admits semiregular automorphisms (see [20]). It is the object of this paper to prove the existence of semiregular automorphisms in vertex-transitive graphs of order 3p 2 , where p is a prime. We hope that this will motivate further research, leading eventually to the solution of the semiregularity problem in the case of vertex-transitive (di)graphs of cube-free order.
Theorem 1. A vertex-transitive graph of order 3p 2 , where p is a prime, admits a semiregular automorphism.
Theorem 1 is proved in Section 2 after a series of propositions each of which considers vertex-transitive graphs in question with particular (im)primitivity actions of their automorphism groups. A comment is in order. There are two reasons for the restriction to vertex-transitive graphs in the main theorem. First, in the proof of Theorem 1 we use certain results from [19], proved within a restricted setting of vertex-transitive graphs. The second reason is somewhat more philosophical and reflects author's personal bias. If one's goal is a complete solution of the semiregularity problem, then rather than worrying over the distinction between the original question and its generalization to 2-closed groups, one should primarily aim at advancements for groups which are not quasiprimitive -say be it solvable or of particular degrees -even if only in the context of vertex-transitive (di)graphs.
2 Vertex-transitive graphs of order 3p 2 Let us first recall the concept imprimitive groups. Given a transitive permutation group G on a set V , we say that a partition B of V is a G-invariant if the elements of G permute the parts, that is, blocks of B, setwise. If the trivial partitions {V } and {{v} | v ∈ V } are the only G-invariant partitions of V , then G is said to be primitive, and is said to be imprimitive otherwise. In the latter case we shall refer to a corresponding Ginvariant partition as to a complete imprimitivity block system, in short an imprimitivity block system, of G. A transitive permutation group is quasiprimitive if each of its nonidentity normal subgroups is transitive, and is said to be genuinely imprimitive otherwise. Note that in the latter case there exists an imprimitivity block system of G arising from orbits of an intransitive normal subgroup of G. A vertex-transitive graph is primitive if its automorphism group is primitive. Otherwise it is called an imprimitive graph.
The following proposition, proved by Giudici in [9], implies the existence of semiregular automophisms in a vertex-transitive graph in case its automorphism group is quasiprimitive. (A finite transitive permutation group is said to be elusive if it has no semiregular element.) Proposition 2.
Proposition 2 implies that only those graphs with genuinely imprimitive automorphism groups need to be considered. In particular, let X be a vertex-transitive graph of order 3p 2 , where p is a prime. We may assume that there exists an intransitive normal subgroup N of the automorphism group Aut(X) of X. In fact, we may, without loss of generality, assume that N is a minimal normal subgroup of Aut(X). The size of the blocks arising from the orbits of N divides the order of X, and is therefore 3, p, p 2 or 3p. The proposition below was recently proved in [19], where semiregularity in vertex-transitive graphs with a solvable automorphism group is considered. Note, however, that this particular result does not require the permutation group to be solvable. The proposition implies the existence of semiregular automorphisms in X in case the blocks arising from the orbits of N are of prime size. The remaining two cases, that is graphs whose automorphism groups admit an intransitive normal subgroup giving rise to imprimitivity block system consisting of blocks of size p 2 and 3p, are considered in Propositions 6 and 7. For the sake of completeness we first state the following classical result which will be used in the proofs. It implies that in a vertex-transitive graph of order 3p 2 , where p is a prime, the orbits of a Sylow p-subgroup of the automorphism group are of length p 2 .

Proposition 4. [27, Theorem 3.4]
Let p be a prime and let P be a Sylow p-subgroup of a permutation group G acting on a set Ω. Let ω ∈ Ω. If p m divides the length of the G-orbit containing ω, then p m also divides the length of the P -orbit containing ω.
Before considering the remaining two cases let us recall a recent result about existence of semiregular automorphisms in a vertex-transitive graph with solvable automorphism group of order mp 2 , where m satisfies certain conditions. This result implies that only vertex-transitive graphs of order 3p 2 with non-solvable automorphism groups need to be considered.

Proposition 5. [19, Theorem 2.4]
Let X be a connected vertex-transitive graph of order p 2 q, where p and q are primes, and either q p or p 2 < q. Then either (i) X admits a semiregular automorphism, or (ii) 2 < q < p and Aut(X) is nonsolvable with an intransitive non-abelian minimal normal subgroup whose orbits are either of length p 2 or of length pq.
In the next proposition we consider the case where the blocks of imprimitivity are of size p 2 .
Proposition 6. Let p be a prime and let X be a vertex-transitive graph of order 3p 2 , where p > 3 is a prime, admitting an imprimitivity block system consisting of three blocks of size p 2 arising from orbits of an intransitive normal subgroup N of Aut(X). Then X admits a semiregular automorphism.
Proof. We may assume that X is connected as otherwise a semiregular automorphism in X can be easily constructed via semiregular automorphisms in the connected components. Namely, if X is disconnected then its connected components are vertex-transitive graphs of order 3, p, p 2 or 3p, and it is well known that such graphs admit semiregular automorphisms.
Let B = {A, B, C} be the imprimitivity block system arising from the orbits of N , each of length p 2 . Clearly, in view of Proposition 4, the orbits A, B and C coincide with the orbits of a Sylow p-subgroup P of Aut(X). Observe also that there must exist an automorphism π ∈ Aut(X) which cyclically permutes the three blocks in B. We may, without loss of generality, assume that π| B = (A B C).
The center Z(P ) of the Sylow p-subgroup P is non-trivial and thus there exists a central element α ∈ Z(P ) of order p. Clearly, for each Y ∈ {A, B, C} either α Y is trivial or α Y is semiregular of order p. If α is not semiregular then there are essentially only two possibilities that need to be considered, depending on the number of orbits Y ∈ {A, B, C} for which the restriction α Y is trivial. Case 1. α A is semiregular and α B = α C = 1.
Then (παπ −1 ) B and (π 2 απ −2 ) C are semiregular, and implying that α · παπ −1 · π 2 απ −2 = (απ) 3 π −3 is the desired semiregular automorphism. and X[A, C] are all isomorphic to the complete bipartite graph K p 2 ,p 2 , and X clearly admits a semiregular automorphism. (For disjoint subsets U, W of the vertex set V (X) the subgraph of X induced by the set U is denoted by X[U ], and similarly, the bipartite subgraph of X induced by the edges having one endvertex in U and the other endvertex in W is denoted by X[U, W ].) Hence, we may assume that the orbits of α on C are blocks of imprimitivity for N C . Note that (αβ) C either fixes the orbits of α or cyclically permutes them. We deal with these two cases in the two subcases below.
It follows that the orbits of α and β on C coincide. Denote these orbits by C i , i ∈ Z p . If all of the restrictions (αβ) C i , i ∈ Z p , are of order p then αβ is a semiregular automorphism of X. If not, then there exists r ∈ Z p such that the restrictions (αβ) C j are of order p for j ∈ {r + 1, . . . , p − 1} and are not of order p for j ∈ {0, 1, . . . , r}. We now define a semiregular automorphism σ of X in the following way: To show that σ is indeed an automorphism of X observe first that the bipartite graph X[C i , C j ], where i ∈ {0, 1, . . . , r} and j ∈ {r + 1, . . . , p − 1}, is either isomorphic to the complete bipartite graph K p,p or is totally disconnected. Combining this with the fact that (αβ) B = α B we obtain that σ B∪C is an automorphism of the subgraph of X induced on B ∪ C. To complete the proof we need to check the edges of the induced bipartite graph X[A, C]. Since for each j ∈ {0, 1, . . . , r} there exists k j coprime with p such that ((αβ) k j ) C j = 1 if follows that each of the bipartite graphs X[A i , C j ], where A = {A i | i ∈ Z p } is a partition of A into the orbits of β and j ∈ {0, 1, . . . , r}, is either isomorphic to the complete bipartite graph K p,p or is totally disconnected. It follows that σ preserves the edges of X[A, C], and consequently σ is an automorphism of X.

Subcase 2.2. (αβ) C cyclically permutes the orbits of α.
Then either (αβ) C is of order p and clearly semiregular, in which case αβ is a semiregular automorphism of X. Alternatively, (αβ) C is of order p 2 in which case (αβ) p is trivial on A ∪ B and semiregular of order p on C. In this case take (αβ) p α π 2 to get the desired semiregular automorphism.
In the next proposition we deal with blocks of size 3p.
Proposition 7. Let X be a vertex-transitive graph of order 3p 2 , where p > 3 is a prime, admitting an imprimitivity block system consisting of p blocks of size 3p arising from orbits of an intransitive normal subgroup N of Aut(X). Then X admits a semiregular automorphism.
Proof. We may again assume that X is connected. Let B be the imprimitivity block system arising from orbits of N . Let P be a Sylow p-subgroup of Aut(X) with orbits the electronic journal of combinatorics 25(2) (2018), #P2.25 A, B and C. Observe that each block in B intersects each of A, B and C in exactly p vertices. We have The center Z(P ) of P is non-trivial and thus there exists a central element α ∈ Z(P ) of order p such that for each Y ∈ {A, B, C} either α Y is trivial or α Y is semiregular of order p. If α is not semiregular then there are essentially only two possibilities depending on the number of orbits Y ∈ {A, B, C} for which the restriction α Y is trivial. Case 1. α A is semiregular of order p, and α B = α C = 1 .
First observe that, since B is an imprimitivity block system, the set of orbits of α A is equal to the set {A i | i ∈ Z p }. By [18,Proposition 3.2], every transitive group of degree p 2 contains a regular (abelian) subgroup, and so there exists Q P such that Q B is either a cyclic or an elementary abelian subgroup acting regularly on B. Thus we distinguish two subcases.
There exists ρ ∈ Q of order p 2 such that ρ B is also of order p 2 and maps according to the rule ρ : Let e, f , and g denote the respective orders of ρ A , ρ B , and ρ C . Then (e, f, g) is one of the following ordered triples: (p 2 , p 2 , p 2 ), (p, p 2 , p 2 ), (p 2 , p 2 , p) or (p, p 2 , p).
In the first case ρ is semiregular. In the second case αρ p is semiregular. In the third and the fourth case, the existence of automorphisms α and σ p implies that each of the bipartite subgraphs X[A i , B j ], X[B i , C j ], and X[A i , C j ] (i, j ∈ Z p ) is either isomorphic to the complete bipartite graph K p,p or is totally disconnected. Consequently, any permutation ω fixing each of A i , B i and C i , i ∈ Z p , set-wise and satisfying the property that ω A , ω B , and ω C is, respectively, an automorphism of X[A], X[B], and X[C], is in fact an automorphism of X. As in the case of the orbit B and the subgroup Q P there exists a subgroup R P such that R C is a regular abelian group. Since C i , i ∈ Z p , are the intersections of the blocks Y i , i ∈ Z p , with the orbit C there must exist an automorphism τ ∈ R fixing these blocks and such that τ C is semiregular (and of order p on each C i ). We now define ω as follows the electronic journal of combinatorics 25(2) (2018), #P2. 25 Clearly, ω is the desired semiregular automorphism of X.
p . There exist ρ, σ ∈ Q such that both ρ B and σ B are of order p. Furthermore, we may assume that the orders of ρ C and σ C are either p or 1, for otherwise an argument analogous to the one used in Subcase 1. 1, with B replaced by C, applies. As for the orders of ρ A and σ A they can be 1, p or p 2 .
We may assume that ρ B permutes the sets B i and that σ B fixes the sets B i . Consequently, ρ C permutes the sets C i , and so ρ is semiregular on both B and C. If ρ A is of order p then it permutes the sets A i , and so ρ A is semiregular, and thus ρ is semiregular. Hence we may assume that ρ A is of order p 2 . Consider now σ. Clearly, σ B is semiregular. If σ C is of order p and semiregular then we are done because we can construct the desired automorphism ω as follows: The mapping ω is an automorphism of X since each of the bipartite subgraphs X[A i , B j ] and X[A i , C j ] is either isomorphic to the complete bipartite graph K p,p or is totally disconnected.
Finally, suppose that σ C is not semiregular. In this case apart from the bipartite subgraphs X[A i , B j ] and X[A i , C j ] also any of the induced bipartite subgraphs X[B i , C j ] is either isomorphic to the complete bipartite graph K p,p or is totally disconnected. Recall that σ B is semiregular with orbits B i . Analogously, we may assume that there exits τ ∈ P such that τ C is semiregular with orbits C i . Hence the permutation defined by the rule is a semiregular automorphism of X.
Case 2. α A = 1, and α B and α C are semiregular of order p.
There exists Q P such that Q A is abelian and regular, and so either cyclic or elementary abelian. Observe also that the set of orbits of α B is equal to the set {B i | i ∈ Z p } and that the set of orbits of α C coincides with the set {C i | i ∈ Z p }.
Note that non-identity elements of P are all of order p or p 2 . There exists ρ ∈ Q of order p 2 such that ρ A is also of order p 2 . Let σ = ρ p . Hence σ A is semiregular of order p. We now analyze possibilities for σ B and σ C : they are either trivial or semiregular of order p. If σ B and σ C are both semiregular then σ is the desired automorphism. If σ B = σ C are both trivial then σα is the desired automorphism. Finally, without loss of generality, assume that σ B is semiregular and σ C = 1. Then ρ B ∼ = Z p 2 and being contained in α, ρ B the electronic journal of combinatorics 25(2) (2018), #P2.25 which is abelian (since α ∈ Z(P )), it follows that ρ B = α, ρ B . Therefore α B ∈ ρ B . In particular α B = (ρ pj ) B , for some j ∈ Z * p . It follows that αρ pj is the desired automorphism. Subcase 2.2. Q A ∼ = Z 2 p . There are elements σ, ρ ∈ P such that σ, ρ A ∼ = Z 2 p . Of course, both ρ A and σ A are semiregular. Moreover, since the sets {A i ∪ B i ∪ C i }, i ∈ Z p are blocks, we may assume that ρ A maps A i to A i+1 , and similarly ρ B maps B i to B i+1 and ρ C maps C i to C i+1 , whereas σ A , σ B and σ C fix these sets. In particular, this means that σ B and σ C fix the orbits of α on B and C. Consider now the action of the conjugates k = ρ −k σρ k (k ∈ Z p ) on B and C. Clearly, A k = σ A . If σ is semiregular on B and C then we are done. If σ is not semiregular on B then there exists B i such that σ B i = 1. Consequently, every bipartite subgraph X[A j , B i ], i, j ∈ Z p , is either isomorphic to the complete bipartite graph K p,p or is totally disconnected. Applying the automorphisms k we see that each of the bipartite subgraphs X[A j , B k ] is isomorphic to the complete bipartite graph K p,p or is totally disconnected. An analogous argument holds for the case when σ is not semiregular on C, implying that X[A j , C k ] is either isomorphic to the complete bipartite graph K p,p or is totally disconnected. Then the permutation ω mapping according to the rule: is a semiregular automorphism of X. We are now left with the case where σ is semiregular on one of the two orbits B and C and not semiregular on the other. Without loss of generality we assume that σ B is semiregular and σ C is not semiregular. Then applying the same argument as above it follows that each of the bipartite subgraphs X[A j , C i ] and X[B j , C i ], j ∈ Z p , is either isomorphic to the complete bipartite graph K p,p or is totally disconnected. Applying then the automorphisms k it follows that the same holds for all of the subgraphs X[A j , C k ] and X[B j , C k ], j, k ∈ Z p . Then the permutation ω mapping according to the rule: ω(u) =    σ(u), u ∈ A σ(u), u ∈ B α(u), u ∈ C is the desired semiregular automorphism of X.
We are now ready to prove Theorem 1.
Proof of Theorem 1: Let X be a vertex-transitive graph of order 3p 2 , where p is a prime, and let Aut(X) be its automorphism group. If p ∈ {2, 3} then X is of order 12 or 3 3 , and the existence of semiregular automorphisms follows from the fact that X is a Cayley graph in both of these two cases (see [18]). We may therefore assume that p > 3.
If Aut(X) is quasiprimitive then Proposition 2 implies the existence of semiregular automorphisms in Aut(X). We may thus assume that Aut(X) is genuinely imprimitive. Let N be an intransitive minimal normal subgroup of Aut(X), and let B be an Aut(X)invariant partition of V (X) arising from the orbits of N . Then the blocks in B are of the electronic journal of combinatorics 25(2) (2018), #P2.25 size 3, p, p 2 or 3p. If the blocks in B are of prime size then the existence of semiregular automorphisms is assured by Proposition 3. If the blocks in B are of prime-squared size then the existence of semiregular automorphisms follows from Proposition 6. If, however, the blocks in B are of size 3p then semiregular automorphisms in Aut(X) exist by Proposition 7.