ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 14 (2018) 67-82
https://doi.org/10.26493/1855-3974.1339.0ee
(Also available at http://amc-journal.eu)
Vertex-quasiprimitive 2-arc-transitive digraphs*
We study vertex-quasiprimitive 2-arc-transitive digraphs, and reduce the problem of vertex-primitive 2-arc-transitive digraphs to almost simple groups. This includes a complete classification of vertex-quasiprimitive 2-arc-transitive digraphs where the action on vertices has O'Nan-Scott type SD or CD.
Keywords: Digraphs, vertex-quasiprimitive, 2-arc-transitive. Math. Subj. Class.: 05C20, 05C25
1 Introduction
A digraph r is a pair (V, with a set V (of vertices) and an antisymmetric irreflexive binary relation ^ on V. All digraphs considered in this paper will be finite. For a nonnegative integer s, an s-arc of r is a sequence v0, v\,..., vs of vertices with vi ^ vi+l for each i = 0,..., s - 1. A 1-arc is also simply called an arc. We say r is s-arc-transitive if the group of all automorphisms of r (that is, all permutations of V that preserve the relation acts transitively on the set of s-arcs. More generally, for a group G of automorphisms of r, we say r is (G, s)-arc-transitive if G acts transitively on the set of s-arcs of r.
A transitive permutation group G on a set Q is said to be primitive if G does not preserve any nontrivial partition of Q, and is said to be quasiprimitive if each nontrivial normal subgroup of G is transitive. It is easy to see that primitive permutation groups are necessarily quasiprimitive, but there are quasiprimitive permutation groups that are not primitive. We say a digraph is vertex-primitive if its automorphism group is primitive on the vertex set. The aim of this paper is to investigate finite vertex-primitive s-arc transitive digraphs with s > 2. However, we will often work in the more general quasiprimitive setting.
*The authors would like to thank the anonymous referee for helpful comments.
^This research was supported by Australian Research Council grant DP150101066.
* Corresponding author
E-mail addresses: michael.giudici@uwa.edu.au (Michael Giudici), binzhou.xia@uwa.edu.au (Binzhou Xia)
Michael Giudici Binzhou Xia *
University of Western Australia, Crawley, Australia
Received 2 March 2017, accepted 2 May 2017, published online 10 May 2017
Abstract
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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There are many s-arc-transitive digraphs, see for example [2, 6, 7, 8]. In particular, for every integer k > 2 and every integer s > 1 there are infinitely many k-regular (G, s)-arc-transitive digraphs with G quasiprimitive on the vertex set (see the proof of Theorem 1 of [2]). On the other hand, the first known family of vertex-primitive 2-arc-transitive digraphs besides directed cycles was only recently discovered in [3]. The digraphs in this family are not 3-arc-transitive, which prompted the following question:
Question 1.1. Is there an upper bound on s for vertex-primitive s-arc-transitive digraphs that are not directed cycles?
The O'Nan-Scott Theorem divides the finite primitive groups into eight types and there is a similar theorem for finite quasiprimitive groups, see [9, Section 5]). For four of the eight types, a quasiprimitive group of that type has a normal regular subgroup. Praeger [8, Theorem 3.1] showed that if r is a (G, 2)-arc-transitive digraph and G has a normal subgroup that acts regularly on V, then r is a directed cycle. Thus to investigate vertex-primitive and vertex-quasiprimitive 2-arc-transitive digraphs, we only need to consider the four remaining types. One of these types is where G is an almost simple group, that is, where G has a unique minimal normal subgroup T, and T is a nonabelian simple group. The examples of primitive 2-arc-transitive digraphs constructed in [3] are of this type. This paper examines the remaining three types, which are labelled SD, CD and PA, and reduces Question 1.1 to almost simple vertex-primitive groups (Corollary 1.6). We now define these three types and state our results.
We say that a quasiprimitive group G on a set Q is of type SD if G has a unique minimal normal subgroup N, there exists a nonabelian simple group T and positive integer k > 2 such that N = Tk, and for w G Q, N is a full diagonal subgroup of N (that is, Nw = T and projects onto T in each of the k simple direct factors of N). It is incorrectly claimed in [8, Lemma 4.1] that there is no 2-arc-transitive digraph with a vertex-primitive group of automorphisms of type SD. However, there is an error in the proof which occurs when concluding "ax also fixes Dt-1". Indeed, given a nonabelian simple group T, our Construction 3.1 yields a (G, 2)-arc-transitive digraph r(T) with G primitive of type SD. These turn out to be the only examples.
Theorem 1.2. Let r be a connected (G, 2)-arc-transitive digraph such that G is quasiprimitive of type SD on the set of vertices. Then there exists a nonabelian simple group T such that r = r(T), as obtained from Construction 3.1. Moreover, Aut(r) is vertex-primitive of type SD and r is not 3-arc-transitive.
The remaining two quasiprimitive types, CD and PA, both arise from product actions. For any digraph E and positive integer m, Em denotes the direct product of m copies of E as in Notation 2.6. The wreath product Sym(A) I Sm = Sym(A)m x Sm acts naturally on the set Am with product action. Let G1 be the stabiliser in G of the first coordinate and let H be the projection of G1 onto Sym(A). If G projects onto a transitive subgroup of Sm, then a result of Kovacs [4, (2.2)] asserts that up to conjugacy in Sym(A)m we may assume that G < H I Sm. A reduction for 2-arc-transitive digraphs was sought in [8, Remark 4.3] but only partial results were obtained. Our next result yields the desired reduction.
Theorem 1.3. Let H < Sym(A) with transitive normal subgroup N and let G < HI Sm acting on V = Am with product action such that G projects to a transitive subgroup of Sm and G has component H. Moreover, assume that Nm < G. If r is a (G, s)-arc-transitive
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digraph with vertex set V such that s > 2, then V = Em for some (H, s)-arc-transitive digraph E with vertex set A.
A quasiprimitive group of type CD on a set Q is one that has a product action on Q and the component is quasiprimitive of type SD, while a quasiprimitive group of type PA on a set Q is one that acts faithfully on some partition P of Q and G has a product action on P such that the component H is an almost simple group. When G is primitive of type PA, H is primitive and the partition P is the partition into singletons, that is, G has a product action on Q. As a consequence, we have the following corollaries.
Corollary 1.4. Suppose r is a connected (G, 2)-arc-transitive digraph such that G is vertex-quasiprimitive of type CD. Then there exists a nonabelian simple group T and positive integer m > 2 such that r = r(T )m, where r(T) is as obtained from Construction 3.1. Moreover, r is not 3-arc-transitive.
Corollary 1.5. Suppose r is a (G, s)-arc-transitive digraph such that G is vertex-primitive of type PA. Then r = Em for some (H, s)-arc-transitive digraph E and integer m > 2 for some almost simple primitive permutation group H < Aut(E).
We give an example in Section 2.3 of an infinite family of (G, 2)-arc-transitive digraphs r with G vertex-quasiprimitive of PA type such that r is not a direct power of a digraph E (indeed the number of vertices of r is not a proper power). We leave the investigation of such digraphs open.
We note that Theorem 1.2 and Corollaries 1.4 and 1.5, reduce Question 1.1 to studying almost simple primitive groups.
Corollary 1.6. There exists an absolute upper bound C such that every vertex-primitive s-arc-transitive digraph that is not a directed cycle satisfies s < C, if and only if for every (G, s)-arc-transitive digraph with G a primitive almost simple group we have s < C.
Theorem 1.2 follows immediately from a more general theorem (Theorem 3.15) given at the end of Section 3. Then in Section 4, we prove Theorem 1.3 as well as Corollaries 1.41.5 after establishing some general results for normal subgroups of s-arc-transitive groups.
2 Preliminaries
We say that a digraph r is k-regular if both the set r-(v) = {u e V | u ^ v} of in-neighbours of v and the set r+(v) = {w e V | v ^ w} of out-neighbours of v have size k for all v e V, and we say that r is regular if it is k-regular for some positive integer k. Note that any vertex-transitive digraph is regular. Moreover, if r is regular and (G, s)-arc-transitive with s > 2 then it is also (G, s — 1)-arc-transitive.
Recall that a digraph is said to be connected if and only if its underlying graph is connected. A vertex-primitive digraph is necessarily connected, for otherwise its connected components would form a partition of the vertex set that is invariant under digraph automorphisms.
2.1 Group factorizations
All the groups we consider in this paper are assumed to be finite. An expression of a group G as the product of two subgroups H and K of G is called a factorization of G. The following lemma lists several equivalent conditions for a group factorization, whose proof is fairly easy and so is omitted.
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Lemma 2.1. Let H and K be subgroups of G. Then the following are equivalent.
(a)	G = HK.
(b)	G = KH.
(c)	G = (x-iHx)(y-iKy) for any x,y G G.
(d)	|H n K||G| = |H||K|.
(e)	H acts transitively on the set of right cosets of K in G by right multiplication.
(f)	K acts transitively on the set of right cosets of H in G by right multiplication.
The s-arc-transitivity of digraphs can be characterized by group factorizations as follows:
Lemma 2.2. Let r be a G-arc-transitive digraph, s > 2 be an integer, and v0 ^ vi ^ • • • ^ vs-i ^ vs be an s-arc of r. Then r is (G, s)-arc-transitive if and only if GVl...Vi = GV0Vl...ViGVl...ViVi+1 for each i in {1,..., s — 1}.
Proof. For any i such that 1 < i < s — 1, the group GVl ...Vi acts on the set r+(vj) of out-neighbours of v^ Since vi+i G r+(vj) and GVl...ViVi+1 is the stabilizer in GVl...Vi of vi+i, by Frattini's argument, the subgroup GVoVl...Vi of GVl...Vi is transitive on r+(vj) if and only if GVl...Vi = GVoVl...ViGVl ...V.V.+l. Note that r is (G, s)-arc-transitiveifandonly if r is (G, s — 1)-arc-transitive and GVoVl...Vi is transitive on r+(vj). One then deduces inductively that r is (G, s)-arc-transitive if and only if GVl...Vi = GVoVl...Vi GVl...ViVi+l for each i in {1,..., s — 1}.	□
If r is a G-arc-transitive digraph and u ^ v is an arc of r, then since G is vertex-transitive we can write v = ug for some g G G and it follows that
— l	s —2	s —l
vg ^ v ^----> vg ^ vg	(2.1)
is an s-arc of r. In this setting, Lemma 2.2 is reformulated as follows.
Lemma 2.3. Let r be a G-arc-transitive digraph, s > 2 be an integer, v be a vertex of r, and g G G such that v ^ vg. Then r is (G, s)-arc-transitive if and only if
i-i / i \ ( i n g-j gv gj=i n g-(j-i)GV gj-i) i n g-j GV gj
j=0	\j=0	J \j=0
for each i in {1,..., s — 1}.
j—l
Proof. Let vj = vg for any integer j such that 0 < j < s — 1. Then the s-arc (2.1) of r turns out to be v0 ^ vi ^ • • • ^ vs-i ^ vs, and for any i in {1,..., s} we have
i i i- i
Gvl...vi=n Gv, = n g-(j-i)Gv gj-i=n g-j GV gj
j=i j=i j=0
and
ii
Gvovl...vi=n Gvj = n g-(j-i)Gv gj-i. j=0 j=0
Hence the conclusion of the lemma follows from Lemma 2.2.	□
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2.2 Constructions of s-arc-transitive digraphs
Let G be a group, H be a subgroup of G, V be the set of right cosets of H in G and g be an element of G \ H such that g-1 G HgH. Define a binary relation ^ on V by letting Hx ^ Hy if and only if yx-1 G HgH for any x, y G G. Then (V, is a digraph, denoted by Cos(G, H, g). Right multiplication gives an action RH of G on V that preserves the relation so that RH (G) is a group of automorphisms of Cos(G, H, g).
Lemma 2.4. In the above notation, the following hold.
(a)	Cos(G, H, g) is |H:H n g-1Hg|-regular.
(b)	Cos(G, H, g) is Rh(G)-arc-transitive.
(c)	Cos(G, H, g) is connected if and only if (H, g) = G.
(d)	Cos(G, H, g) is Rh (G)-vertex-primitive if and only if H is maximal in G.
(e)	Let s > 2 be an integer. Then Cos(G, H, g) is (RH (G), s)-arc-transitive if and only if for each i in {1,..., s — 1},
Proof. Parts (a)-(d) are folklore (see for example [2]), and part (e) is derived in light of
Remark 2.5. Lemma 2.4 establishes a group theoretic approach to constructing s-arc-transitive digraphs. In particular, Cos(G, H, g) is (RH(G), 2)-arc-transitive if and only if
Next we show how to construct s-arc-transitive digraphs from existing ones. Let r be a digraph with vertex set U and S be a digraph with vertex set V. The direct product of r and S, denoted r x S, is the digraph (it is easy to verify that this is indeed a digraph) with vertex set U x V and (ui, vi) ^ (u2, v2) if and only if u ^ u2 and vi ^ v2, where uj G U and vj G V for i = 1, 2.
Notation 2.6. For any digraph S and positive integer m, denote by Sm the direct product of m copies of S.
Lemma 2.7. Let s be a positive integer, r be a (G, s)-arc-transitive digraph and S be a (H, s)-arc-transitive digraph. Then r x S is a (G x H, s)-arc-transitive digraph, where G x H acts on the vertex set of r x S by product action.
Proof. Let (wo,vo) ^ (u1,v1 ) ^ ••• ^ (us,vs) and (u0,v0) ^ (ui,vi) ^ ••• ^ (u', v' ) beany two s-arcs of r x S. Then u0 ^ u1 ^ • • • ^ us and u0 ^ ui ^ • • • ^ u's are s-arcs of r while v0 ^ vi ^ • • • ^ vs and v'0 ^ v' ^ • • • ^ v' are s-arcs of S. Since r is (G, s)-arc-transitive, there exists g G G such that ug = uj for each i with 0 < i < s. Similarly, there exists h G H such that vh = v' for each i with 0 < i < s. It follows that (uj, vj)(g'h) = (uj, v') for each i with 0 < i < s. This means that r x S is a (G x H, s)-arc-transitive.	□
Lemma 2.3.
□
H = (gHg-1 n H)(H n g-1Hg).
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2.3 Example
In this subsection we give an example of an infinite family of (G, 2)-arc-transitive digraphs r with G vertex-quasiprimitive of PA type such that r is not a direct power of a digraph E. In fact, we prove in Lemma 2.9 that the number of vertices of r is not a proper power.
Let n > 5 be odd, Gi = Alt({1, 2, ...,n}) and G2 = Alt({n + 1,n + 2,..., 2n}). Take permutations
a = (1, n +1)(2,n + 2) ••• (n, 2n), b = (1, 2)(3, 4)(n +1,n + 2)(n + 3,n + 4) and
g = (1, n + 2, 2, n + 3, 5, n + 6, 7, n + 8,. .., 2« - 1, n + 2i,. .., n - 2, 2n - 1, n, n + 1, 3, n + 4, 4,n + 5, 6,n + 7,.. ., 2j, n + 2 j + 1, .. . ,n - 1, 2n).
In fact, g = ac with
c = (1, 3, 5, 6, 7, .. ., n)(n + 1, n + 2, ..., 2n).
Let G = (G1 x G2) x (a), and note that g G G as c G G1 x G2. Let H = (a, b) = (a) x (b) and r„ = Cos(G,H,g).
Lemma 2.8. For all odd n > 5, rn is a connected (G, 2)-arc-transitive digraph with G quasiprimitive of PA type on the vertex set.
Proof. As (G1 x G2) n H = (b) we see that G is quasiprimitive of PA type on the vertex set. To show that rn is connected, we shall show (H, g) = G in light of Lemma 2.4(c). Let M = (H, g) n (G1 x G2). Then we only need to show M = G1 x G2 since a G (H, g).
Denote the projections of G1 x G2 onto G1 and G2, respectively, by n1 and n2. Note that g2 fixes both {1,..., n} and {n + 1,..., 2n} setwise with
n1(g2) = (1, 2, 5, 7,..., 2i - 1,..., n, 3,4, 6,..., 2j,..., n - 1)
and
n1(gn+1) = (1, 3, 2, 4, 5, .. ., n).
We have g2 G M and
n1(g-(n+1)bgn+1b) = n1(g-(n+1)bgn+1)n1(b) = (3,4)(2, 5)(1, 2)(3,4) = (1, 2, 5), which implies
n1(M) > n1((g2,b)) > n1((g2,g-(n+1)bgn+1b)) = (^(g2), ^(g-^bg^b)) = G1
using the fact that the permutation group generated by a 3-cycle (a, ß, 7) and an n-cycle with first 3-entries a, ß, 7 is An. It follows that
n2 (M) = n2(Ma) = (n2(M ))a = G? = G2,
and so M is either G1 x G2 or a full diagonal subgroup of G1 x G2. However, c = ag G M while n1(c) and n2(c) have different cycle types. We conclude that M is not a diagonal subgroup of G1 x G2, and so M = G1 x G2 as desired.
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Now we prove that rn is (G, 2)-arc-transitive, which is equivalent to proving that H = (gHg-1 n H)(H n g-1Hg) according to Lemma 2.4(e). In view of
(ah)9 = (ab)ac = (ab)c = a	(2.2)
we deduce that a e H n H9. Since H is not normal in G = (H, g), we have H9 = H. Consequently, H n H9 = (a). Then again by (2.2) we deduce that
H n H9-1 = (H n H9)9-1 = (a)9-1 = (a9-1) = (ah).
This yields
(gHg-1 n H)(H n g-1Hg) = (a)(ah) = H.	(2.3)
Finally, the condition g-1 £ HgH holds as a consequence (see [3, Lemma 2.3]) of (2.3) and the conclusion H9 = H. This completes the proof.	□
Lemma 2.9. The number of vertices of rn is not a proper power for any odd n > 5.
Proof. Suppose that the number of vertices of rn is mk for some m > 2 and k > 2. Then we have
mk = ]GI = 2nW = W	(2.4)
|H |	4	8	V y
If k = 2, then (2.4) gives (n!)2 = 2(2m)2, which is not possible. Hence k > 3. By Bertrand's Postulate, there exists a prime number p such that n/2 < p < n. Thus, the largest p-power dividing n! is p, and so the largest p-power dividing the right hand side of (2.4) is p2. However, this implies that the largest p-power dividing mk is p2, contradicting the conclusion k > 3.	□
2.4 Normal subgroups
Lemma 2.10. Let r be a (G, s)-arc-transitive digraph with s > 2, M be a vertex-transitive normal subgroup of G, and v1 ^ ■ ■ ■ ^ vs be an (s — l)-arc of r. Then G = MGVl.Vi for each i in {1,..., s}.
Proof. Since M is transitive on the vertex set of r, there exists m e M such that v™ = v2.
i-1
m
-1
Denote u = v™ for each i such that 0 < i < s. Then GUoul...ui = mGUl...Uiui+1 for each i such that 0 < i < s - 1, and u0 ^ u1 ^ ■ ■ ■ ^ us is an s-arc of r since v1 ^ v2 and m is an automorphism of r. For each i in {1,..., s - 1}, we deduce from Lemma 2.2 that
gui...ui GU0U1 ...ui gui...uiui+i	(mgui ...uiui+1 m )gu1 ...uiui+1 .
Let ( be the projection from G to G/M .It follows that
u1 ...ui ) = f(m )((GU1 ...UiUi+1 )((m) ((gu1 ...uiui+1 ) f(G u1 ...uiui+1 )((gu1 ...uiui+1 ) = f(gu1...uiui+1 )
and so GU1...UiM = GU1.UiUi+1 M for each i in {1,..., s - 1}. Again as M is transitive on the vertex set of r, we have G = MGU1. Hence
G = mgu1 = mgu1u2 = ■ ■ ■ = mgu1...ui = ■ ■ ■ = mgu1...us .
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Now for each i in {1,..., s}, the digraph r is (G, i —1)-arc-transitive, so there exists g G G such that (vg,..., vg) = (u,..., uj). Hence
G = MGMl...„i = M (g-1Gv1...Vi g) = MG„1...„i
by Lemma 2.1(c).	□
By Frattini's argument, we have the following consequence of Lemma 2.10:
Corollary 2.11. Let r be a (G, s)-arc-transitive digraph with s > 2, and M be a vertex-transitive normal subgroup of G. Then r is (M, s — 1)-arc-transitive.
To close this subsection, we give a short proof of the following result of Praeger [8, Theorem 3.1] using Lemma 2.10.
Proposition 2.12. Let r be a (G, 2)-arc-transitive digraph. If G has a vertex-regular normal subgroup, then r is a directed cycle.
Proof. Let N be a vertex-regular normal subgroup of G, and u ^ v be an arc of r. Then |G|/|N| = |Gv |, and G = GuvN by Lemma 2.10. Hence by Lemma 2.1(d), |Guv | > IG | /1N | = | Gv | and so | Guv | = | Gv |. Consequently, r is 1 -regular, which means that r is a directed cycle.	□
2.5 Two technical lemmas
Lemma 2.13. Let A be an almost simple group with socle T and L be a nonabelian simple group. Suppose Ln < A and |T| < |Ln| for some positive integer n. Then n = 1 and L = T.
Proof. Note that Ln/(Ln n T) = (LnT)/T < A/T, which is solvable by the Schreier conjecture. If Ln n T = Ln, then Ln/(Ln n T) = Lm for some positive integer m, a contradiction. Hence Ln n T = Ln, which means Ln < T. This together with the condition that |T| < |Ln| implies Ln = T. Hence n =1 and L = T, as the lemma asserts.	□
Lemma 2.14. Let A be an almost simple group with socle T and S be a primitive permutation group on |T | points. Then S is not isomorphic to any subgroup of A.
Proof. Suppose for a contradiction that S < A. Regard S as a subgroup of A, and write Soc(S) = Ln for some simple group L and positive integer n. Since S is primitive on |T| points, Soc(S) is transitive on |T| points, and so |T| divides |Soc(S)| = |L|n. Consequently, L is nonabelian for otherwise T would be solvable. Then by Lemma 2.13 we have Soc(S) = L = T .It follows that S is an almost simple primitive permutation group with Soc(S) regular, contradicting [5].	□
3 Vertex-quasiprimitive of type SD 3.1 Constructing the graph r(T)
Construction 3.1. Let T be a nonabelian simple group of order k with T = {t1,..., tk }. Let D = {(t,... ,t) | t G T} be a full diagonal subgroup of Tk and let g = (t1,... ,tk). Define r = Cos(Tk, D, g) and let V be the set of right cosets of D in Tk, i.e. the vertex set of r(T).
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Lemma 3.2. r(T) is a |T\-regular digraph.
Proof. Suppose that Dng-1 Dg = 1. Then there exist s,t G T\{1} such that (s,..., s) = (t-1 tt1,... ,t-1ttk). Thus s = t-1tti for each i such that 1 < i < k. Since {tj | 1 < i < k} = T, we have tj = 1 for some 1 < j < k. It then follows from the equality s = t-1ttj that s = t. Thus t = t—1tti for each i such that 1 < i < k. Hence t lies in the center of T, which implies t = 1 as T is nonabelian simple, a contradiction. Consequently, D n g-1Dg = 1, and so Cos(Tk,D,g) is |T|-regularas |D|/|D n g-1Dg| = |D| = |T|.
Suppose that g-1 G DgD. Then there exist s,t G T such that (t-1,... ,t-1) = (st1t,..., stkt). It follows that t-1 = stjt for each i such that 1 < i < k. Since {tj | 1 < i < k} = T, we have tj = 1 for some 1 < j < k. Then the equality t-1 = stjt leads to s = t-1. Thus t-1 = t-1 tjt for each i such that 1 < i < k. This implies that the inverse map is an automorphism of T and so T is abelian, a contradiction. Hence g-1 G DgD, from which we deduce that Cos(Tk, D, g) is a digraph, completing the proof.	□
Next we show that up to isomorphism, the definition of r(T) does not depend on the order of t1, t2,..., tk.
Lemma 3.3. Let g' = (ti,..., tk) such that T = {ti,. . . , tk}. Then Cos(Tk, D, g) = Cos(Tk, D, g').
Proof. Since {t1,..., t'k } = {t1,..., tk }, there exists x G Sk such that t^ = tj for each i with 1 < i < k. Define an automorphism A of Tk by (g1,..., gk)A = (g1x,..., gk*) for all (g1,..., gk) G Tk. Then A normalizes D and A-1gA = g'. Hence the map Dh ^ DhA gives an isomorphism from Cos(Tk, D, g) to Cos(Tk, D, g').	□
For any t G T, let x(t) and y(t) be the elements of Sk such that tjx(t) = ttj and tjy(t) = tjt-1 for any 1 < i < k, and define permutations A(t) and p(t) of V by letting
D(g1,..., gk)A(t) = D(g1x(t),..., gfcx(t))
and
D(g1,..., gk)p(t) = D(g1„(t),..., gfc„(t))
for any (g1,..., gk) G Tk. For any ^ G Aut(T), let z(y) G Sk such that t^^ = tf for any 1 < i < k, and define S(<£>) G Sym(V) by letting
D(g1,..., gk)^) = D((g1,(,-i) ,..., (gk,(,-i) )
for any (g1,..., gk) G Tk. In particular, £(y>) both permutes the coordinates and acts on each entry.
Lemma 3.4. A and p are monomorphisms from T to Sym(V), and S is a monomorphism from Aut(T) to Sym(V).
Proof. For any s, t G T, noting that x(t)x(s) = x(st), we have
D(g1, . . . , gk )A(S)A(t) = D(g1x(s) ,...,gkx(s) )A(i) = D(g1x(t)x(s) , . . . , gkx(t)x(s) )
= D(g1x(st) , . . . , gkx(st) )
= D(g1,..., gk)A(si)
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for each (gi,..., € Tk, and so A(st) = A(s)A(t). This means that A is a homomor-phism from T to Sym(V). Moreover, since A(t) acts on V as the permutation x(t) on the entries, A(t) = 1 if and only if x(t) = 1, which is equivalent to t = 1. Hence A is a monomorphism from T to Sym(V). Similarly, p is a monomorphism from T to Sym(V). For any ^ € Aut(T), since	=	= z((y>^)_i), we have
D(gi,...,gk)i(*)i(' = D((g^-i) )*..., (g^-i)
= D((giz(^-1)z(^-1) Y'^ , . . . , (gkz(i-1)z(y-1) Y'^ )
= D(gi,...,gk
for all (gi,..., gk) € Tk. This means that S is a homomorphism from Aut(T) to Sym(V). Next we prove that S is a monomorphism. Let ^ € Aut(T) such that
D((giz(,-1) )*..., (g^-i) Y) = D(gi,..., gk)'(v) = D(gi,..., gk) (3.1)
for each (gi,..., gk) € Tk. Take any i € {1,..., k} and (gi,..., gk) € Tk such that gj = 1 for all j = i and g» = 1. By (3.1), there exists t € T such that (gj-z(^-i) = tgj for each j € {1,..., k}. As a consequence, we obtain t = 1 by taking any j € {1,..., k}\{i} such that jz(^-1) = i. Also, for j € {1,..., k}, (g^-i) = t if and only if j = i. It
follows that iz(v 1) = i. As i is arbitrary, this implies that	= 1, and so f = 1.
This shows that S is a monomorphism from Aut(T) to Sym( V).	□
Let M be the permutation group on V induced by the right multiplication action of Tk. For any group X, the holomorph of X, denoted by Hol(X), is the normalizer of the right regular representation of X in Sym(X). Note that (x(T), y(T), z(Aut(T))) = x(T) x z(Aut(T)) = y(T) x z(Aut(T)) is primitive on {1,..., k} and permutation isomorphic to Hol(T). Thus,
X := (M,A(T), p(T),S(Aut(T)))	(3.2)
is a primitive permutation group on V of type SD with socle M, and the conjugation action of X on the set of k factors of M = Tk is permutation isomorphic to Hol(T). Let v = D € V, a vertex of r(T). For any t € T let <r(t) € M be the permutation of V induced by right multiplication by (t,..., t). Then
Xv/a(T) = Xv/(Xv n M) = XvM/M = X/M = Hol(T)
since M acts transitively on V, and therefore
|Xv | = |a(T)||Hol(T )| = |T |3|Out(T)|.	(3.3)
Lemma 3.5. X < Aut(r(T)).
Proof. Clearly M < Aut(r(T)), so it remains to verify that A(T), p(T) and S(Aut(T)) are subgroups of Aut(r(T)). Let D(gi,..., gk) € V and D(gi,..., gk) € V. Then we have D(gi,..., gk) ^ D(gi,..., g'k) in r(T) if and only if
(gig-i,...,gkg-) € D(ti,...,tk)D.	(3.4)
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Let t G T. Since (3.4) holds if and only if
(g'l x(t) g1x1(t)	x(t) gfcx1(t) ) G D(tix(t) ,...,tfcx(t) )D
= D(tt1; ...,ttfc)D = D(ti,...,tfc )D,
we conclude that D(g1;...,gk) ^ D(gi,..., gi) if and only if D(g1;..., gk)A(i) ^ D(gi,... ,gi)A(i). This shows A(t) G Aut(r(T)) for any t G T. Similarly, we have p(t) G Aut(r(T )) for any t G T .Let y G Aut(T ). Then (3.4) holds if and only if
((si.(,-1)sr^-1)..., K-(,-!)gfci-i))") G D((t1,(,-!))*..., (tfcz(^))*)D
= D((t^-1 )*,..., (t^-1 )V)D = D(t1,...,tfc)D.
It follows that D(g1,..., gk ) ^ D(gi,..., gi) if and only if
D(g1,...,gk )5(v) ^ D(gi,..., si )5(^, and so ¿(y) G Aut(r(T)) for any y G Aut(T). This completes the proof.	□
Denote H = (M, A(T)) = M x A(T) < X. Lemma 3.6. r(T) is (H, 2)-arc-transitive.
Proof. It is readily seen that Hv = a(T ) x A(T ) ^ T2 .Let K = Mt)A(t) 11 G T}. For any t G T and any (g1,..., gk ) G Tk we have
D(g1,..., gfc)g-V(i)A(i)g = D(g1t-11,..., gkt-1 t)A(i)g
= D(g1x(t)t-x1(t)t,... ,gkx(t)t-x1(t)t)g
=	D(g1x(t) (tt1)-1t, . . . , gkx(t) (ttk)-1 t)g
=	D(g1x(t) t-1,...,gkx( t)t-1)g
=	D(g1x(t) ,...,gkx(t))
=	D(g1,...,gk)A(i).
Hence g-V(t)A(t)g = A(t) for all t G T. Consequently, g-1Kg = A(T) < Hv and so K < Hv n gHvg-1. Now for any elements s and t of T,
a(s)A(t) = (a(s)A(s))A(s-1t) G KA(T) = K(g-1Kg).
It follows that
Hv < K(g-1Kg) < (Hv n gHvg-1)(Hv n g-1Hvg),
so Hv = (Hv n gHvg-1)(Hv n g-1Hvg). Thus by Remark 2.5, r(T) is (H, 2)-arc-transitive, as the lemma asserts.	□
An immediate consequence of Lemma 3.6 is that r(T) is (X, 2)-arc-transitive. However, X is not transitive on the set of 3-arcs of r(T), as we shall see in the next lemma.
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Lemma 3.7. r(T) is not (X, 3)-arc-transitive.
Proof. Suppose that r(T) is (X, 3)-arc-transitive. Then since M is a vertex-transitive normal subgroup of X, Corollary 2.11 asserts that r(T) is (M, 2)-arc-transitive. As a consequence, Mv is transitive on A2(v) := {(v^ v2) G V2 | v ^ v1 ^ v2}, the set of 2-arcs starting from v. However, M | = |T| while |A2(v)| = |T|2 as r(T) is |T|-regular. This is not possible.	□
3.2 Classification
Throughout this subsection, let T be a nonabelian simple group, k > 2 be an interger, D = {(t,..., t) 1t g T} be a full diagonal subgroup of Tk, V be the set of right cosets of D in Tk, and M be the permutation group induced by the right multiplication action of Tk on V. Suppose that G is a permutation group on V with M < G < M.(Out(T) x Sk), and r is a connected (G, 2)-arc-transitive digraph. Let v = D g V and w be an out-neighbour of v. Then w = D(ti,...,tk) g V for some elements ti,..., tk of T which are not all equal. Without loss of generality, we assume tk = 1. Let u = D(t-i,... , t-i) g V and g g M be the permutation of V induced by right multiplication by (ti,..., tk) g Tk. Moreover, define {Qi,..., Q„} to be the partition of {1,..., k} such that tj = tj if and only if i and j are in the same part of {Qi,..., 0„|. Note that Gv < Aut(T) x Sk. Let a be the projection of Gv into Aut(T) and ft be the projection of Gv into Sk. Let A = a(Gv) and S = ,0(Gv), so that Gv < A x S, where each element a of A is induced by an automorphism of T acting on V as
D(gi,..., gk )f = D(gf ,...,gf)
and each element x of S is induced by a permutation on {1,..., k| acting on V as
D(gi,..., gk)x = D(gix-i,..., gkx-i).
As G > M we have Inn(T) < A < Aut(T). Moreover, since G is 2-arc-transitive, Lemma 2.2 implies that Gv = Guv Gvw. Let R be the stabilizer in S of k in the set
{1, .. ., k|.
Take any a g A and x g S. Then ax g Gu if and only if x ia i fixes u, that is
D((t-xi)ff-1,..., (t(k— i )x)ff-1, (t-X)ff-1) = D(t - i,..., t-ii, 1), or equivalently,
D(tfc* t -x , . . . , ifcx t
(fc- 1)x
,1)= D((t- 1 )-,..., (t-1 , 1).
(3.5)
Similarly, ax G Gw if and only if x 1a 1 fixes w, which is equivalent to
D(t-x111 x , . . . ,tfcx1 t(fc- 1 )x , 1) = D(tl , . . . ,t1- 1, 1).
.-1
Lemma3.8. (t 1,...,tk) = T.
(3.6)
Proof. For all a g a(Guv), there exists x g S such that ax g Gu. Then (3.5) implies that tkxtiXi = (t-i)CT and thus tf = tj*t-X for all i such that 1 < i < k. This shows that a(Guv) stabilizes (ti,... ,tk}. Similarly, for all a g a(Gvw), there exists
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x G S such that ax G Gw. Then (3.6) implies that if = i-*1^* for all i such that
1 < i < k. Accordingly, a(Gvw) also stabilizes (t1;... , tk}. It follows that A = a(Gv) = a(GuvGvw) = a(Guv)a(Gvw) stabilizes (i1;...,ifc). Hence (t1,...,ifc} = T since Inn(T) < A < Aut(T).	□
Lemma 3.9. Guv n (A x R) = Gvw n (A x R).
Proof. Let a G A and x G R. Then tk* = tk = 1, and thus (3.6) shows that ax G Gw if and only if t(* = tf for all i such that 1 < i < k. Similarly, (3.5) shows that ax G Gu if and only if t_ 1 = (i-1)f for all i such that 1 ^ i ^ k. Since this is equivalent to t(* = if for all i, we conclude that ax G Gw if and only if ax G Gu. As a consequence,
Guv n (A x R) = Gvw n (A x R).	□
Lemma 3.10. Guv n A = Gvw n A = 1.
Proof. In view of Lemma 3.9 we only need to prove that Gvw n A =1. For any a G Gvw n A, (3.6) shows that D(t1,..., tfc_1,1) = D(tf,..., if_1; 1), and so tf = tj for all i such that 1 < i < k. By Lemma 3.8, this implies that a =1 and so Gvw n A = 1, as desired.	□
Lemma 3.11. Both P(Guv) and P(Gvw) preserve the partition ...,
Proof. Let x G P(Guv). Then there exists a G A such that ax G Gu, and so (3.5) gives
tk* t-*1 = (t_1)f	(3.7)
for all i such that 1 < i < k. For any i, j G {1,..., k}, if i and j are in the same part of ..., then t( = tj and so (t_1)f = (t-1)f, which leads to t(* = j by (3.7). Since t(* = tj* if and only if ix and jx are in the same part of {fi1,..., this shows that x, hence P(Guv), preserves the partition {fi1,..., }. The proof for P(Gvw) is similar.	□
Lemma 3.12. t1,..., tk are pairwise distinct.
Proof. Let U be the subset of V consisting of the elements D(g1,..., gk) with g( = gj whenever i and j are in the same part of {fi1,..., By Lemma 3.11, both P(Guv) and P(Gvw) preserve the partition ,..., Then since S = P(Gv) = P(GuvGvw) = P(Guv )P(Gvw), we derive that S preserves the partition {fi1,..., As a consequence, S stabilizes U setwise. Meanwhile, A and g stabilize U setwise. Hence G = (Gv, g) < (A x S, g) stabilizes U setwise, which implies U = V. Thus each has size 1 and so t1,...,tk are pairwise distinct.	□
Lemma 3.13. Guv n R = Gvw n R =1.
Proof. In view of Lemma 3.9 we only need to prove that Gvw n R =1. Let x G Gvw n R. Then tk* = tk = 1, and so (3.6) shows that t(* = t( for all i such that 1 < i < k. Note that t1,..., tk are pairwise distinct by Lemma 3.12. We conclude that x = 1 and so Gvw n R = 1, as desired.	□
Lemma 3.14. k = |T|, {t1,... ,tk} = T and r = r(T) as given in Construction 3.1. Moreover, if G is vertex-primitive, then the induced permutation group of G on the k copies of T is a subgroup of Hol(T) containing Soc(Hol(T)).
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Proof. It follows from Lemma 3.9 that Guvw n (A x R) = Guv n (A x R). Then as G is 2-arc-transitive on r, we have
|Gv1 _ |Guv1 < _|Guv1__(3 8)
|Guv1 |Guvw1	|Guvw n (A x R)|
|G„vl _ |G„v(A x R)| < |A x S| = k
|Guv n (A x R)| |A x R| ^ |A x R|
We thus obtain |Gv | < k|Guv | = k|Gvw |. From Lemma 3.10 we deduce P(Guv) = Guv and P(Gvw) = Gvw. Moreover, 11,... ,tk are pairwise distinct by Lemma 3.12, which implies |T | > k. Therefore,
k|S| < |T||S| < |Gv n A||S| = |Gv| < k|G„v| = k|^(G„v)| < k|S|
and
k|S| < |T||S| < |Gv n A||S| = |Gv| < k|Gvw| = k|p(Gvw)| < k|S|.
Hence |G„ n A| = |T| = k, |G„ | = k|Guv | = k|Gvw | and P(Guv) = P(Gvw) = S. As a consequence, T = {ti,... ,tfc} by Lemma 3.12, and so r = Cos(Tk,D,g) = r(T). Also, (3.8) implies that Guvw = Guvw n (A x R). If Guv n S =1 or Gvw n S = 1, then Lemma 3.10 implies S = P(Guv) = Guv < A or S = p(Gvw) = Gvw < A, contradicting Lemma 2.14. Thus Guv nS and Gvw nS are both nontrivial normal subgroups of p (G„v ) = P(Gvw ) = S.
From now on suppose that G is primitive and so S is a primitive subgroup of Sk. By Lemma 3.13, Guv n R = Gvw n R = 1, so we derive that Guv n S and Gvw n S are both regular normal subgroups of S. Moreover, Guv n S = Gvw n S for otherwise Guvw n S = Guv n S would be a regular subgroup of S, contrary to the condition Guvw = Guvw n (A x R) < A x R. This indicates that S has at least two regular normal subgroups, and so Soc(S) = N2n for some nonabelian simple group N and positive integer n such that k = |N |n and S/(Guv n S) has a normal subgroup isomorphic to Nn. It follows that
Nn < S/(G„v n S) = p (G„v)/(G„v n S) = a (Guv )/(Gu„ n A) = a(Guv) < A,
and then Lemma 2.13 implies that n =1 and N = T. Thus, Soc(S) = T2, and so
Soc(Hol(T)) < S < Hol(T).	□
We are now ready to give the main theorem of this section. Recall X defined in (3.2).
Theorem 3.15. Let T be a nonabelian simple group, k > 2 be an interger, and Tk < G < T k.(Out(T) x Sk) with diagonal action on the set V of right cosets of {(t,..., t) 11 G T} in Tk. Suppose r is a connected (G, 2)-arc-transitive digraph with vertex set V. Then k = |T|, r = r(T), Aut(r) = X is vertex-primitive of type SD with socle Tk and the conjugation action on the k copies of T permutation isomorphic to Hol(T), and r is not 3-arc-transitive.
Proof. We have by Lemma 3.14 that k = |T|, {ti,..., tfc} = T and r = r(T). In the following, we identify r with r(T). Let X be as in (3.2) and Y = Aut(r(T)). Then X is vertex-primitive of type SD with socle T|T 1, and the conjugation action of X on the |T| copies of T is permutation isomorphic to Hol(T). Also, X < Y by Lemma 3.5. It follows from [1, Theorem 1.2] that Y is vertex-primitive of type SD with the same socle of X. Then again by Lemma 3.14 we have Yv < Aut(T) x Hol(T). Thus by (3.3) Yv = XK. Since X is vertex-transitive, it follows that Y = XYv = X, and so r is not 3-arc-transitive by Lemma 3.7.	□
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4 Product action on the vertex set
In this section, we study (G, s)-arc-transitive digraphs with vertex set Am such that G acts on Am by product action. We first prove Theorem 1.3.
Proof of Theorem 1.3. Let G1 be the stabiliser in G of the first coordinate and n1 be the projection of Gi into Sym(A). Then n1(G1) = H. Since N is normal in H and transitive on A, Nm is normal in G and transitive on Am = V. Hence Corollary 2.11 implies that r is (Nm, s — 1)-arc-transitive. In particular, since s > 2, Nm is transitive on the set of arcs of r, and so r has arc set A = {un ^ vn | n e Nm} for any arc u ^ v of r.
Let a e A, u = (a,..., a) e V and v = (p1,..., pm) be an out-neighbour of u in r. By Lemma 2.10 we have G = NmGuv. Let p be the projection of G to Sm, and we regard p(G) as a subgroup of Sym(V). Then
p(G) < HmG = Hm(NmGuv) = HmGuv.
Take any i in {1,..., m}. Since p(G) is transitive on {1,..., m}, there exists x e p(G) such that 1x = i and x = yz with y = (y1,..., ym) e Hm and z e Guv. Note that z e Guv and x e Sm both fix u. We conclude that y fixes u and hence yj e Ha for each
j in {1,..., m}. Also, y—1x = z e Guv < Gv implies pvii = pi. It follows that for each i in {1,... ,m} there exists hi e Ha with phi = p1. Let w = (p1,... ,p1) e V, h = (h1,..., hm) e (Ha)m and rh be the digraph with vertex set V and arc set Ah := {unh ^ vnh I n e Nm}. It is evident that uh = u, vh = w, and h gives an isomorphism from r to rh. Let E be the digraph with vertex set A and arc set I := {an ^ fin I n e N}. Then N < Aut(E), and viewing Nmh = hNm we have
Ah = {uhn ^ vhn I n e Nm} = {un ^ wn I n e Nm}
= {(ani ,...,anm) ^ (pn, .. ., ) | n1,...,nm e N}.
This implies that rh = Em. Consequently, r = Em.
For any p e A, denote by S(ft) the point in V = Am with all coordinates equal to p. Then S(a) ^ S(p1) in rh since a ^ p1 in E. Let x be any element of H. Then since
H = h-1Hh1 = h-1n1(G1)h1 = n1(h)-1n1(G1)n1(h) = n1(h-1G1h),
there exists g e h-1G1h such that x = n1(g). As g is an automorphism of rh and S(a) ^ S(p1) in rh, we have S(a)g ^ S(p1)g in rh. Comparing first coordinates, this implies that ani(s) ^ pni(g) in E, which turns out to be ax ^ PX in E. In other words, ax ^ PX is in I. It follows that
I = {axn ^ Pxn | n e N} = {anx ^ p\iX | n e N}
as xN = Nx. Hence x preserves I, and so H < Aut(E).
Let a0 ^ a1 ^ ■ ■ ■ ^ as be an s-arc of E. Since r is (Nm, s — 1)-arc-transitive and Nm = h-1Nmh, it follows that rh is (Nm, s — 1)-arc-transitive. Then for any (s — 1)-arc a[ ^ ■ ■ ■ ^ a's of E, since S(a1) ^ ■ ■ ■ ^ S(as) and S(a1) ^ ■ ■ ■ ^ S(a's) are both (s — 1)-arcs of rh, there exists (n1, ...,nm) e Nm such that S(ai)(ni'--n™) = S(ai) for each i with 1 < i < s. Hence an = ai for each i with 1 < i < s. Therefore, E is (N, s — 1)-arc-transitive. Let E+(as-1) be the set of out-neighbours of
as — 1 in
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E. Take any ft G E+(as_1). As S(as) and S(ft) are both out-neighbours of J(as_i) in rh and rh is (h_1Gh, s)-arc-transitive, there exists g G h_1Gh < H I Sm such that g fixes S(a0), J(a1),..., ¿(as-1) and maps S(as) to S(ft). Write g = (x1;..., xm)a with (x1;..., xm) G Hm and a G Sm. Then x1 fixes a0, a1,..., as-1 and maps as to ft. This shows that E is (H, s)-arc-transitive, completing the proof.	□
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