ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.02 / 207–216
https://doi.org/10.26493/1855-3974.2593.1b7
(Also available at http://amc-journal.eu)
Two-distance transitive normal Cayley graphs*
Jun-Jie Huang , Yan-Quan Feng † , Jin-Xin Zhou
School of mathematics and statistics, Beijing Jiaotong University,
Beijing 100044, P. R. China
Received 3 April 2021, accepted 10 July 2021, published online 14 April 2022
Abstract
In this paper, we construct an infinite family of normal Cayley graphs, which are 2-
distance-transitive but neither distance-transitive nor 2-arc-transitive. This answers a ques-
tion proposed by Chen, Jin and Li in 2019.
Keywords: Cayley graph, 2-distance-transitive graph, simple group.
Math. Subj. Class. (2020): 05C25, 05E18, 20B25
1 Introduction
In this paper, all graphs are finite, simple, and undirected. For a graph Γ , let V (Γ ), E(Γ ),
A(Γ ) or Aut(Γ ) denote its vertex set, edge set, arc set and its full automorphism group, re-
spectively. The graph Γ is called G-vertex-transitive, G-edge-transitive or G-arc-transitive,
with G ≤ Aut(Γ ), if G is transitive on V (Γ ), E(Γ ) or A(Γ ) respectively, and G-semi-
symmetric, if Γ is G-edge-transitive but not G-vertex-transitive. It is easy to see that a
G-semisymmetric graph Γ must be bipartite such that G has two orbits, namely the two
parts of Γ , and the stabilizer Gu for any u ∈ V (Γ ) is transitive on the neighbourhood of u
in Γ . An s-arc of Γ is a sequence v0, v1, . . . , vs of s+1 vertices of Γ such that vi−1, vi are
adjacent for 1 ≤ i ≤ s and vi−1 ̸= vi+1 for 1 ≤ i ≤ s− 1. If Γ has at least one s-arc and
G ≤ Aut(Γ ) is transitive on the set of s-arcs of Γ , then Γ is called (G, s)-arc-transitive,
and Γ is said to be s-arc-transitive if it is (Aut(Γ ), s)-arc-transitive.
For two vertices u and v in V (Γ ), the distance d(u, v) between u and v in Γ is the
smallest length of paths between u and v, and the diameter diam(Γ ) of Γ is the maximum
distance occurring over all pairs of vertices. For i = 1, 2, . . . ,diam(Γ ), denote by Γi(u)
*The work was supported by the National Natural Science Foundation of China (11731002, 12011540376,
12011530455, 12071023, 12161141005) and the 111 Project of China (B16002).
†Corresponding author.
E-mail addresses: 20118006@bjtu.edu.cn (Jun-Jie Huang), yqfeng@bjtu.edu.cn (Yan-Quan Feng),
jxzhou@bjtu.edu.cn (Jin-Xin Zhou)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
208 Ars Math. Contemp. 22 (2022) #P2.02 / 207–216
the set of vertices at distance i with vertex u in Γ . A graph Γ is called distance transitive
if, for any vertices u, v, x, y with d(u, v) = d(x, y), there exists g ∈ Aut(Γ ) such that
(u, v)g = (x, y). The graph Γ is called (G, t)-distance-transitive with G ≤ Aut(Γ ) if,
for each 1 ≤ i ≤ t, the group G is transitive on the ordered pairs of form (u, v) with
d(u, v) = i, and Γ is said to be t-distance-transitive if it is (Aut(Γ ), t)-distance-transitive.
Distance-transitive graphs were first defined by Biggs and Smith in [2], and they showed
that there are only 12 trivalant distance-transitive graphs. Later, distance-transitive graphs
of valencies 3, 4, 5, 6 and 7 were classified in [2, 10, 14, 25], and a complete classification
of distance-transitive graphs with symmetric or alternating groups of automorphisms was
given by Liebeck, Praeger and Saxl [18]. The 2-distance-transitive but not 2-arc-transitive
graphs of valency at most 6 were classified in [4, 16], and the 2-distance-primitive graphs
(a vertex stabilizer of automorphism group is primitive on both the first step and the second
step neighbourhoods of the vertex) with prime valency were classified in [15]. By defini-
tion, a 2-arc-transitive graph is 2-distance-transitive, but a 2-distance-transitive graph may
not be 2-arc-transitive; an example is the Kneser graph KG6,2, see [16]. Furthermore, Corr,
Jin and Schneider [5] investigated properties of a connected (G, 2)-distance-transitive but
not (G, 2)-arc-transitive graph of girth 4, and they applied the properties to classify such
graphs with prime valency. For more information about 2-distance-transitive graphs, we
refer to [6, 7].
For a finite group G and a subset S ⊆ G \ {1} with S = S−1 := {s−1 | s ∈ S}, the
Cayley graph Cay(G,S) of the group G with respect to S is the graph with vertex set G
and with two vertices g and h adjacent if hg−1 ∈ S. For g ∈ G, let R(g) be the permutation
of G defined by x 7→ xg for all x ∈ G. Then R(G) := {R(g) | g ∈ G} is a regular group
of automorphisms of Cay(G,S). It is known that a graph Γ is a Cayley graph of G if and
only if Γ has a regular group of automorphisms on the vertex set which is isomorphic to G;
see [1, Lemma 16.3] and [24]. A Cayley graph Γ = Cay(G,S) is called normal if R(G)
is a normal subgroup of Aut(Γ ). The study of normal Cayley graphs was initiated by Xu
[27] and has been investigated under various additional conditions; see [8, 22].
There are many interesting examples of arc-transitive graphs and 2-arc-transitive graphs
constructed as normal Cayley graphs. However, the status for 2-distance-transitive graphs
is different. Recently, 2-distance-transitive circulants were classified in [3], where the fol-
lowing question was proposed:
Question 1.1 ([3, Question 1.2]). Is there a normal Cayley graph which is 2-distance-
transitive, but neither distance-transitive nor 2-arc-transitive?
In this paper, we answer the above question by constructing an infinite family of such
graphs, which are Cayley graphs of the extraspecial p-groups of exponent p of order p3.
Theorem 1.2. For an odd prime p, let G = ⟨a, b, c | ap = bp = cp = 1, [a, b] = c, [c, a] =
[c, b] = 1⟩ and S = {ai, bi | 1 ≤ i ≤ p − 1}. Then Cay(G,S) is a 2-distance-transitive
normal Cayley graph that is neither distance-transitive nor 2-arc-transitive.
A clique of a graph Γ is a maximal complete subgraph, and the clique graph Σ of Γ is
defined to have the set of all cliques of Γ as its vertex set with two cliques adjacent in Σ
if the two cliques have at least one common vertex. Applying Theorem 1.2, we can obtain
the following corollary.
Corollary 1.3. Under the notation given in Theorem 1.2, let Cos(G, ⟨a⟩, ⟨b⟩) be the graph
with vertex set {⟨a⟩g | g ∈ G} ∪ {⟨b⟩h | h ∈ G} and with edges all these coset pairs
J.-J. Huang et al.: Two-distance transitive normal Cayley graphs 209
{⟨a⟩g, ⟨b⟩h} having non-empty intersection in G. Then Cos(G, ⟨a⟩, ⟨b⟩) is the clique
graph of Cay(G,S), and Cay(G,S) is the line graph of Cos(G, ⟨a⟩, ⟨b⟩). Furthermore,
Cos(G, ⟨a⟩, ⟨b⟩) is 3-arc-transitive.
The graph Cos(G, ⟨a⟩, ⟨b⟩) was first constructed in [19] as a regular cover of Kp,p,
where it is said that Cos(G, ⟨a⟩, ⟨b⟩) is 2-arc-transitive in [19, Theorem 1.1], but not 3-arc-
transitive generally for all odd primes p in a remark after [19, Example 4.1]. However, this
is not true and Corollary 1.3 implies that Cos(G, ⟨a⟩, ⟨b⟩) is always 3-arc-transitive for each
odd prime p. In fact, Cos(G, ⟨a⟩, ⟨b⟩) is 3-arc-regular, that is, Aut(Cos(G, ⟨a⟩, ⟨b⟩)) is
regular on the set of 3-arcs of Cos(G, ⟨a⟩, ⟨b⟩). Some more information about the structure
and symmetry properties of Cos(G, ⟨a⟩, ⟨b⟩) are given in Lemma 3.2.
2 Preliminaries
In this section we list some preliminary results used in this paper. The first one is the
well-known orbit-stabilizer theorem (see [9, Theorem 1.4A]).
Proposition 2.1. Let G be a group with a transitive action on a set Ω and let α ∈ Ω. Then
|G| = |Ω||Gα|.
The well-known Burnside paqb theorem was given in [12, Theorem 3.3].
Proposition 2.2. Let p and q be primes and let a and b be positive integers. Then a group
of order paqb is soluble.
The next proposition is an important property of a non-abelian simple group acting
transitively on a set with cardinality a prime-power, whose proof depends on the finite
simple group classification, and we refer to [13, Corollary 2] or [26, Proposition 2.4].
Proposition 2.3. Let T be a nonabelian simple group acting transitively on a set Ω with
cardinality a p-power for a prime p. If p does not divide the order of a point-stabilizer of
T , then T acts 2-transitively on Ω .
Let Γ = Cay(G,S) be a Cayley graph of a group G with respect to S. Then R(G)
is a regular subgroup of Aut(Γ ), and Aut(G,S) := {α ∈ Aut(G) | Sα = S} is also a
subgroup of Aut(Γ ), which fixes 1. Furthermore, R(G) is normalized by Aut(G,S), and
hence we have a semiproduct R(G) ⋊ Aut(G,S), where R(g)α = R(gα) for any g ∈ G
and α ∈ Aut(G,S). Godsil [11] proved that the semiproduct R(G)⋊Aut(G,S) is in fact
the normalizer of R(G) in Aut(Γ ). By Xu [27], we have the following proposition.
Proposition 2.4. Let Γ = Cay(G,S) be a Cayley graph of a finite group G with respect
to S, and let A = Aut(Γ ). Then the following hold:
(1) NA(R(G)) = R(G)⋊Aut(G,S);
(2) Γ is a normal Cayley graph if and only if A1 = Aut(G,S), where A1 is the stabilizer
of 1 in A.
Let Γ be a G-vertex-transitive graph, and let N be a normal subgroup of G. The normal
quotient graph ΓN of Γ induced by N is defined to be the graph with vertex set the orbits
of N and with two orbits B,C adjacent if some vertex in B is adjacent to some vertex in
C in Γ . Furthermore, Γ is called a normal N -cover of ΓN if Γ and ΓN have the same
valency.
210 Ars Math. Contemp. 22 (2022) #P2.02 / 207–216
Proposition 2.5. Let Γ be a connected G-vertex-transitive graph and let N be a normal
subgroup of G. Suppose that either Γ is an N -cover of ΓN , or Γ is G-arc-transitive of
prime valency and N has at least three orbits on vertices. Then the following statements
hold:
(1) N is semiregular on V Γ and is the kernel of G acting V (ΓN ), so G/N ≤ Aut(ΓN );
(2) Γ is (G, s)-arc-transitive if and only if ΓN is (G/N, s)-arc-transitive;
(3) Gα ∼= (G/N)δ for any α ∈ V Γ and δ ∈ V (ΓN ).
Proposition 2.5 was given in many papers by replacing the condition that Γ is a normal
N -cover of ΓN by one of the following assumptions: (1) N has at least 3-orbits and G
is 2-arc-transitive (see [21, Theorem 4.1]); (2) N has at least 3-orbits, G is arc-transitive
and Γ has a prime valency (see [20, Theorem 2.5]); (3) N has at least 3-orbits and G is
locally primitive (see [17, Lemma 2.5]). The first step for these proofs is to show that for
any two vertices B,C ∈ V (ΓN ), the induced subgraph [B] of B in Γ has no edge and if
B and C are adjacent in ΓN then the induced subgraph [B ∪ C] in Γ is a matching, which
is equivalent to that Γ is a normal N -cover of ΓN . Then Proposition 2.5(1) - (3) follows
from these proofs.
3 Proof Theorem 1.2
For a positive integer n and a prime p, we use Zn and Zrp to denote the cyclic group of
order n and the elementary abelian group of order pr, respectively. In this section, we
always assume that p is an odd prime, and denote by Z∗p the multiplicative group of Zp
consisting of all non-zero numbers in Zp. Note that Z∗p ∼= Zp−1. Furthermore, we also set
the following assumptions in this section:
G = ⟨a, b, c | ap = bp = cp = 1, [a, b] = c, [c, a] = [c, b] = 1⟩,
S = {ai, bi | 1 ≤ i ≤ p− 1},
Γ = Cay(G,S), A = Aut(Γ ), N = NA(R(G)) = R(G)⋊Aut(G,S), and Z∗p = ⟨t⟩.
By Proposition 2.4, NA(R(G)) = R(G) ⋊ Aut(G,S), and R(g)δ = R(gδ) for any
R(g) ∈ R(G) and δ ∈ Aut(G,S). Since G = ⟨S⟩, Γ is a connected Cayley graph of
valency 2(p− 1). Let
α : a 7−→ at, b 7−→ b, c 7−→ ct;
β : a 7−→ a, b 7−→ bt, c 7−→ ct;
γ : a 7−→ b, b 7−→ a, c 7−→ c−1.
It is easy to check that at, b, ct satisfy the same relations as a, b, c in G, that is,
[at, b] = ct, [ct, at] = [ct, b] = 1. By the von Dyck’s Theorem (see [23, 2.2.1]), α in-
duces an epimorphism from G to ⟨at, b, ct⟩, which must be an automorphism of G because
⟨at, b, ct⟩ = G. Similarly, β and γ are also automorphisms of G.
Lemma 3.1. Aut(G,S) = ⟨α, β, γ⟩ ∼= (Zp−1 × Zp−1) ⋊ Z2, and Γ is N -arc-transitive.
Furthermore, N has no normal subgroup of order p2.
J.-J. Huang et al.: Two-distance transitive normal Cayley graphs 211
Proof. Since Z∗p = ⟨t⟩, it is easy to check that αp−1 = βp−1 = γ2 = 1, αβ = βα and
αγ = β. Thus ⟨α, β, γ⟩ ∼= (Zp−1 × Zp−1)⋊ Z2. Clearly, α, β, γ ∈ Aut(G,S). To prove
Aut(G,S) = ⟨α, β, γ⟩ ∼= (Zp−1 × Zp−1) ⋊ Z2, it suffices to show that |Aut(G,S)| ≤
2(p− 1)2.
Clearly, ⟨α, β, γ⟩ is transitive on S, and hence Γ is N -arc-transitive. Since G = ⟨S⟩,
Aut(G,S) is faithful on S. By Proposition 2.1, |Aut(G,S)| = |S||Aut(G,S)a|, where
Aut(G,S)a is the stabilizer of a in Aut(G,S). Note that Aut(G,S)a fixes ai for each
1 ≤ i ≤ p− 1. Again by Proposition 2.1, |Aut(G,S)a| ≤ (p− 1)|Aut(G,S)a,b|, where
Aut(G,S)a,b is the subgroup of Aut(G,S) fixing a and b. Since G = ⟨a, b⟩, we obtain
Aut(G,S)a,b = 1, and then |Aut(G,S)| ≤ 2(p− 1)2, as required.
Let H ≤ N be a subgroup of order p2. Since R(G) is the unique normal Sylow p-
subgroup of N = R(G) ⋊ Aut(G,S), we have H ≤ R(G), and since |R(G) : H| = p,
we have H ⊴ R(G). Note that the center C := Z(R(G)) = ⟨R(c)⟩ and C ∩ H ̸= 1.
Thus, C ∩ H = C as |C| = p, implying C ≤ H . Since H/C is a subgroup of order p,
and R(G)/C = ⟨R(a)C⟩ × ⟨R(b)C⟩ ∼= Z2p, we have H/C = ⟨R(b)C⟩ or ⟨R(a)R(b)iC⟩
for some 0 ≤ i ≤ p − 1. It follows that H = ⟨R(b)⟩ × C or ⟨R(abi)⟩ × C for some
0 ≤ i ≤ p− 1.
Suppose H ⊴N . Since C is characteristic in R(G) and R(G) ⊴N , we have C ⊴N .
Recall that R(a)γ = R(aγ) = R(b). Then (⟨R(a)⟩×C)γ = ⟨R(b)⟩×C. This implies that
both ⟨R(a)⟩×C and ⟨R(b)⟩×C are not normal in N . Thus, H = ⟨R(abi)⟩×C for some
1 ≤ i ≤ p − 1. Since H ⊴ N , we have Hβ = H , that is, ⟨R(abti)⟩ × C = Hβ = H =
⟨R(abi)⟩ × C. It follows that ⟨R(abti)⟩ = ⟨R(abi)⟩ and then R(abti) = R(abi), which
further implies bti = bi. This gives rise to p | i(t− 1), and since (i, p) = 1, we have t = 1,
contradicting that Z∗p = ⟨t⟩ ∼= Zp−1. Thus, N has no normal subgroup of order p2.
For a positive integer n, np denotes the largest p-power diving n. By Lemma 3.1,
Γ = Cay(G,S) is N -arc-transitive.
Lemma 3.2. The clique graph Σ of Γ is a connected p-valent bipartite graph of order 2p2,
A has a faithful natural action on Σ , and Σ is R(G)-semisymmetric and N -arc-transitive.
Furthermore, |A|p = p3.
Proof. Recall that G = ⟨a, b, c | ap = bp = cp = 1, [a, b] = c, [c, a] = [c, b] = 1⟩ and
S = {ai, bi | 1 ≤ i ≤ p − 1}. Then Γ = Cay(G,S) has exactly two cliques passing
through 1, that is, the induced subgraphs of ⟨a⟩ and ⟨b⟩ in Γ . Since R(G) ≤ Aut(Γ )
is transitive on vertex set, each clique of Γ is an induced subgraph of the coset ⟨a⟩x or
⟨b⟩x for some x ∈ G. Thus, we may view the vertex set of Σ as {⟨a⟩x, ⟨b⟩x | x ∈ G}
with two cosets adjacent in Σ if they have non-empty intersection. It is easy to see that
⟨a⟩x ∩ ⟨b⟩y ̸= ∅ if and only if |⟨a⟩x ∩ ⟨b⟩y| = 1, and any two distinct cosets, either in
{⟨a⟩x | x ∈ G} or in {⟨b⟩x | x ∈ G}, have empty intersection. Furthermore, ⟨a⟩ has
non-empty intersection with exactly p cosets, that is, ⟨b⟩ai for 0 ≤ i ≤ p− 1. Thus, Σ is a
p-valent bipartite graph of order 2p2. The connectedness of Σ follows from that of Γ .
Clearly, A has a natural action on Σ . Let K be the kernel of A on Σ . Then K fixes
each coset of ⟨a⟩x and ⟨b⟩x for all x ∈ G. Since ⟨a⟩x ∩ ⟨b⟩x = {x}, K fixes x and hence
K = 1. Thus, A is faithful on Σ and we may let A ≤ Aut(Σ ).
Note that R(G) is not transitive on {⟨a⟩x, ⟨b⟩x | x ∈ G}, but transitive on
{⟨a⟩x | x ∈ G} and {⟨b⟩x | x ∈ G}. Furthermore, R(⟨a⟩) fixes ⟨a⟩ and is transitive
on {⟨b⟩ai | 0 ≤ i ≤ p− 1}, the neighbourhood of ⟨a⟩ in Σ , and similarly, R(⟨b⟩) fixes ⟨b⟩
212 Ars Math. Contemp. 22 (2022) #P2.02 / 207–216
and is transitive on the neighbourhood {⟨a⟩bi | 0 ≤ i ≤ p−1} of ⟨b⟩ in Σ . It follows that Σ
is R(G)-semisymmetric. Recall that N = R(G)⋊Aut(G,S) and Aut(G,S) = ⟨α, β, γ⟩.
Since aγ = b and bγ = a, γ interchanges {⟨a⟩x | x ∈ G} and {⟨b⟩x | x ∈ G}. This yields
that Σ is R(G)⋊ ⟨γ⟩-arc-transitive and hence N -arc-transitive.
Since Σ is a connected graph with prime valency p, we have p2 ∤ |Aut(Σ )u| for
any u ∈ V (Σ ), and in particular, p2 ∤ |Au|. Note that p | |Au|. By Proposition 2.1,
|A| = |Σ ||Au| = 2p2|Au|. This implies that |A|p = p3.
Lemma 3.3. A = Aut(Γ ) = R(G)⋊Aut(G,S).
Proof. By Lemma 3.2, |A|p = p3, and since |V (Γ )| = p3 and A is vertex-transitive on
V (Γ ), the vertex stabilizer A1 is a p′-group, that is, p ∤ |A1|. To prove the lemma, by
Proposition 2.4 we only need to show that R(G)⊴A, and since R(G) is a Sylow p-subgroup
of A, it suffices to show that A has a normal Sylow p-subgroup.
Let M be a minimal normal subgroup of A. Then M = T1×T2 · · ·×Td, where Ti ∼= T
for each 1 ≤ i ≤ d with a simple group T . Since |V (Γ )| = p3, each orbit of M has length
a p-power and hence each orbit of Ti has length a p-power. It follows that p | |T |. Assume
that |T |p = pℓ. Then |M |p = pdℓ and dℓ = 1, 2 or 3 as |A|p = p3.
We process the proof by considering the two cases: M is insoluble or soluble.
Case 1: M is insoluble.
In this case, T is a non-abelian simple group. We prove that this case cannot happen by
deriving contradictions. Recall that dℓ = 1, 2 or 3.
Assume that dℓ = 1. Then |M |p = p. By Lemma 3.2, M ⊴ A ≤ Aut(Σ ), and since
|V (Σ )| = 2p2, M has at least three orbits. Since Σ has valency p, Proposition 2.5 implies
that M is semiregular on V (Σ ) and hence |M | | 2p2. By Proposition 2.2, M is soluble, a
contradiction.
Assume that dℓ = 2. Since R(G) is a Sylow p-subgroup of A and M ⊴ A, R(G) ∩M
is a Sylow p-subgroup of M and hence |R(G) ∩M | = |M |p = p2. Since R(G)⊴N and
M ⊴ A, M ∩R(G) is a normal subgroup of order p2 in N , contradicting to Lemma 3.1.
Assume that dℓ = 3. Then (d, ℓ) = (1, 3) or (3, 1). Since |M |p = p3 = |A|p, we
deduce R(G) ≤ M and hence M is transitive on Γ .
For (d, ℓ) = (1, 3), M is a non-abelian simple group. Since M1 ≤ A1 is a p′-group,
Proposition 2.3 implies that M is 2-transitive on Γ , forcing that Γ is the complete graph of
order p3, a contradiction.
For (d, ℓ) = (3, 1), we have M = T1 × T2 × T3. Then |M |p = p3, and since M ⊴ A,
we derive R(G) ≤ M . By Lemma 3.2 M ≤ Aut(Σ ), and Σ is R(G)-semisymmetric.
Since M has no subgroup of index 2, M fixes the two parts of Σ setwise, and hence Σ is
M -semisymmetric. Noting that γ interchanges the two parts of Σ , we have that Σ is M⟨γ⟩-
arc-transitive. Since γ is an involution, under conjugacy it fixes Ti for some 1 ≤ i ≤ 3, say
T1. Then T1 ⊴ ⟨M,γ⟩ and by Proposition 2.5, T1 is semiregular on Σ . This gives rise to
|T1| | 2p2, contrary to the simplicity of T1.
Case 2: M is soluble.
Since p | |M |, we have M = Zdp with 1 ≤ d ≤ 3. If d = 3 then A has a normal Sylow
p-subgroup, as required. If d = 2 then M ≤ R(G) ≤ N and N has a normal subgroup
of order p2, contrary to Lemma 3.1. Thus, we may let d = 1, and since M ≤ R(G) and
R(G) has a unique normal subgroup of order p that is the center of R(G), we derive that
M = ⟨R(c)⟩.
J.-J. Huang et al.: Two-distance transitive normal Cayley graphs 213
Now it is easy to see that the quotient graph ΓM = Cay(G/M,S/M) with S/M =
{aiM, biM | 1 ≤ i ≤ p − 1}. Note that G/M = ⟨aM⟩ × ⟨bM⟩ ∼= Z2p. Then ΓM is a
connected Cayley graph of order p2 with valency 2(p − 1), so Γ is a normal M -cover of
ΓM . By Proposition 2.5, we may let A/M ≤ Aut(ΓM ) and ΓM is A/M -arc-transitive.
Let H/M be a minimal normal subgroup of A/M . Then H ⊴A and H/M = L1/M ×
· · · × Lr/M , where Li ⊴H and Li/M (1 ≤ i ≤ r) are isomorphic simple groups. Since
|ΓM | = p2, we infer p | |H/M | and similarly, p | |Li/M |. Let |Li/M |p = ps. Then
|H/M |p = prs, and since |A/M |p = p2, we obtain that sr = 1 or 2.
We finish the proof by considering the two subcases: H/M is insoluble or soluble.
Subcase 2.1: H/M is insoluble.
In this subcase, Li/M are isomorphic non-abelain simple groups. We prove this sub-
case cannot happen by deriving contradictions. Recall that sr = 1 or 2.
Let sr = 1. Then |H/M |p = p, and therefore |H|p = p2. Since H ⊴ A, H ∩ R(G)
is a Sylow p-subgroup of H , implying |H ∩ R(G)| = p2, and then R(G) ⊴N yields that
H ∩R(G) is a normal subgroup of order p2 in N , contrary to Lemma 3.1.
Let rs = 2. Then |H/M |p = p2 and |H|p = p3. This yields R(G) ≤ H and H is
transitive on Γ , so H/M is transitive on V (ΓM ). Note that (r, s) = (1, 2) or (2, 1).
For (r, s) = (1, 2), H/M is a nonabelian simple group. By Propostion 2.5, (H/M)u
for u ∈ V (ΓM ) is a p′-group because H1 ≤ A1 is a p′-group, and by Proposition 2.3, H/M
is 2-transitive on V (ΓM ), forcing that ΓM is a complete group of order p2, a contradiction.
For (r, s) = (2, 1), H/M ∼= L1/M × L2/M , where L1/M and L2/M are isomorphic
nonabelain simple groups and |Li/M |p = p. It follows that |H|p = p3 and |Li|p = p2 for
1 ≤ i ≤ 2. Since H ⊴ A, we derive R(G) ≤ H . Note that H has no subgroup of index 2.
Since Σ is bipartite, it is H-semisymmetric. Let ∆1 and ∆2 be the two parts of Σ . Then
|∆1| = |∆2| = p2, and H is transitive on both ∆1 and ∆2.
Suppose (L1)u = 1 for some u ∈ V (Σ ) = ∆1∪∆2. By Proposition 2.1, |L1| = |uL1 |,
and since L1 ⊴ H and |∆1| = |∆2| = p2, we derive |L1| = p or p2, contrary to the
insolubleness of L1. Thus (L1)u ̸= 1. Since Σ has prime valency p, Hu is primitive on the
neighbourhood Σ (u) of u in Σ , and since (L1)u ⊴Hu, (L1)u is transitive on Σ (u), which
implies that |(L1)u|p = p. Since |L1|p = p2, each orbit of L1 on ∆1 or ∆2 has length p.
Let x ∈ ∆1 and y ∈ ∆2 be adjacent in Σ , and let ∆11 and ∆21 be the orbits of L1
containing x and y, respectively. Then |∆11| = |∆21| = p. Since (L1)x is transitive on
Σ (x), x is adjacent to each vertex in ∆21, and therefore, each vertex in ∆11 is adjacent
to each vertex in ∆21, that is, the induced subgroup [∆11 ∪∆21] is the complete bipartite
graph Kp,p. It follows that Σ ∼= pKp,p, contrary to the connectedness of Σ .
Subcase 2.2: H/M is soluble.
In this case, |H| = p2 or p3. Recall that H ⊴ A. If |H| = p2 then H ≤ R(G) and
N has normal subgroup of order p2, contradicts Lemma 3.1. Thus, |H| = p3 and A has a
normal Sylow p-subgroup, as required. This completes the proof.
Now we are ready to finish the proof.
Proof of Theorem 1.2. By Lemmas 3.1 and 3.3, Γ is a arc-transitive normal Cayley graph.
In particular, Γ is 1-distance transitive. Since S = {ai, bi | 1 ≤ i ≤ p− 1}, Γ has girth 3,
so it is not 2-arc-transitive.
214 Ars Math. Contemp. 22 (2022) #P2.02 / 207–216
Recall that G = ⟨a, b, c | ap = bp = cp = 1, [a, b] = c, [c, a] = [c, b] = 1⟩. Clearly,
Γ1(1) = S = {ai, bi | 1 ≤ i ≤ p− 1},
Γ2(1) = {bjai, ajbi | 1 ≤ i, j ≤ p− 1}.
Note that Aut(G,S) = ⟨α, β, γ | αp−1 = βp−1 = γ2 = 1, αβ = α, αγ = β⟩, where
aα = at, bα = b, cα = ct, aβ = a, bβ = bt, cβ = ct, aγ = b, bγ = a and cγ = c−1.
Then (ba)α
iβj = bt
i
at
j
, and since Z∗p = ⟨t⟩, we obtain that ⟨α, β⟩ is transitive on the set
{bjai | 1 ≤ i, j ≤ p − 1}. Similarly, ⟨α, β⟩ is transitive on {ajbi | 1 ≤ i, j ≤ p − 1}.
Furthermore, γ interchanges the two sets {bjai | 1 ≤ i, j ≤ p− 1} and {ajbi | 1 ≤ i, j ≤
p−1}. It follows that Aut(G,S) is transitive on Γ2(1) and hence Γ is 2-distance transitive.
Noting that ab = bac, we have that b−1ab = ac ∈ Γ3(1) and aba = ba2c ∈ Γ3(1).
Also it is easy to see that (ac)Aut(G,S) = (ac)⟨α,β,γ⟩ = {aicj , bicj | 1 ≤ i, j ≤ p−1}. Now
it is easy to see that ba2c ̸∈ (ac)Aut(G,S), and since A1 = Aut(G,S) by Proposition 2.4,
Γ is not distance-transitive.
Proof of Corollary 1.3. Recall that Σ is the clique graph of Γ . By the first paragraph
in the proof of Lemma 3.2 and the definition of Cos(G, ⟨a⟩, ⟨b⟩) in Corollary 1.3, we
have Σ = Cos(G, ⟨a⟩, ⟨b⟩). Again by Lemma 3.2, Σ is R(G)-semisymmetric, and since
|E(Σ )| = (2p2 · p)/2 = p3 = |R(G)|, R(G) is regular on the edge set E(Σ ) of Σ . Thus,
the line graph of Σ is a Cayley graph on G.
For a given edge {⟨a⟩x, ⟨b⟩y} ∈ E(Σ ), we have |⟨a⟩x ∩ ⟨b⟩y| = 1, and then we may
identify this edge with the unique element in ⟨a⟩x∩⟨b⟩y. Note that Σ has valency 2(p−1).
Then the edge 1 = ⟨a⟩ ∩ ⟨b⟩ in Σ is exactly incident to all edges in S = {ai, bi | 1 ≤ i ≤
p− 1}, because {ai} = ⟨a⟩∩ ⟨b⟩ai and {bi} = ⟨b⟩∩ ⟨a⟩bi. It follows that Γ = Cay(G,S)
is exactly the line graph of Σ .
If α ∈ Aut(Σ ) fixes each edge in Σ then α fixes all vertices of Σ , that is, Aut(Σ ) acts
faithfully on Γ . Thus, we may view Aut(Σ ) as a subgroup of Aut(Γ ). By Lemmas 3.2
and 3.3, we have Aut(Γ ) = Aut(Σ ) = R(G)⋊Aut(G,S).
Recall that Aut(G,S) = ⟨α, β, γ⟩ and Σ is arc-transitive. Since aβ = a, bβ = bt and
cβ = ct, where Z∗p = ⟨t⟩, ⟨β⟩ fixes the arc (⟨a⟩, ⟨b⟩) in Σ and is transitive on the vertex set
{⟨a⟩bi | 1 ≤ i ≤ p − 1}, where {⟨a⟩} ∪ {⟨a⟩bi | 1 ≤ i ≤ p − 1} is the neighbourhood of
⟨b⟩ in Σ . Thus, Σ is 2-arc-transitive. Since aα = at, bα = b and cα = ct, ⟨α⟩ fixes the
2-arc (⟨a⟩, ⟨b⟩, ⟨a⟩b) and is transitive on the vertex set {⟨b⟩aib | 1 ≤ i ≤ p − 1}, where
{⟨b⟩} ∪ {⟨b⟩aib | 1 ≤ i ≤ p − 1} is the neighbourhood of ⟨a⟩b in Σ . It follows that Σ is
3-arc-transitive. It is easy to see that the number of 3-arcs in Σ equals to |A| = 2p3(p−1)2,
A is regular on the set of 3-arcs of Σ .
ORCID iDs
Jun-Jie Huang https://orcid.org/0000-0002-9250-0672
Yan-Quan Feng https://orcid.org/0000-0003-3214-0609
Jin-Xin Zhou https://orcid.org/0000-0002-8353-896X
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