ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 23 (2023) #P2.10
https://doi.org/10.26493/1855-3974.2857.07b
(Also available at http://amc-journal.eu)
Locally s-arc-transitive graphs arising from
product action*
Michael Giudici
Department of Mathematics and Statistics, The University of Western Australia,
Perth WA 6009, Australia
Eric Swartz †
Department of Mathematics, William & Mary,
P.O. Box 8795, Williamsburg, VA 23187-8795, USA
Received 30 March 2022, accepted 14 September 2022, published online 13 December 2022
Abstract
We study locally s-arc-transitive graphs arising from the quasiprimitive product action
(PA). We prove that, for any locally (G, 2)-arc-transitive graph with G acting quasiprimi-
tively with type PA on both G-orbits of vertices, the group G does not act primitively on
either orbit. Moreover, we construct the first examples of locally s-arc-transitive graphs of
PA type that are not standard double covers of s-arc-transitive graphs of PA type, answering
the existence question for these graphs.
Keywords: Locally s-arc-transitive graph, quasiprimitive group, product action.
Math. Subj. Class. (2020): 20B25, 05C25, 05E18
1 Introduction
For an integer s ⩾ 1, an s-arc in a graph Γ is an (s+ 1)-tuple (α0, α1, . . . , αs) of vertices
such that αi ∼ αi+1 and αi ̸= αi+2 for each i. We say that Γ is s-arc-transitive if Γ
contains an s-arc and the automorphism group of Γ acts transitively on the set of all s-arcs.
*This paper formed part of the Australian Research Council’s Discovery Project DP120100446 of the first au-
thor. The authors would also like to thank Ákos Seress, who made this collaboration possible in 2012 by allowing
the second author to visit Australia, and Luke Morgan, for providing a proof of Lemma 3.2 and encouraging the
completion of the project. Finally, the authors wish to thank the anonymous referees for their detailed comments
and suggestions that greatly improved the final version of this paper.
†Corresponding author.
E-mail addresses: michael.giudici@uwa.edu.au (Michael Giudici), easwartz@wm.edu (Eric Swartz)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 23 (2023) #P2.10
If Γ is s-arc-transitive and each (s − 1)-arc can be extended to an s-arc then any s-arc-
transitive graph is also (s − 1)-arc-transitive. The study of s-arc-transitive graphs goes
back to the pioneering work of Tutte [32, 33], who showed that if Γ has valency three then
s ⩽ 5. Weiss [35] later showed that if the valency restriction is relaxed to allow valency at
least three then s ⩽ 7, with equality holding for the generalised hexagons arising from the
groups G2(q) for q = 3f .
Praeger [25] initiated a programme for the study of finite connected s-arc-transitive
graphs by first showing that if G ⩽ Aut(Γ) acts transitively on the set of all s-arcs of Γ
and N ◁ G has at least three orbits on the set of vertices, then the quotient graph ΓN
whose vertices are the orbits of N is also s-arc-transitive. Moreover, Γ is a cover of ΓN .
This reduces the study of finite connected (G, s)-arc-transitive graphs to two basic types:
• those where G is quasiprimitive on the set of vertices, that is, where all nontrivial
normal subgroups of G are transitive on vertices;
• those where G is biquasiprimitive on the set of vertices, that is, where all nontrivial
normal subgroups of G have at most two orbits on vertices and there is a normal
subgroup with two orbits.
Praeger showed that of the eight types of finite quasiprimitive groups, only four — HA
(affine), TW (twisted wreath), AS (almost simple) and PA (product action) — can act 2-
arc-transitively on a graph [25]. We use the types of quasiprimitive groups as given in [27]
and define type PA, the main focus of this paper, in Section 2. These are slight variations
on the types of primitive permutation groups given by the O’Nan–Scott Theorem. All
graphs of type HA were classified by Praeger and Ivanov [18] while those of type TW were
studied by Baddeley [1]. The 2-arc-transitive graphs for some families of almost simple
groups have all been classified, for example the Suzuki groups [9], Ree groups [8] and
PSL(2, q) [16]. The first examples of 2-arc-transitive graphs of PA type were given by
Li and Seress [22] and studied further by Li, Seress, and Song [23]. Another family of
quasiprimitive 2-arc-transitive graphs of PA type were constructed by Li, Ling, and Wu in
[21].
In the biquasiprimitive case the graph is bipartite and such graphs were investigated in
[26, 28]. An alternative way to study such graphs is via the notion of local s-arc-transitivity.
We say that a graph Γ is locally (G, s)-arc-transitive for a group G ⩽ Aut(Γ) if for each
vertex α, the vertex stabiliser Gα acts transitively on the set of all s-arcs starting at α.
If G also acts transitively on the set of vertices then Γ is s-arc-transitive. If Γ is locally
(G, s)-arc-transitive but G is intransitive on the set of vertices, then G has two orbits on
vertices and Γ is bipartite. One way to construct locally s-arc-transitive graphs is to start
with an s-arc-transitive graph Γ and take its standard double cover Σ, which has vertex set
V Γ × {1, 2} and (α, i) ∼ (β, j) precisely when i ̸= j and α ∼ β in Γ. Then Aut(Γ)
acts as automorphisms on Σ with two orbits on vertices and Σ is locally (Aut(Γ), s)-arc-
transitive [11].
If Γ is a bipartite graph and G ⩽ Aut(Γ) acts transitively on the set of vertices, then
Γ is locally (G+, s)-arc-transitive where G+ is the index two subgroup that stabilises each
part of the bipartition. Hence the study of locally s-arc-transitive graphs encompasses the
study of all bipartite s-arc-transitive graphs and hence the biquasiprimitive case in Praeger’s
programme. It is also a wider class of graphs as the known generalised octagons are locally
9-arc-transitive but not vertex-transitive, and it has been shown by van Bon and Stellmacher
[34] that this is best possible.
M. Giudici and E. Swartz: Locally s-arc-transitive graphs arising from product action 3
A programme for the study of finite connected locally s-arc-transitive graphs was mapped
out by Giudici, Li and Praeger [11]. If Γ is locally (G, s)-arc-transitive with G having two
orbits on vertices and N ◁ G is intransitive on both G-orbits, then the quotient graph ΓN
is also locally s-arc-transitive. Moreover, Γ is a cover of ΓN . This reduces the study of
finite connected locally (G, s)-arc-transitive graphs for which G is vertex-intransitive into
two basic types:
• those where G is quasiprimitive on each of its two orbits on vertices;
• those where G is quasiprimitive on only one of its two orbits on vertices.
In the second case, it was shown [11] that the quasiprimitive action must be of type HA, HS,
AS, PA or TW. These were further studied in [12] where all examples where the quasiprim-
itive action has type PA preserving a product structure or type HS were classified. An in-
finite family of examples where the quasiprimitive action has type TW was given by Kaja
and Morgan [19]. In the first case, either the two quasiprimitive actions have the same
quasiprimitive type and are one of HA, AS, TW or PA, or they are different with one of
type SD and one of type PA [11]. All 2-arc-transitive graphs of the latter type were classi-
fied in [13] and there are locally 5-arc-transitive examples in this case [14]. It was shown in
[17, Lemma 3.2] that all locally 2-arc-transitive graphs where the quasiprimitive action is
of type HA on both orbits are actually vertex-transitive but a complete classification has not
been obtained – see [18, Section 2] for further discussion. All locally (G, 2)-arc-transitive
graphs have been classified in the cases where G is an almost simple group whose socle is
a Ree group [7], Suzuki group [31], or PSL(2, q) [3], while the sporadic group case was
studied in [20]. Examples also exist in the PA and TW cases as we can take standard double
covers of s-arc-transitive graphs of type PA and TW respectively.
The aim of this paper is to study locally s-arc-transitive graphs of PA type. We prove
that, for any locally (G, 2)-arc-transitive graph with G acting quasiprimitively with type PA
on both G-orbits of vertices, the group G does not act primitively on either orbit. Moreover,
in the spirit of [22], we solve the existence problem for locally 2-arc-transitive graphs of
PA type. In particular, we construct the first examples of locally s-arc-transitive graphs of
PA type that are not standard double covers of s-arc-transitive graphs of PA type.
2 PA type
Let G act quasiprimitively on a set Ω. We say that G has type PA if there exists a G-invariant
partition B of Ω such that G acts faithfully on B and we can identify B with ∆k for some
set ∆ and k ⩾ 2 such that G ⩽ H wrSk acts in the usual product action of a wreath
product on ∆k, where H ⩽ Sym(∆) is an almost simple group acting quasiprimitively on
∆. Moreover, if T = soc(H) then G has a unique minimal normal subgroup N = T k.
Note that since G is quasiprimitive, N acts transitively on Ω and hence on B. Thus G =
NGα = NGB , where B ∈ B is a block containing α ∈ Ω. As N is minimal normal
in G we have that G transitively permutes the simple direct factors of N and hence so do
both Gα and GB . Thus given B = (δ, . . . , δ) ∈ B we may assume that NB = T kδ and for
α ∈ B we have that Nα is a subdirect subgroup of NB , that is, the projection of Nα onto
each direct factor is isomorphic to Tδ .
Let R = Tδ . Following the terminology of [22], if Nα ∼= R then we call Nα a diagonal
subgroup of NB = Rk. Then there exists automorphisms φ2, φ3, . . . , φk of R such that
Nα = {(t, tφ2 , . . . , tφk) | t ∈ R}.
4 Ars Math. Contemp. 23 (2023) #P2.10
If each of the φi is the trivial automorphism then we call Nα a straight diagonal subgroup
while if some φi is nontrivial then we call Nα a twisted diagonal subgroup. Furthermore, if
Nα ̸∼= R then we refer to Nα as being a nondiagonal subgroup. We refer to the quasiprim-
itive permutation group G of type PA as being of straight diagonal, twisted diagonal, or
nondiagonal type according to the type of Nα.
Note that unlike for primitive groups of type PA, G does not necessarily preserve a
product structure on Ω, only on some G-invariant partition B. Indeed the following result
shows that for locally 2-arc-transitive graphs this partition must be nontrivial on each of the
bipartite halves.
Theorem 2.1. Let Γ be a locally (G, 2)-arc-transitive connected graph with G quasiprim-
itive of type PA on both orbits Ω1 and Ω2. Let N = T k = soc(G) and for i = 1, 2, let Bi
be a G-invariant partition of Ωi such that G preserves a product structure ∆ki on each Bi.
Then Bi ̸= Ωi for each i.
Proof. Suppose that Bi = Ωi for some i. Without loss of generality suppose that i = 1.
Also note that there is an almost simple group H with socle T such that G ⩽ H wrSk.
Let α = (ω, . . . , ω) ∈ Ω1. Then Nα = T kω with Tω ̸= 1 and Gα = G ∩ (Hω wrSk).
By [11, Lemma 3.2], GΓ(α)α is 2-transitive so either all neighbours of α lie in the same
block of B2 or in distinct blocks. If they all lie in the same block then for each β ∈ Ω1
we have that the neighbours of β lie in the same block. However, this contradicts Γ being
connected. Hence for each α ∈ Ω1, the neighbours of α lie in distinct blocks. Hence
Gα acts 2-transitively on the set X of blocks of B2 that contain neighbours of α. By [11,
Lemma 6.2], NΓ(α)α is a transitive subgroup of the 2-transitive group G
Γ(α)
α and so Nα also
acts transitively on X . Let B = (δ1, δ2, . . . , δk) ∈ B2 be a block containing a neighbour γ
of α. Then X = (δ1, δ2, . . . , δk)Nα = δTω1 × δ
Tω
2 × · · · × δ
Tω
k . By [29, Theorem 1.1(b)],
the stabiliser G1 in G of the first simple direct factor of N projects onto H in the first
coordinate and so (G1)α projects onto Hω in the first coordinate. Hence δTω1 = δ
Hω
1 . Since
Gα ⩽ Hω wrSk and transitively permutes the k simple direct factors of N , it follows that
δTωi = δ
Tω
1 for each i. In particular, X = A
k for some set A and we could have chosen
B = (δ, . . . , δ) for some δ ∈ ∆2. Thus Gαγ ⩽ Gα,B ⩽ Hωδ wrSk. However, for
δ′ ∈ A\{δ} there is no element of Hωδ wrSk mapping (δ′, δ, . . . , δ) to (δ′, δ′, δ, . . . , δ),
contradicting Gα acting 2-transitively on X . Thus B1 ̸= Ω1.
Corollary 2.2. Let Γ be a locally (G, 2)-arc-transitive connected graph with G quasiprim-
itive of type PA on both orbits. Then G is not primitive on either orbit.
3 Constructions
Let G be a finite group with subgroups L and R. Let ∆1 be the set [G : L] of right cosets
of L in G and ∆2 be the set [G : R] of right cosets of R in G. We define the coset graph
Γ = Cos(G,L,R) to be the bipartite graph with vertex set the disjoint union ∆1 ∪ ∆2
such that {Lx,Ry} is an edge if and only if Lx ∩ Ry ̸= ∅, or equivalently xy−1 ∈ LR.
Then G acts by right multiplication on both ∆1 and ∆2, and induces automorphisms of
Γ. Note that the vertices in ∆1 have valency |L : L ∩ R| while the vertices in ∆2 have
valency |R : L ∩ R|. We say that Γ has valency {|L : L ∩ R|, |R : L ∩ R|}. Conversely,
if Γ is a graph and G ⩽ Aut(Γ) acts transitively on the set of edges of Γ but not on the set
of vertices then Γ can be constructed in this way [11, Lemma 3.7]. We refer to the triple
(L,R,L ∩R) as the associated amalgam.
M. Giudici and E. Swartz: Locally s-arc-transitive graphs arising from product action 5
We collect the following properties of coset graphs. We say that a subgroup H of a
group G is core-free if ∩g∈GHg = 1.
Lemma 3.1 ([11, Lemma 3.7]). Let G be a group with proper subgroups L and R, and let
Γ = Cos(G,L,R).
(1) Γ is connected if and only if G = ⟨L,R⟩.
(2) G acts faithfully on both [G : L] and [G : R] if and only if both L and R are core
free in G.
(3) G acts transitively on the set of edges of Γ.
(4) Γ is locally (G, 2)-arc-transitive if and only if L acts 2-transitively on [L : L ∩ R]
and R acts 2-transitively on [R : L ∩R].
We also need the following result, which essentially follows from the definition of a
completion (and the universal completion) of an amalgam (see [15]) and results on covers
of graphs (see, e.g., [2, Chapter 19]). The result is truly “folklore”: while it seems to be
taken for granted in the field, we also cannot find an explicit proof in the literature. We
have included a proof here provided by Luke Morgan [24].
Lemma 3.2. If Γ is a locally s-arc-transitive graph with amalgam (L,R,L ∩ R) and
s ⩾ 2, then any other graph with amalgam (L,R,L ∩R) is locally s-arc-transitive.
Proof. Let G := L ∗L∩RR be the universal completion of (L,R,L∩R) and let Γ∗ denote
the universal tree on which G acts edge-transitively. We identify L and R with their images
in G, and label an edge {α, β} so that Gα = L, Gβ = R, and Gαβ = L ∩ R. Since Γ
is locally s-arc-transitive for s ⩾ 2, it is locally 2-arc-transitive and so the actions of L on
the set of right cosets of L ∩ R in L, and of R on the set of right cosets of L ∩ R in R are
2-transitive [11, Lemma 3.2]. In particular, Γ∗ is locally (G, 2)-arc-transitive.
Now let Σ be a graph with edge-transitive group of automorphisms H such that the
amalgam (Hγ , Hδ, Hγδ) is isomorphic to (L,R,L∩R), where {γ, δ} is an edge of Σ. By
the universal property of G and of Γ∗, there is a map ϕ : G → H such that the following
diagrams commute:
L G R G L ∩R G
Hγ H Hδ H Hγδ H
.
Let N be the kernel of ϕ. Then, Σ = Γ∗N , the quotient graph, and the kernel of the
action of G on Σ is exactly N .
In particular, ϕ(Gα) = Hγ and ϕ(Gβ) = Hδ . Further, since ϕ(Gαβ) = Hγδ, we have
commutative diagrams of the following groups:
Gα G
Γ∗(α)
α Gβ G
Γ∗(β)
β
Hγ H
Σ(γ)
γ Hδ H
Σ(δ)
δ
,
where GΓ
∗(α)
α denotes the induced action of Gα on Γ∗(α), etc.
6 Ars Math. Contemp. 23 (2023) #P2.10
We now claim that for ε = γ, δ and ζ ∈ Γ∗(ε), we have ζN ∩ Γ∗(ε) = {ζ}. Indeed,
this follows since |Gα : Gαβ | = |Hγ : Hγδ| and |Gβ : Gαβ | = |Hδ : Hγδ|.
Now suppose Γ∗ is locally (G, r)-arc-transitive and Σ is locally (H, t)-arc-transitive.
By [11, Lemma 5.1(3)], we have t ⩾ r.
Assume that r < t. We will show that Γ∗ would be locally (G, r + 1)-arc-transitive in
this case, contradicting the maximality of r.
Suppose P and P ′ are (r + 1)-paths in Γ∗ with initial vertex α or β. Since r ⩾ 1,
without loss of generality we may assume P = (α, β1, . . . , βr, βr+1) and P ′ = (α, β1, . . . ,
βr, β
′
r+1), where β1 = β.
Consider the images of PN and (P ′)N in Σ. Note that the images are two (r+1)-paths,
since the equality βNi−1 = β
N
i+i would contradict our claim above. Hence, there is h ∈ Hγ
such that (PN )h = (P ′)N . Since ϕ(Gα) = Hγ , we can take h = ϕ(g) for g ∈ Gα, so g
fixes α. Now, (PN )h = (P ′)N implies (βN )g = βN . Thus, g fixes βN , and, since g fixes
α, g fixes the unique vertex in Γ∗(α) ∩ βN , which is β; so, g ∈ Gαβ . Continuing in this
way, we see that g ∈ Gαβ1...βr . Now, (βNr+1)h = (β′r+1)N , and so β
g
r+1 lies in the N -orbit
of β′r+1, and at the same time must be adjacent to βr, since g ∈ Gβr . Once more, the claim
implies βgr+1 = β
′
r+1.
We have thus shown that Gα is transitive on (r + 1)-arcs with initial vertex u. A
similar argument establishes that same result for Gβ , and hence Γ∗ is locally (G, r + 1)-
arc-transitive. This contradicts the maximality of r, and, therefore, r = t, as desired. In
particular, taking Σ = Γ we see that r = s. Hence Γ∗, and so any graph with amalgam
(L,R,L ∩R), is locally s-arc-transitive.
Lemma 3.1 enables us to construct locally (G, 2)-arc-transitive graphs where G has
two orbits ∆1 and ∆2 on vertices and acts quasiprimitively of type PA on each. Recall the
three types straight diagonal, twisted diagonal and nondiagonal of quasiprimitive groups of
type PA. Analogously to [22], we refer to a locally (G, 2)-arc-transitive graph Γ where G
is quasiprimitive of type PA on each orbit by the type of the two PA actions. For example,
if G is of straight diagonal type on ∆1 and twisted diagonal type on ∆2 then we refer to Γ
as being of straight-twisted type.
3.1 Straight-twisted type
Construction 3.3. We begin with the following: let (L,R,L ∩ R) be an amalgam for a
locally s-arc-transitive graph, and suppose further that L = L1⋊K and R = R1⋊K such
that K acts trivially on R1. Note that this implies L ∩R = (L1 ∩R1)K.
Let H be an almost simple group with socle T , and subgroups H1 and H2 such that
• H1 ∼= L1, H2 ∼= R1, H1 ∩ H2 ∼= L1 ∩ R1, i.e., ϕ : H1 → L1, τ : H2 → R1 are
isomorphisms with restrictions each sending H1 ∩H2 → L1 ∩R1,
• H = ⟨H1, H2⟩, and
• not all automorphisms of L1 in K extend to automorphisms of T .
We will abuse notation slightly and assume L1, R1 ⩽ H. Let k = |K| and let
F = {f : K → H} ∼= Hk.
M. Giudici and E. Swartz: Locally s-arc-transitive graphs arising from product action 7
For each ℓ ∈ L1 and r ∈ R1, define fℓ, fr ∈ F such that
fℓ(κ) = ℓ
κ,
fr(κ) = r
for all κ ∈ K. Furthermore, we let
Nα := {fℓ | ℓ ∈ L1} ∼= L1,
Nβ := {fr | r ∈ R1} ∼= R1.
Since K acts trivially on R1, we have that
Nα ∩Nβ = {fr | r ∈ R1 ∩ L1} ∼= L1 ∩R1.
Let N := ⟨Nα, Nβ⟩.
Now K acts on F via fσ(κ) = f(σκ) for each σ, κ ∈ K. Then for ℓ ∈ L1 we have
that
(fℓ)
σ(κ) = fℓ(σκ) = ℓ
σκ = fℓσ (κ).
Hence (fℓ)σ = fℓσ and so K normalises Nα. Similarly, (fr)σ = fr for all r ∈ R1 so K
normalises Nβ and hence also N . Define
Gα := Nα ⋊K,
Gβ := Nβ ⋊K,
G := ⟨Gα, Gβ⟩.
Finally, we define Γ := Cos(G,Gα, Gβ).
Lemma 3.4. Let Γ be a graph yielded by Construction 3.3. Then Γ is a connected locally
(G, s)-arc-transitive graph such that G acts quasiprimitively with type PA on each orbit of
vertices. Moreover, the action of G on [G : Gβ ] is straight diagonal, and the action of G
on [G : Gα] is twisted diagonal, that is, Γ is of straight-twisted type.
Proof. Let FT = {f ∈ F | f(κ) ∈ T for all κ ∈ K} ∼= T k. For each κ ∈ K, let
πκ : F → H
f 7→ f(κ).
Since ⟨R1, L1⟩ = H , we have that πκ(N) = H for all κ ∈ K and so by
[30, page 328, Lemma], N ∩ FT is a direct product of diagonal subgroups, each isomor-
phic to T . Since there are elements κ ∈ K that do not extend to an automorphism of T , it
follows that N ∩FT is not itself a diagonal subgroup and so N ∩FT ∼= T j for some integer
2 ⩽ j ⩽ k.
Since the action of K on Nα is isomorphic to the action of K on L1 we see that
Gα ∼= L and similarly, Gβ ∼= R. Moreover, Gα∩Gβ ∼= ⟨L1∩R1,K⟩ = L∩R. Therefore
Γ := Cos(G,Gα, Gβ) is a connected graph with amalgam (L,R,L ∩ R) and is thus a
locally s-arc-transitive graph.
Finally, since K transitively permutes the simple direct factors of FT it also transitively
permutes the simple direct factors of N ∩ FT . Thus soc(G) ∼= T j and G ≲ H wrSj
for some integer j ⩾ 2. Since πκ(Nα) = L1 for all κ ∈ K it follows that Nα is a
8 Ars Math. Contemp. 23 (2023) #P2.10
subdirect subgroup of Lj1 and similarly, Nβ is a subdirect subgroup of R
j
1. Therefore, G
acts quasiprimitively with type PA on both [G : Gα] and [G : Gβ ], and, by construction,
the action of G on [G : Gβ ] is straight diagonal, and the action of G on [G : Gα] is twisted
diagonal.
Example 3.5. This example is based on [22, Example 4.1]. First, (AGL(1, 5)× C2, S3 ×
C4, C4×C2) is an amalgam admitting a locally 2-arc-transitive connected graph of valency
{3, 5}: indeed, a GAP computation shows that in the group S7 we can take
L = ⟨(4, 5, 6, 7), (3, 4, 5, 7, 6), (1, 2)⟩ ∼= AGL(1, 5) × C2 and R = ⟨(1, 2), (1, 2, 3),
(4, 5, 6, 7)⟩ ∼= S3 × C4 such that ⟨L,R⟩ = S7, and L ∩R ∼= C4 × C2 [10].
Let T = PSL(2, p), where p is a prime and p ≡ ±1 (mod 60). Thus we may select
D < T such that D ∼= D60, with D = ⟨h, d | h30 = d2 = 1, hd = h−1⟩. First, define
L1 := ⟨h3⟩ ∼= C10 ∼= C5 × C2. Noting that D has a subgroup B := ⟨h15, d⟩ ∼= C22 , there
exists an element x of T such that Bx = B and dx = h15 [6]. Define R1 := ⟨(h10)x, dx⟩
to be a subgroup of Hx isomorphic to S3. Hence ⟨L1, R1⟩ = T and L1 ∩ R1 = C2.
Finally, the order four elements of AGL(1, 5) cannot be extended to automorphisms of T
since Aut(T ) = PGL(2, p) has no elements of order four normalising but not centralising
a subgroup of order five. Thus we let K = ⟨k⟩ ∼= C4 and L = L1 ⋊ K. Note, as
in [22, Example 4.1], that the action of k2 on elements of T is the same as conjugation
by d. Therefore, by Lemma 3.4, there is a locally 2-arc-transitive graph with amalgam
(AGL(1, 5)× C2, S3 × C4, C4 × C2) of straight-twisted type.
Theorem 3.6. There is an infinite family of locally 5-arc-transitive graphs with valencies
{4, 5} of straight-twisted type.
Proof. By [20], there is an amalgam admitting a locally 5-arc-transitive connected graph
of valency {4, 5} from the Mathieu group M24, with L = C42 ⋊ (A4 ×C3), R = A5 ×A4,
and L ∩ R = A4 × A4. Note that L = L1 ⋊K and R = R1 ×K where L1 = C42 ⋊ C3,
R1 = A5 and K = A4.
Let n ⩾ 2 be an integer and T = PSL(2, 22n). Then T contains a subgroup R1 ∼=
A5 ∼= PSL(2, 4) (see [6], for instance). Furthermore, T contains a subgroup Y isomorphic
to C2n2 ⋊ C22n−1, and 22n − 1 ≡ 0 (mod 3). Let Y = Y2 ⋊ Y1, where Y2 ∼= C2n2 and
Y1 = ⟨y1⟩ ∼= C22n−1. Thus Y1 has a cyclic group of order three, which we will denote
by Y3 = ⟨y(2
2n−1)/3
1 ⟩, acting semiregularly on the nonidentity elements of Y2. Moreover,
we may choose R1 such that Y0 := R1 ∩ Y ∼= A4 and Y3 ⩽ Y0. By [6, Theorem 260],
we see that NT (Y0) ⩽ Y , and, noting that Y1 acts regularly on the nonidentity elements of
Y2, we see that NT (Y0) = Y0. By [6, Theorem 255], for each divisor m of 2n, all subfield
subgroups of T isomorphic to PSL(2, 2m) are conjugate. This implies that Y0 is contained
in a unique subfield subgroup Tm isomorphic to PSL(2, 2m) for each divisor m of 2n, m
even (if m is odd, then 22 − 1 = 3 does not divide 2m − 1). Note also that this implies
that the maximal subgroup of Tm isomorphic to Cm2 ⋊ C2m−1 is actually Tm ∩ Y . We
claim that no subfield subgroup Tm containing Y0, for m a proper even divisor of 2n, also
contains Y y10 . If some Tm contains Y
y1
0 , then, since the elements of order two in Y0 and
Y y10 commute and Y0 ∩ Y
y1
0 = Y3, we have that ⟨Y0, Y
y1
0 ⟩ ⩽ Tm ∩ Y ∼= Cm2 ⋊ C2m−1,
where Tm ∩ Y1 acts regularly on the nonidentity elements of Tm ∩ Y2. However, Y1 acts
regularly on the nonidentity elements of Y2, so y1 is the unique element of Y1 mapping, say,
y2 ∈ Y0 ∩ Y2 to yy12 ∈ Y
y1
0 ∩ Y2. On the other hand, y1 ̸∈ Tm ∩ Y1 = ⟨y
(22n−1)/(2m−1)
1 ⟩,
so we have a contradiction.
M. Giudici and E. Swartz: Locally s-arc-transitive graphs arising from product action 9
Let L1 := ⟨Y0, Y ′y10 ⟩. Then L1 ∼= 24:3 (SmallGroup(48,50) in the GAP [10] small
groups library) which is isomorphic to the subgroup L1 in L, hence the abuse of notation.
Moreover, L1 ∩ R1 ∼= A4 and, since L1 is not contained in any subfield subgroup, we
have that T = ⟨L1, R1⟩. Since PΓL(2, 22n) does not contain a subgroup isomorphic to L
([6, Theorem 260] and noting that the outer automorphism group of PSL(2, 22n) is cyclic),
it follows that not all automorphisms of L1 in L extend to automorphisms of T . Hence
by Lemma 3.4, Construction 3.3 yields a locally 5-arc-transitive graph of straight-twisted
type.
3.2 Twisted-twisted type
If G acts quasiprimitively with straight PA type on a set Ω, then there exists α ∈ Ω such that
Nα = {(r, r, . . . , r) | r ∈ R}, where N = T k is the unique minimal normal subgroup of
G. If g = (t1, t2, . . . , tk) ∈ Rk ⩽ N then Nαg = (Nα)g = {(rt1 , rt2 , . . . , rtk) | r ∈ R},
which is a twisted diagonal subgroup if ti /∈ CT (R) for some i. Thus the examples given in
the previous section can also be viewed as being of twisted-twisted type. However, if G acts
quasiprimitively of type twisted PA on a set Ω then Nα is a twisted diagonal subgroup of
Rk for some R but there may not be a β ∈ Ω such that Nβ is a straight diagonal subgroup.
Thus not all twisted-twisted type examples arise in this way. In this section we give an
alternative construction.
Construction 3.7. Let (L,R,L ∩ R) be an amalgam for a locally s-arc-transitive graph,
and suppose further that L = L1 ⋊K and R = R1 ⋊K such that K = KL ×KR where
KL ⩽ Aut(L1) such that KL ∩ Inn(L1) = {1}, KL acts trivially on R1, KR ⩽ Out(R1)
and KR acts trivially on L1. Let H be an almost simple group with socle T , and subgroups
H1 and H2 such that
• H1 ∼= L1, H2 ∼= R1, H1 ∩H2 ∼= L1 ∩R1,
• H = ⟨H1, H2⟩, and
• not all elements of K extend to automorphisms of T .
We will abuse notation slightly and assume L1, R1 ⩽ H. Let k = |K| and let F =
{f : K → H} ∼= Hk. For each ℓ ∈ L1 ∪ R1, define fℓ ∈ F such that fℓ(κ) = ℓκ for all
κ ∈ K. Furthermore, we let Nα := {fℓ | ℓ ∈ L1} ∼= L1 and Nβ = {fr | r ∈ R1} ∼= R1.
Moreover, Nα ∩Nβ = {fr | r ∈ R1 ∩ L1} ∼= L1 ∩R1. Let N := ⟨Nα, Nβ⟩.
Now K acts on F via fσ(κ) = f(σκ) for each σ, κ ∈ K. As in Construction 3.3, K
normalises both Nα and Nβ , and hence also N . Define Gα := Nα ⋊K, Gβ := Nβ ⋊K
and G := ⟨Gα, Gβ⟩. Let Γ = Cos(G,Gα, Gβ).
Lemma 3.8. Let Γ be a graph yielded by Construction 3.7. Then Γ is a connected locally
(G, s)-arc-transitive graph such that G acts quasiprimitively with type PA on each orbit
on vertices. Moreover, the action of G on both [G : Gα] and [G : Gβ ] is twisted diagonal,
that is, Γ is of twisted-twisted type.
Proof. The proof is analogous to that of Lemma 3.4.
Example 3.9. First, (C71:C70 × C9, C19:C18 × C35, C630) is an amalgam that admits a
10 Ars Math. Contemp. 23 (2023) #P2.10
locally 2-arc-transitive graph; indeed, if G = A89,
L := ⟨(1, 2, 8, 28, 14, 30, 34, 3, 20, 54, 36, 33, 40, 41, 9, 56, 26, 51, 60, 18, 42, 29, 39, 17, 46, 58,
47, 10, 15, 70, 62, 13, 32, 59, 57, 31, 66, 22, 24, 67, 48, 27, 35, 50, 45, 12, 23, 11, 52, 4, 64,
7, 53, 25, 16, 61, 21, 44, 6, 5, 68, 71, 19, 55, 38, 69, 65, 49, 63, 43, 37),
(2, 3, 4, . . . , 71)(72, 73, . . . , 89)⟩,
and
R := ⟨(1, 72, 73, 85, 74, 88, 86, 78, 75, 80, 89, 84, 87, 77, 79, 83, 76, 82, 81),
(2, 3, 4, . . . , 71)(72, 73, . . . , 89)⟩,
then, using GAP, we see that L ∼= C71:C70 × C9, R ∼= C19:C18 × C35, L ∩ R ∼= C630,
⟨L,R⟩ = G, and by Lemma 3.1, the coset graph Cos(G,L,R) is a connected locally
(G, 2)-arc-transitive graph.
Let T = M, the Monster Group. By [4], T contains subgroups L1 ∼= D142 and R1 ∼=
D38, and L1 and R1 may be selected such that L1 ∩ R1 ∼= C2 (here, the element of order
two is of type 2B). By [36] we see that M does not have a maximal subgroup of order
divisible by 71 and 19. Thus ⟨L1, R1⟩ = T . Let K = C315 = C35 × C9, and since T
does not contain an element of order 315 [5], not all elements of K lift to an automorphism
of T . Therefore, by Lemma 3.8, Construction 3.7 yields a locally 2-arc-transitive graph Γ
with amalgam (C71:C70×C9, C19:C18×C35, C630) of twisted-twisted type with valencies
{71, 19}.
3.3 Straight-nondiagonal type
We first include an example of an equidistant linear code from [22], which proves useful
in later constructions. A linear (n,k)-code C over GF(q) is a k-dimensional subspace of
GF(q)n, a codeword has weight w if it has exactly w nonzero coordinates, and a code C is
equidistant if all nonzero codewords have the same weight.
Example 3.10 ([22, Example 5.1]). Let V = GF(3)4, and let
C = ⟨(1, 1, 1, 0), (1, 2, 0, 1)⟩ < V.
Then, C is a linear (4, 2)-code, and it contains eight nonzero code words:
(1, 1, 1, 0), (1, 2, 0, 1), (2, 0, 1, 1), (0, 2, 1, 2), (2, 2, 2, 0), (2, 1, 0, 2), (1, 0, 2, 2), (0, 1, 2, 1),
and hence C is equidistant of weight 3.
Let τ = (σ, 1, σ, σ)(1, 2, 3, 4) ∈ GL(1, 3)wrS4 < GL(V ). Then, τ4 = (σ, σ, σ, σ),
|τ | = 8, and τ permutes the eight nonzero words of C in the order given above.
Our next result constructs examples of straight-nondiagonal type.
Theorem 3.11. For each integer n ⩾ 3, there exists a locally 2-arc-transitive graph of
straight-nondiagonal type with valencies {n, 9}.
Proof. We adapt the construction of [22, Lemma 5.2]. Let H = Sn+2. Then H contains
subgroups L ∼= S2×Sn and R ∼= S3×Sn−1 such that ⟨L,R⟩ = H and L∩R ∼= S2×Sn−1
M. Giudici and E. Swartz: Locally s-arc-transitive graphs arising from product action 11
(this is realized by letting L be the stabilizer of {1, 2} and letting R be the stabilizer of
{1, 2, 3}).
Based on the equidistant linear code defined in Example 3.10, we define Nα :=
⟨(ℓ, ℓ, ℓ, ℓ) | ℓ ∈ L⟩. Moreover, if R = R1 × R2, where R1 ∼= S3, R2 ∼= Sn−1, and R1 =
⟨h, σ|h3 = σ2 = hhσ = 1⟩, we define Nβ := ⟨(h, h, h, 1), (h, h−1, 1, h), (x, x, x, x)|x ∈
⟨σ⟩×R2⟩. By choosing σ ∈ L we have Nα∩Nβ ∼= S2×Sn−1, and, as in [22, Lemma 5.2],
Nβ ∼= (C23 :C2) × Sn−1 ̸∼= R. Let N := ⟨Nα, Nβ⟩. Since ⟨L,R⟩ ∼= Sn+2 it follows that
N projects onto Sn+2 in each of its four coordinates. Moreover, given any two of the four
coordinates, Nβ contains an element that is the identity in one coordinate and a nonidentity
element of An+2 in another. Thus A4n+2 ◁ N . Note that N is not necessarily all of S
4
n+2;
indeed, the elements of Nβ that do not have all entries equal have even permutations as
their entries.
Define τ := (σ, 1, σ, σ)(1, 2, 3, 4). Then τ4 = (σ, σ, σ, σ) and so τ8 = 1. Furthermore,
τ centralizes Nα and normalises Nβ . Let Gα := ⟨Nα, τ⟩, Gβ := ⟨Nβ , τ⟩, and G :=
⟨Gα, Gβ⟩. By similar reasoning as in [22, Lemma 5.2], A4n+2 ≲ G and G induces C4
on the 4 simple direct factors. Moreover, Gβ ∼= AGL(1, 32) × Sn−1. We also see that
Gα ∼= C8 × Sn, and Gα ∩Gβ ∼= C8 × Sn−1.
Let Γ := Cos(G,Gα, Gβ). Since Gα acts on [Gα:Gα ∩ Gβ ] as Sn does on n points
and Gβ acts on [Gβ :Gα∩Gβ ] as AGL(1, 32) does on GF(32), we see that Γ is a connected
locally 2-arc-transitive graph with valencies {n, 9}. Clearly, the action of G on [G:Gα] is
straight diagonal, and the action of G on [G:Gβ ] is nondiagonal (as in [22, Lemma 5.2]).
Therefore, Γ is a locally 2-arc-transitive graph of straight-nondiagonal type with vertex
valencies {n, 9}.
3.4 Twisted-nondiagonal type
As discussed at the start of Section 3.2, the straight-nondiagonal examples given by Theo-
rem 3.11 can also be viewed as twisted-nondiagonal examples. We also have the following
construction of a graph of twisted-nondiagonal type.
Example 3.12. Let T = PSL(2, 61). By [6], T contains a maximal subgroup M ∼= D60.
Now, M contains a subgroup X isomorphic to C22 , and NT (X) ∼= A4. Now, NT (X)
contains an element g of order three that is not in M . Thus we may select subgroups
L ⩽ M and R ⩽ Mg such that L ∼= C10 ∼= C5 × C2, R ∼= C3:C2, ⟨L,R⟩ = T and
L ∩ R = X ∼= C2. Note that we may select presentations L = ⟨ℓ, x|ℓ5 = x2 = 1⟩ and
R = ⟨r, x|r3 = x2 = rrx = 1⟩.
Note that L has an isomorphism ϕ defined by ϕ : ℓ 7→ ℓ2, x 7→ x. We define ℓ :=
(ℓ, ℓϕ, ℓϕ
2
, ℓϕ
3
) = (ℓ, ℓ2, ℓ4, ℓ3) and x := (x, x, x, x). Furthermore, we define Nα :=
⟨ℓ, x⟩, Nβ := ⟨(r, r, r, 1), (r, r−1, 1, r), x⟩, and N := ⟨Nα, Nβ⟩. As in [22, Lemma 5.2],
none of the coordinates of Nβ can be linked, so N ∼= T 4. Moreover, Nα ∼= L ∼= C5 × C2,
Nβ ∼= C23 :C2 and Nα ∩Nβ ∼= C2.
Define τ := (x, 1, x, x)(1, 2, 3, 4). Then τ4 = (x, x, x, x) and so τ8 = 1. Let Gα :=
⟨Nα, τ⟩, Gβ := ⟨Nβ , τ⟩, and G := ⟨Gα, Gβ⟩. We note that τ centralizes x, whereas
ℓ
τ
= (ℓ3, ℓ, ℓ2, ℓ4) = ℓ3, and so Gα ∼= C2.AGL(1, 5). By similar reasoning as in [22,
Lemma 5.2], we deduce that G ∼= PSL(2, 61)wrC4 and Gβ ∼= AGL(1, 32). We also see
that Gα ∩Gβ ∼= C8.
Let Γ := Cos(G,Gα, Gβ). Since Gα acts on [Gα:Gα ∩ Gβ ] as AGL(1, 5) does on
GF(5) and Gβ acts on [Gβ :Gα ∩ Gβ ] as AGL(1, 32) does on GF(32), we see that Γ is a
12 Ars Math. Contemp. 23 (2023) #P2.10
connected locally 2-arc-transitive graph with vertex valencies {5, 9}. Clearly, the action of
G on [G:Gα] is twisted diagonal, and the action of G on [G:Gβ ] is nondiagonal (as in [22,
Lemma 5.2]). Therefore, Γ is a locally 2-arc-transitive graph of twisted-nondiagonal type
with valencies {5, 9}.
3.5 Nondiagonal-nondiagonal type
Finally, in this subsection, we include a construction of a graph of nondiagonal-nondiagonal
type.
Example 3.13. Let T = J2, the second Janko group. By [5], T has two conjugacy classes
of elements of order three, labelled 3A and 3B, and two conjugacy classes of involutions,
labelled 2A and 2B. Moreover, the elements of type 3A are contained in a maximal sub-
group isomorphic to A5×D10 which contains involutions from class 2B, and the elements
of type 3B are contained in a maximal subgroup isomorphic to A5 which also contains
involutions of type 2B. Furthermore, within each of these maximal subgroups the elements
of order three are normalized by an involution of type 2B. Using GAP, there are subgroups
L,R < T , each isomorphic to S3, such that L ∩ R ∼= C2, L contains an element of or-
der three of type 3A, R contains an element of order three of type 3B, and ⟨L,R⟩ = T.
Furthermore, by [5], the two conjugacy classes of order three are not fused by any outer au-
tomorphism of T . Let L = ⟨ℓ, x|ℓ3 = x2 = ℓℓx = 1⟩ and R = ⟨r, x|r3 = x2 = rrx = 1⟩.
We again use the equidistant linear code as defined in Example 3.10. Define Nα :=
⟨(ℓ, ℓ, ℓ, 1), (ℓ, ℓ−1, 1, ℓ), (x, x, x, x)⟩ and Nβ := ⟨(r, r, r, 1), (r, r−1, 1, r), (x, x, x, x)⟩.
Note that L ∩ R ∼= C2, and, reasoning as in [22, Lemma 5.2], we deduce that Nα ∼=
Nβ ∼= C23 :C2 ̸∼= L,R. Also, given any two of the four coordinates, both Nα and Nβ con-
tain an element that is the identity in one coordinate and a nonidentity element in another,
so N := ⟨Nα, Nβ⟩ ∼= J42 .
Define τ := (x, 1, x, x)(1, 2, 3, 4). Then τ4 = (x, x, x, x) and so τ8 = 1. Let Gα :=
⟨Nα, τ⟩, Gβ := ⟨Nβ , τ⟩, and G := ⟨Gα, Gβ⟩. By similar reasoning as in [22, Lemma 5.2],
G ∼= J2 wrC4 and Gα ∼= Gβ ∼= AGL(1, 32). We also see that Gα ∩Gβ ∼= C8.
Let Γ := Cos(G,Gα, Gβ). Since Gα (respectively Gβ) acts on [Gα:Gα ∩ Gβ ] (re-
spectively [Gβ :Gα ∩ Gβ ]) as AGL(1, 32) does on GF(32), we see that Γ is a connected
locally (G, 2)-arc-transitive graph with valencies {9, 9}. Moreover, Γ cannot be a standard
double cover of a (G, 2)-arc-transitive graph since L and R are not conjugate subgroups
in Aut(J2). Clearly, the action of G on both [G:Gα] and [G:Gβ ] is nondiagonal (as in
[22, Lemma 5.2]). Therefore, Γ is a locally (G, 2)-arc-transitive graph of nondiagonal-
nondiagonal type that is regular of valency 9.
ORCID iDs
Michael Giudici https://orcid.org/0000-0001-5412-4656
Eric Swartz https://orcid.org/0000-0002-1590-1595
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