ISSN 2590-9770
The Art of Discrete and Applied Mathematics 5 (2022) #P3.06
https://doi.org/10.26493/2590-9770.1391.f46
(Also available at http://adam-journal.eu)
On automorphisms of Haar graphs
of abelian groups
Ted Dobson*
FAMNIT and IAM, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia
Received 21 October 2020, accepted 16 February 2022, published online 18 July 2022
Abstract
Let G be a group and S ⊆ G. In this paper, a Haar graph of G with connection set S
has vertex set Z2 × G and edge set {(0, g)(1, gs) : g ∈ G and s ∈ S}. Haar graphs are
then natural bipartite analogues of Cayley digraphs, and are also called BiCayley graphs.
We first examine the relationship between the automorphism group of the Cayley digraph
of G with connection set S and the Haar graph of G with connection set S. We establish
that the automorphism group of a Haar graph contains a natural subgroup isomorphic to the
automorphism group of the corresponding Cayley digraph. In the case where G is abelian,
we show there are exactly four situations in which the automorphism group of the Haar
graph can be larger than the natural subgroup corresponding to the automorphism group
of the Cayley digraph together with a specific involution, and analyze the full automor-
phism group in each of these cases. As an application, we show that all s-transitive Cayley
graphs of generalized dihedral groups have a quasiprimitive automorphism group, can be
constructed from digraphs of smaller order, or are Haar graphs of abelian groups whose
automorphism groups have a particular permutation group theoretic property.
Keywords: Groups, graphs.
Math. Subj. Class.: 05C15, 05C10
A Haar graph of a group G with connection set S has vertex set Z2 × G and edge
set {(0, g)(1, gs) : g ∈ G and s ∈ S}, where S ⊆ G. Haar graphs are natural bipartite
analogues of Cayley digraphs, and these graphs have appeared in a variety of contexts
and under a variety of names. To the author’s knowledge, Haar graphs were introduced
in [18], where some of their elementary properties were studied, including some results on
isomorphic Haar graphs. Recently there has been a fair amount of work on the isomorphism
problem for Haar graphs [4, 5, 21–23, 25, 44], some of it motivated by applications (see
*The author is indebted to the referees for their careful reading of the paper, and their suggestions for improve-
ments
E-mail address: ted.dobson@upr.si (Ted Dobson)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Art Discrete Appl. Math. 5 (2022) #P3.06
[7, 27, 41]). Most of this work falls into the two categories of considering the isomorphism
problem for graphs of small valency or determining the structure of groups which are BCI-
groups. Intuitively, these are groups where isomorphism is determined in the “nicest”
possible way.
Our work in this paper was motivated by the isomorphism problem, but in the end we
will not consider this problem here. It is well known that the isomorphism problem for a
Cayley digraph Γ of a group G depends upon the conjugacy classes of natural subgroups of
Aut(Γ) (see [6, Lemma 3.1]). A similar result also holds for Haar graphs [26, Lemma 2.2]
or [4, Theorem C]. Thus, information about a Cayley digraph’s automorphism group is
crucial in determining which other Cayley digraphs of G are isomorphic to it, and similarly,
for Haar graphs. Typically, more information is known about the automorphism group of
Cayley digraphs of a group G (see for example [9,11,13,17]) than of Haar graphs of G, and
for every Haar graph there is a corresponding Cayley digraph with the same connection set.
It is thus of natural interest to determine the relationship, if any, between the automorphism
group of a Haar graph and the automorphism group of its corresponding Cayley digraph
(again, much more is known about the automorphism groups of Cayley digraphs). This is
the main focus of the work in this paper.
We will show in Corollary 2.16 that the automorphism group of a Haar graph of an
abelian group A falls into four natural families. In two of these families, the automorphism
groups are a wreath product and so can be found provided one knows the automorphism
groups of the graphs involved in the wreath products (which are always of smaller order
and so presumably easier to find). In the third family, the automorphism group of the Haar
graph of A is determined, up to conjugacy by |A| natural and explicitly defined permu-
tations, by the automorphism group of the corresponding Cayley digraph. For the fourth
and final family the situation is more interesting in that there does not seem to be a natural
or obvious relationship between the automorphism group of the Haar graph and its cor-
responding Cayley digraph. We do, though, give a group theoretic construction for all of
these graphs, but unfortunately the group theoretic information needed for the construction
does not seem easy to obtain. We should also mention that there is some related work on
finding automorphism groups of Haar graphs - see [46].
As an application, we next consider the implications of the above automorphism group
results to s-arc-transitive Haar graphs. In particular, we characterize s-arc-transitive Cay-
ley graphs of generalized dihedral groups with abelian subgroup of odd order. In all cases
except one (which corresponds to the case in the preceding paragraph where there was
no obvious relationship between the automorphism group of the Haar graph and its cor-
responding Cayley digraph), such s-arc-transitive graphs can be constructed from other
highly symmetric graphs and digraphs of smaller order without the use of graph covers.
There is another problem in the literature related to this work. For a graph Γ, its canon-
ical double cover is the graph K2 × Γ and is denoted B(Γ). So V (B(Γ)) = Z2 × V (Γ)
and E(B(Γ)) = {(0, x)(1, y) : xy ∈ E(Γ)}. If Γ = Cay(G,S) is a Cayley graph, then
B(Γ) = Haar(G,S). Automorphism groups of B(Γ) for Γ a Cayley graph were first stud-
ied in [34] and subsequently by several other authors [28,35,39,43]. The main question is,
for a graph Γ (not necessarily a Cayley graph), is Aut(Γ) = Z2 × Aut(Γ)? If so, such a
graph is stable, and if not, unstable. Corollary 2.16, for example, refines [28, Theorem 3.2]
in the case where Γ is a Cayley graph of an abelian group. Our results also hold for di-
graphs, so one can consider the Haar graph construction for Cayley digraphs of a group G
to be a natural generalization of the canonical double cover of graphs to digraphs.
T. Dobson: On automorphisms of Haar graphs of abelian groups 3
Some words about notation should be mentioned. Haar graphs are special cases of
bi-Cayley graphs of G, which are usually defined as graphs that contain a semiregular
subgroup with two orbits that is isomorphic to G [2, 19, 33, 45]. Bi-Cayley graphs need
not be bipartite, with the prefix bi referring to the two orbits of the semiregular subgroup
isomorphic to G, not to the graph being bipartite, while Haar graphs are always bipartite.
Additionally, some authors refer to Haar graphs as bi-Cayley graphs with the prefix bi
presumably referring to the fact that they are bipartite. To make the terminology issues
more complex, some authors refer to bi-Cayley graphs as defined here as semi-Cayley
graphs. We prefer the term Haar graph to bi-Cayley graph as defined here (this definition
will be formally stated below to hopefully eliminate all ambiguity and henceforth we will
not use the term bi-Cayley graph), simply because this choice of terminology causes less
confusion in that there is only one use of the term “Haar graph” in the literature, at least as
far as the author knows!
1 Basic definitions and results on automorphism groups
All groups and graphs are finite. In this section, we define Cayley digraphs and Haar graphs
and determine a relationship between their automorphism groups. This section is mainly
concerned with Haar graphs of general finite groups G, not necessarily abelian.
Definition 1.1. Let G be a group and S ⊆ G. Define a Cayley digraph of G, denoted
Cay(G,S), to be the digraph with V (Cay(G,S)) = G and A(Cay(G,S)) = {(g, gs) :
g ∈ G, s ∈ S}. We call S the connection set of Cay(G,S).
Note that the map gL : G 7→ G given by gL(x) = gx is always an automorphism of
Cay(G,S) for every group G and connection set S. Thus the group GL = {gL : g ∈ G} ≤
Aut(Cay(G,S)) for every group G and connection set S. Here, if Γ is a graph or digraph,
then Aut(Γ) denotes its automorphism group. Also note that by [6, Proposition 2.1] if
α ∈ Aut(G), then α(Cay(G,S)) = Cay(G,α(S)). In Figure 1, we give an example of a
Cayley digraph of the cyclic group Z7.
•
0
•6
•5
•
4
•
3
• 2
•1''

oo
\\
KK
77

ww
SS //

gg
BB

BB
ww //
gg

SS
Figure 1: The Cayley digraph Cay(Z7, {1, 2, 4}).
4 Art Discrete Appl. Math. 5 (2022) #P3.06
Definition 1.2. Let G be a group and S ⊆ G. Define the Haar graph Haar(G,S) with
connection set S to be the graph with vertex set Z2 × G and edge set {(0, g)(1, gs) : g ∈
G and s ∈ S}.
Some authors use H(G,S) for Haar(G,S). We prefer the somewhat longer but more
descriptive notation. In Figure 2 we give Haar(Z7, {1, 2, 4}). We call Haar(G,S) the
Haar graph corresponding to Cay(G,S). The graph in Figure 2 is the Haar graph corre-
sponding to the Cayley digraph in Figure 1, and is the Heawood graph.
•(0, 0)
•(0, 1)
•(0, 2)
•(0, 3)
•(0, 4)
•(0, 5)
•(0, 6)
• (1, 0)
• (1, 1)
• (1, 2)
• (1, 3)
• (1, 4)
• (1, 5)
• (1, 6)
Figure 2: The Heawood graph as Haar(Z7, {1, 2, 4}).
Clearly every Haar graph is a bipartite graph. Notice also that it is very much allowed
that 1G ∈ S, while for Cayley digraphs this is usually not allowed. This is because the
effect of including 1G in the connection set of a Cayley digraph is to put a loop at each
vertex, and doing this does not usually affect the symmetry properties of Cayley digraphs
(e.g. adding a loop at each vertex does not change the automorphism group of a Cayley
digraph). In some situations, though, allowing 1G ∈ S for a Cayley digraph is not only
advantageous, but crucial (see for example [3]). In this paper we allow loops in Cayley
digraphs.
Definition 1.3. Throughout this paper, if Γ = Haar(G,S), then the natural bipartition
of V (Γ) will be denoted by B, where B = {B0, B1}, B0 = {(0, g) : g ∈ G}, and
B1 = {(1, g) : g ∈ G}.
Notice that the map ĝL : Z2 × G 7→ Z2 × G given by ĝL(i, j) = (i, gj) is an auto-
morphism of Haar(G,S) for every group G and connection set S, and corresponds to the
subgroup GL ≤ Aut(Cay(G,S)).
It is easily determined using results in [1] that Cay(Z7, {1, 2, 4}) has automorphism
group {x 7→ ax + b : a = 1, 2, 4, b ∈ Z7} which is a metacyclic group of order 21. The
Heawood graph is a Haar graph of Z7, as shown in Figure 2. While the Heawood graph
does have a metacyclic subgroup of order 21 corresponding to Aut(Cay(Z7, {1, 2, 4})), the
T. Dobson: On automorphisms of Haar graphs of abelian groups 5
automorphism group of the Heawood graph is actually Z2⋉PGL(3, 2) ∼= PGL(2, 7) which
has order 336 and is an almost simple group. Our first result shows that the automorphism
group of a Haar graph always has a natural subgroup isomorphic to the automorphism
group of its corresponding Cayley digraph. The Heawood graph example, though, shows
the automorphism group of the Haar graph may be much larger.
Lemma 1.4. Let G be a group, and γ ∈ SG. The map γ̂ : Z2 × G 7→ Z2 × G given by
γ̂(i, j) = (i, γ(j)) is an automorphism of Haar(G,S) if and only if γ is an automorphism
of Cay(G,S).
Proof. The permutation γ ∈ SG is in Aut(Cay(G,S)) if and only if whenever g ∈ G
and s ∈ S, γ(g, gs) = (γ(g), γ(gs)) ∈ A(Cay(G,S)). This occurs if and only if there
exists s′ ∈ S such that γ(gs) = γ(g)s′ which is true if and only if (0, γ(g))(1, γ(g)s′) =
(0, γ(g))(1, γ(gs)) ∈ E(Haar(G,S)). This last statement is true if and only if γ̂ ∈
Aut(Haar(G,S)).
We now give a standard notation for the automorphism of Haar(G,S) induced by an
automorphism of Cay(G,S) which will be used henceforth.
Definition 1.5. Let G be a group and S ⊆ G. Let γ ∈ Aut(Cay(G,S)). The auto-
morphism of Haar(G,S) induced by γ as in Lemma 1.4 will be denoted by γ̂. That
is, if γ ∈ Aut(Cay(G,S)) then the automorphism of Haar(G,S) corresponding to γ
is γ̂ : Z2 × G 7→ Z2 × G given by γ̂(i, j) = (i, γ(j)). If H ≤ Aut(Cay(G,S)), then we
define Ĥ = {ĥ : h ∈ H}. In particular, the natural semiregular subgroup of Haar(G,S)
isomorphic to G is denoted ĜL.
In this paper we focus on Aut(Haar(G,S)) in the special case when G is an abelian
group. The reason to restrict our attention to abelian groups is that the automorphism
groups of Haar graphs of abelian groups and nonabelian groups are different. As implicitly
used in [18], for an abelian group A, Aut(Haar(A,S)) contains the element ι : Z2 ×A 7→
Z2 × A given by ι(i, j) = (i+ 1,−j). The group ⟨ι, ÂL⟩ is transitive and so Haar graphs
of abelian groups have transitive automorphism group while Haar graphs of nonabelian
groups need not. See for example [15, Proposition 11]. For Haar graphs of abelian groups
we may use the automorphism ι defined above to say a little more.
Lemma 1.6. Let A be an abelian group, and S ⊆ A. Then
Z2 ⋉Aut(Cay(A,S)) ≤ Aut(Haar(A,S)).
Proof. As A is an abelian group, straightforward computations will show that the map
ι : Z2 × A 7→ Z2 × A given by ι(i, j) = (i + 1,−j) is an automorphism of Haar(A,S).
Set K = Aut(Cay(A,S)). By Lemma 1.4, K̂ ≤ Aut(Haar(A,S)).
Let a ∈ A. Then (a, a+s) ∈ A(Cay(A,S)) if and only if (a, a−s) ∈ A(Cay(A,−S)),
which occurs if and only if (a + s, a) ∈ A(Cay(A,−S)). So Cay(A,−S) can be ob-
tained from Cay(A,S) by reversing the direction of each arc in Cay(A,S). Now, (a, b) ∈
A(Cay(A,S)) if and only if (γ(a), γ(b)) ∈ A(Cay(A,S)) for every γ ∈ Aut(Cay(A,S)).
As (a, b) ∈ A(Cay(A,S)) if and only if (b, a) ∈ A(Cay(A,−S)), we see (γ(a), γ(b)) =
γ(a, b) ∈ A(Cay(A,S)) if and only if (γ(b), γ(a)) = γ(b, a) ∈ A(Cay(A,−S)). We
conclude Aut(Cay(A,S)) = Aut(Cay(A,−S)).
6 Art Discrete Appl. Math. 5 (2022) #P3.06
Define r : A → A by r(j) = −j. Then r ∈ Aut(A) and r(Cay(A,S)) = Cay(A,−S).
Also r−1 = r, and r(Cay(A,−S)) = Cay(A,S). Hence
Aut(Cay(A,S)) = rAut(Cay(A,−S))r.
Thus γ ∈ Aut(Cay(A,S)) if and only if the map j 7→ −γ(−j) is also contained in
Aut(Cay(A,S)). As ιγ̂ι(i, j) = (i,−γ(−j)), we see ι normalizes K̂. Then the group
⟨ι, K̂⟩ = Z2 ⋉Aut(Cay(G,S)) ≤ Aut(Haar(A,S)) as |ι| = 2 and ι normalizes K̂.
In the case where S = −S and Cay(A,S) is a graph, we have a slightly nicer result,
which is contained in the significantly stronger result [31, Lemma 4.2].
Lemma 1.7. Let A be an abelian group and S ⊆ A such that S = −S. Then
Z2 ×Aut(Cay(A,S)) ≤ Aut(Haar(A,S)).
Proof. Simply observe that if S = −S the map i 7→ −i is an automorphism of Cay(A,S)
and so the map (i, j) 7→ (i+ 1, j) is an automorphism of Haar(A,S).
One circumstance in which Aut(Haar(G,S)) is bigger than Z2⋉Aut(Cay(G,S)) is if
Cay(G,S) is connected but Haar(G,S) is not. For example, for n ≥ 2, Cay(Z2n, {±1})
is a 2n-cycle with automorphism group D2n, while Haar(Z2n, {±1}) is a disjoint union
of two 2n-cycles, with automorphism group isomorphic to Z2 ≀ D2n (the group wreath
product is given in Definition 2.3; here it is enough to observe that this group is bigger).
Notice that the vertex sets of these two 2n-cycles are not the sets B0 and B1. A perhaps
more extreme example is Cay(Z2n, S), where S is all odd elements of Z2n. The graph
Cay(Z2n, S) = Kn,n is connected, but Haar(Z2n, S) consists of two disjoint copies of
Kn,n. The necessary and sufficient condition for Haar(G,S) to be connected is SS−1 =
{st−1 : s, t ∈ S} generates G and is given in [14, Lemma 2.3(iii)]. A more appealing
formulation for Haar graphs of abelian groups is the following:
Lemma 1.8. Let A be an abelian group. Haar(A,S) is disconnected if and only if
S ⊆ a+H for some subgroup H < A and a ∈ A.
Proof. If S ⊆ a + H for some H < A and a ∈ A then for s, t ∈ S we have s − t =
a + h1 − (a + h2) ∈ H < A for some h1, h2 ∈ H , and so Haar(A,S) is disconnected
by [14, Lemma 2.3(iii)]. If Haar(A,S) is disconnected, then ⟨s− t : s, t ∈ S⟩ = H < A
by [14, Lemma 2.3(iii)]. Fix a ∈ S, and let s ∈ S. Then a− s = −h ∈ H and s = a+ h.
Hence S ⊆ a+H .
Before proceeding, we will need some permutation group theoretic terms.
Definition 1.9. Let X be a set and K ≤ SX be transitive. A subset C ⊆ X is a block of
K if whenever k ∈ K, then k(C) ∩ C = ∅ or C. If C = {x} for some x ∈ X or C = X ,
then C is a trivial block. Any other block is nontrivial. The set C = {k(C) : k ∈ K} is a
partition of X , called a block system of K, and is nontrivial if C is nontrivial.
A basic fact about a graph Γ is that Aut(Γ) = Aut(Γ̄), where Γ̄ is the complement of
Γ. But the complement of a Haar graph is not a Haar graph. One could consider “bipartite
complements” (defined below) to avoid this problem, but it still need not be the case that the
automorphism group of the bipartite complement of a Haar graph Γ is the automorphism
group of Γ (although if B is a block system of the automorphism group of the bipartite
T. Dobson: On automorphisms of Haar graphs of abelian groups 7
complement of Γ we do have equality of automorphism groups under bipartite comple-
ments [10, Corollary 4]). Our next result is really an exercise and has certainly appeared as
a comment in the literature [38]. We state and prove this result due to its importance to the
work in this paper.
Lemma 1.10. Let Γ be a vertex-transitive bipartite graph with bipartition B = {B0, B1}.
If Γ is connected then B is a block system of Aut(Γ).
Proof. We prove the contrapositive, and so suppose there exists γ ∈ Aut(Γ) such that
γ(B) = B′ = {B′0, B′1} ̸= B. As B is a bipartition of Γ and γ ∈ Aut(Γ), γ(B) = B′
is also a bipartition of Γ. Let C0 = B0 ∩ B′0, C1 = B0 ∩ B′1, C2 = B1 ∩ B′0, and
C3 = B1 ∩B′1. As B ̸= B′, none of the sets Ci, i ∈ Z4, are empty. If Γ is connected, some
vertex v of C0 ⊂ B0 is adjacent to some vertex w of B1. As C0 is a subset of B0 and B′0,
w must be in both B1 and B′1, so in C3. But by a symmetrical argument, any vertex of C3
can only be adjacent to vertices in C0, and so there is no path in Γ from v to any vertex of
C2. Hence Γ is disconnected.
Definition 1.11. Let Γ be a bipartite graph with bipartition B = {B0, B1}. The bipartite
complement of Γ is the graph with vertex set V (Γ) and two vertices are adjacent if they are
in different bipartition classes and are not adjacent in Γ.
Corollary 1.12. Let G be a group and S ⊆ G. If Haar(G,S) and Haar(G,G \ S) are
both connected then Aut(Haar(G,S)) = Aut(Haar(G,G \ S)).
Proof. By Lemma 1.10, B is a block system of both Aut(Haar(G,S)) and Haar(G,
G \ S)), and so by [10, Corollary 4] Aut(Haar(G,S)) = Aut(Haar(G,G \ S)).
It is not true that if both Haar(A,S) and Haar(A,A − S) are disconnected then their
automorphism groups are the same. Let A = Z2n, and S = ⟨2⟩ so A − S = 1 + ⟨2⟩.
Then both S + (−S) and (A − S) + [−(A − S)] = ⟨2⟩ and so by Lemma 1.8 neither
are connected. Both of these graphs are isomorphic to two copies of Kn,n, but the vertex
sets of the Kn,n’s are different (in one it is even vertices adjacent to even vertices and odd
vertices adjacent to odd vertices while in the other it is even vertices adjacent to odd vertices
and odd vertices adjacent to even vertices), so their automorphism groups are different, but
permutation isomorphic!
2 Characterization of automorphism groups of Haar graphs of abelian
groups
We now, with some exceptions, focus on abelian groups. We will use the symbol A when
the group under consideration is abelian, and G when a nonabelian group is allowed. We
first consider disconnected Haar graphs. We begin with more permutation group terms.
Definition 2.1. Let X be a set and suppose K ≤ SX is a transitive group which has a block
system C. Then K has an induced action on C, denoted K/C. Namely, for k ∈ K, define
k/C : C 7→ C by k/C(C) = C ′ if and only if k(C) = C ′, and set K/C = {k/C : k ∈ K}.
We also define the fixer of C in K, denoted fixK(C), to be {k ∈ K : k/C = 1}. That is,
fixK(C) is the subgroup of K which fixes each block of C set-wise, and is the kernel of the
induced action of K on C.
8 Art Discrete Appl. Math. 5 (2022) #P3.06
We observe that for an abelian group A, H = Z2 ⋉ Aut(Cay(A,S)) has B as a block
system. Here H/B ∼= Z2 and fixH(B) = K̂, where K = Aut(Cay(A,S)). We shall also
need definitions of wreath products of digraphs and groups.
Definition 2.2. Let Γ1 and Γ2 be digraphs. The wreath product of Γ1 and Γ2, denoted Γ1 ≀
Γ2, is the digraph with vertex set V (Γ1) × V (Γ2) and arcs ((u, v), (u, v′)) for
u ∈ V (Γ1) and (v, v′) ∈ A(Γ2) or ((u, v), (u′, v′)) where (u, u′) ∈ A(Γ1) and v, v′ ∈
V (Γ2).
Definition 2.3. Let G ≤ SX and H ≤ SY . Define the wreath product of G and H , denoted
G ≀H , to be the set of all permutations of X×Y of the form (x, y) 7→ (g(x), hx(y)), where
g ∈ G and each hx ∈ H .
It is not hard to show that for vertex-transitive digraphs, Aut(Γ1) ≀ Aut(Γ2) ≤
Aut(Γ1 ≀ Γ2). See [12] for more information regarding wreath products.
Let Γ = Haar(A,S) be connected, so by Lemma 1.10 B is a block system of Aut(Γ).
Then F = fixAut(Γ)(B) has induced actions on B0 and B1. Let f ∈ F with f(i, j) =
(i, γi(j)). The induced action of F on B0 is given by f · (0, j) = (0, γ0(j)), and on B1
by f ∗ (1, j) = (1, γ1(j)). We will be considering these induced actions frequently, and
will abuse notation by considering the induced action of F on B0 as an action simply on
A, in which case f · (0, j) = γ0(j) (i.e. we just delete the first coordinate if it is clear
from context), and similarly for the induced action of F on B1: f ∗ (1, j) = γ1(j). We
will also not write the actions formally, and not use the · and ∗ notation, but instead write
FBi , i ∈ Z2 or analogous notation for a subgroup of F . For example, we simply say that
AL is contained in the image of the actions of F on B0 and B1 for the induced action of
ÂL on B0 and B1. Or more simply, ÂL
Bi
= AL as with the above abuse of notation,
âL · (0, j) = aL(0) and âL(1, j) = aL(0) for every a ∈ A.
Definition 2.4. Let G be a group. We will use the notation ḡR for the permutations of
Z2 ×G given by ḡR(0, j) = (0, j) and ḡR(1, j) = (1, jg) in what follows.
It is shown in the proof of [32, Lemma 2.2] that for any group G, S ⊆ G, and g ∈ G,
ḡR(Haar(G,S)) ∼= Haar(G,Sg) ∼= Haar(G,S).
Theorem 2.5. Let A be an abelian group, and S ⊆ A. If Γ = Haar(A,S) is disconnected,
then there is a ∈ A and H < A such that Γ = ā−1R (Haar(A, a + S)) and Aut(Γ) ∼=
ā−1R (SA/H ≀Aut(Haar(H, a+ S)))āR.
Proof. If Γ is disconnected, then by Lemma 1.8 S ⊆ −a + H for some a ∈ A and
H < A. Set H = ⟨S + (−S)⟩ < A (this is written additively as A is abelian). Then
Haar(H, a+ S) is a connected graph which is a component of Haar(A, a+ S) = āR(Γ).
Thus Γ = ā−1R (Haar(A, a+ S)). Also, there are [A : H] components of Haar(A, a+ S),
and Haar(A, a + S) ∼= K̄A/H ≀ Haar(H, a + S), where K̄A/H is the complement of
the complete graph with vertices the cosets of H in A. Then Aut(Haar(A, a + S)) ∼=
SA/H ≀ Aut(Haar(H, a + S)), and Aut(Haar(A,S)) = ā−1R Aut(Haar(A, a + S))āR as
ā−1R (Haar(A, a+ S)) = Haar(A,S).
Another way in which the automorphism group of a Haar graph Γ of an abelian group
is bigger than expected is if the graph is connected and the action of F = fixAut(Γ)(B) is
not faithful in its action on a block of B. The next result shows that this situation is easy to
spot by examining the connection set of Γ.
T. Dobson: On automorphisms of Haar graphs of abelian groups 9
Lemma 2.6. Let G be a group, and Γ = Haar(G,S) be connected and vertex-transitive.
Then fixAut(Γ)(B) in its action on B0 or B1 is unfaithful if and only if S is a union of left
cosets of some nontrivial subgroup of G.
Proof. Set F = fixAut(Γ)(B). First observe that as Γ is connected, B is a block system
of Aut(Γ). As Aut(Γ) is transitive, the action of F on B0 is unfaithful if and only if the
action of F on B1 is unfaithful.
Suppose S is a union of left cosets of some subgroup H ̸= 1 of A. Clearly for h ∈ H ,
the map h̄R is an automorphism of Γ as it fixes left cosets of H . Then h̄R is contained in
the kernel of the action of F on B0.
Suppose F in its action on B0 is unfaithful with K ̸= 1 the kernel of the action. Then
K ◁ F and so the orbits of K form a block system C of FB1 . As FB1 contains GL acting
regularly, the blocks of C are left cosets of some subgroup H of G. As K stabilizes (0, 0),
it is clear that if (0, 0) is adjacent to some element (1, gh) for g ∈ G and h ∈ H , then
(0, 0) is adjacent to (1, gh) for every h ∈ H . So S is a union of left cosets of H . As K is
nontrivial, H ̸= 1.
We next explicitly determine the automorphism groups of such Haar graphs of abelian
groups. We will need some additional notation.
Definition 2.7. Let K be a transitive group with block system C. If D is also a block system
of K and each block of D is a union of blocks of C, then we write D/C for the set of blocks
of C whose union is D. Let Γ be a vertex-transitive digraph with K ≤ Aut(Γ). Define the
block quotient digraph of Γ with respect to C, denoted Γ/C, to be the digraph with vertex set
C and arc set {(C,C ′) : C ̸= C ′ ∈ C and (u, v) ∈ A(Γ) for some u ∈ C and v ∈ C ′}.
Theorem 2.8. Let A be an abelian group, S ⊆ A, and Γ = Haar(A,S). If Γ is connected,
and the action of fixAut(Γ)(B) is unfaithful on B1, then there exists a subgroup 1 < H ≤ A
such that Γ ∼= Haar(A/H,S) ≀ K̄β where β = |H| and S is interpreted as a set of cosets of
H in A. Also, Aut(Γ) ∼= Aut(Haar(A/H,S)) ≀ Sβ , and denoting the natural bipartition
of Haar(A/H,S) by D, the action of fixAut(Haar(A/H,S))(D) on D ∈ D is faithful.
Proof. By Lemma 2.6 S is a union of cosets of some subgroup 1 < H ≤ A. We choose
H to be maximal with respect to this property - clearly such a maximal subgroup exists.
Let C be the block system of ⟨ι, ÂL⟩ formed by the subgroup ⟨ĥL : h ∈ H⟩. Notice that
C = {{(i, a+ h) : h ∈ H} : i ∈ Z2, a ∈ A}.
Claim 1: In Γ, for any two distinct blocks C,C ′ ∈ C, either every vertex of C is adjacent
to every vertex of C ′ or no vertex of C is adjacent to any vertex of C ′.
Clearly if C,C ′ ⊆ B0 or B1 then this statement is true as there are no edges with both
endpoints inside any Bi, i = 0, 1. If, say, C ⊂ B0 and C ′ ⊂ B1 and some vertex (0, v) of
C is adjacent to some vertex (1, v′) of C ′, then by definition v′ = v + s for some s ∈ S.
Then C = {(0, w) : w ∈ v +H} and C ′ = {(1, w′) : w′ ∈ v + s +H}. Let (0, w) ∈ C
and (1, w′) ∈ C ′ with w = v + h and w′ = v + s+ h′, h, h′ ∈ H . Then
(0, w)(1, w′) = (0, v+ h)(1, v+ s+ h′) = (0, v+ h)(1+ v+ h+ s+ (h′ − h)) ∈ E(Γ),
as s+ (h′ − h) ∈ s+H ⊆ S, and the claim follows.
It is now straightforward to verify using the definition of the wreath product that Γ ∼=
Haar(A/H,S) ≀ K̄β .
10 Art Discrete Appl. Math. 5 (2022) #P3.06
Claim 2: Haar(A/H,S) ̸∼= Γ2 ≀ K̄r with r ≥ 2, where Γ2 is a connected vertex-transitive
graph.
Suppose not, so Haar(A/H,S) ∼= Γ2 ≀ K̄r for some vertex-transitive graph Γ2 and r ≥
2 is maximum. By [12, Theorem 5.7] and the maximality of r, we have Aut(Haar(A,S)) ∼=
Aut(Γ2)≀Srβ , so Aut(Γ) has a block system E with blocks of size rβ such that Aut(Γ)/E =
Γ2. Now, if Γ2 has an odd cycle, then Γ2 ≀ K̄r has an odd cycle. However, Γ2 ≀ K̄r is a
Haar graph and bipartite. So Γ2 is bipartite, with bipartition {F0, F1}. Observe that both
F0 and F1 are a set of cosets of H in A, and so ∪a+H∈F0(a +H) and ∪a+H∈F1(a +H)
are subsets of A.
We now show that {∪a+H∈F0(a + H),∪a+H∈F1(a + H)} = B. It suffices to show
that if E,E′ ∈ E and E,E′ ∈ F0, then E ∪ E′ ⊆ Bj , for some fixed j = 0, 1. As Γ is
connected, Γ2 is connected. Thus there is an EE′ path E0, . . . , Et in Γ2. Then Ei ⊆ Fj if i
is even and Ei ⊆ Fj+1 if i is odd. Similarly, the Ei with even subscripts are all contained in
Bj while the Ei with odd subscripts are contained in Bj+1. As both E and E′ are contained
in F0, t is even and E ∪ E′ ⊆ Bj , so {∪a+H∈F0(a+H),∪a+H∈F1(a+H)} = B.
As ι interchanges the blocks of B, ι also interchanges the union of the bipartition sets
∪F0 and ∪F1 of Γ2. Then ι/E is well defined and is not 1 i.e. ι permutes the blocks of E
nontrivially as well. Finally, by [8, Theorem 1.5A] there exists a subgroup K ≤ ⟨ι, ÂL⟩
such that the block of E that contains (0, 0) is the orbit of K that contains (0, 0). As
ι/E ̸= 1, K ≤ ÂL is of order rβ. This implies the blocks of E are orbits of K, and
K = {ℓ̂ : ℓ ∈ L ≤ A}. Also, it should be clear that H ≤ L as every block of C is
contained in a block of E . As r ≥ 2, H < L. But this implies that S is a union of cosets of
L, contradicting the maximality of H . The claim is established.
By [12, Theorem 5.7] and Claim 2, Aut(Γ) ∼= Aut(Haar(A/H,S)) ≀ K̄β . That
the action of fixAut(Haar(A/H,S))(D) on D ∈ D is faithful follows from Claim 2 and
Theorem 2.6.
We now turn to Haar graphs Γ of abelian groups A that are connected and whose con-
nection set is not a union of cosets of some subgroup of A. These two hypotheses imply
that B is a block system of Aut(Γ) and that the actions of fixAut(Γ)(B) on B0 and B1 are
faithful. To investigate further, we will need the terminology and some results concerning
inequivalent permutation representations.
Definition 2.9. A permutation representation of a group K is a homomorphism
ϕ : K → Sn for some n.
Definition 2.10. Let K be a group, and X and Y sets. Let α : K 7→ SX and
ω : K 7→ SY be permutation representations of K. We say α and ω are equivalent permu-
tation representations of K if there exists a bijection λ : X 7→ Y such that λ(α(k)(x)) =
ω(k)(λ(x)) for all x ∈ X and k ∈ K. In this case, we will say that α(K) and ω(K) are
permutation equivalent.
The examples of Cay(Z7, {1, 2, 4}) and its corresponding Haar graph, the Heawood
graph, provide the next way in which the automorphism group of a Haar graph of an abelian
group can be larger than its expected automorphism group. Namely, the full automorphism
group of the Heawood graph is Z2 ⋉PGL(3, 2), and PGL(3, 2) acts permutation inequiv-
alently on the two blocks B0 and B1 of B. It turns out that this is not a coincidence, as we
will show. Before turning to that result, we prove a preliminary result which will be crucial.
T. Dobson: On automorphisms of Haar graphs of abelian groups 11
It does not depend upon A being abelian or even a group, so we will use the symbol X in
place of A or G. We keep the notation Bi = {(i, x) : x ∈ X}, i = 0, 1.
Lemma 2.11. Let X be a set, and let F ≤ SZ2×X have B0 and B1 as orbits. Additionally,
assume that the actions of F on B0 and B1 are faithful and the action of F on B0 is
permutation equivalent to the action of F on B1. Then there exists ϵ̄ ∈ SZ2×X such that
ϵ̄(B0) = B0, ϵ̄B0 = 1 and every element of ϵ̄−1F ϵ̄ has the form (i, j) 7→ (i, g(j)), where
g ∈ FB0 .
Proof. Let α : F 7→ FB0 and ω : F 7→ FB1 be permutation representations of FB0 and
FB1 , respectively. As F is faithful on B0 and B1, both α and ω are faithful permutation
representations of F . As the action of F on B0 is permutation equivalent to the action of
F on B1, there exists a bijection λ : B0 7→ B1 such that λ(α(g)(0, j)) = ω(g)(λ(0, j)) for
every j ∈ X and g ∈ F . Let β : B0 7→ B1 be defined by β(0, j) = (1, j) for every j ∈ X ,
so that β is a bijection. Set ϵ = λβ−1 so ϵβ = λ and ϵ ∈ SB1 . Substituting ϵβ for λ we
have ϵβ(α(g)(0, j)) = ω(g)(ϵβ(0, j)) or equivalently, β(α(g)(0, j)) = ϵ−1ω(g)ϵ(β(0, j))
for every j ∈ X . As ϵ is a bijection and ω a faithful permutation representation of F , it is
straightforward to verify that the map δ : F 7→ SB1 given by δ(g) = ϵ−1ω(g)ϵ is a faithful
permutation representation of F .
Now, as α(g)(0, j) ∈ B0 and similarly, δ(g)(1, j) ∈ B1 for every j ∈ X , α(g) and
δ(g) induce permutations ᾱ(g) and δ̄(g) in SX defined by α(g)(0, j) = (0, ᾱ(g)(j)) and
δ(g)(1, j) = (1, δ̄(g)(j)) for all j ∈ X . Then
β(α(g)(0, j)) = β(0, ᾱ(g)(j)) = (1, ᾱ(g)(j))
and
ϵ−1ω(g)ϵ(β(0, j)) = δ(g)(1, j) = (1, δ̄(g)(j))
for all j ∈ X . As β(α(g)(0, j)) = ϵ−1ω(g)ϵ(β(0, j)) for every j ∈ X , we see that
ᾱ(g)(j) = δ̄(g)(j) for all j ∈ X , and so ᾱ(g) = δ̄(g) for all g ∈ F . Define ϵ̄ : Z2 ×X 7→
Z2 ×X by ϵ̄(0, j) = (0, j) and ϵ̄(1, j) = ϵ(1, j). Let g ∈ F . Then
ϵ̄−1gϵ̄(0, j) = α(g)(0, j) = (0, ᾱ(g)(j))
and
ϵ̄−1gϵ̄(1, j) = ϵ−1ω(g)ϵ(1, j) = (1, δ̄(g)(j)) = (1, ᾱ(g)(j)),
and the result follows.
Theorem 2.12. Let G be a group, S ⊆ G, Γ = Haar(G,S), and let F ≤ Aut(Γ) be
the largest subgroup of Aut(Γ) that fixes B0 and B1 set-wise. Suppose F satisfies the
following conditions:
(1) F acts faithfully on both B0 and B1, and
(2) the action of F on B0 is permutation equivalent to the action of F on B1.
Let L be the group of all elements of SZ2×G of the form (i, j) 7→ (i, ℓ(j)) where ℓ ∈ FB0 ,
and g ∈ G such that StabF (1, g) = StabF (0, 1G). Then L = ḡ−1R F ḡR.
12 Art Discrete Appl. Math. 5 (2022) #P3.06
Proof. Clearly every element of F can be written as (i, j) 7→ (i, ℓi(j)), where ℓ0, ℓ1 ∈ SG.
By Lemma 2.11 there exists ϵ̄ ∈ SZ2×G such that ϵ̄B0 = 1 and every element of ϵ̄−1F ϵ̄
has the form (i, j) 7→ (i, ℓ(j)), where ℓ ∈ FB0 . Let ϵ ∈ SG such that ϵ̄(1, j) = (1, ϵ(j)).
Define gR : G → G by gR(x) = xg, and set GR = {gR : g ∈ G}. We next show that
ϵ ∈ GR.
Of course, ĜL ≤ F , and ĜL has the form (i, j) 7→ (i, ℓ(j)), where ℓ ∈ ĜL
B0
. As
ϵ̄B0 = 1, we have ϵ̄−1ĝLϵ̄ = ĝL for every g ∈ G. Hence ϵ̄ centralizes ĜL, and so ϵ
centralizes GL as ϵ̄B0 = 1. As the centralizer of GL in SG is GR (this well known fact can
be deduced from [8, Theorem 4.2A(ii)]), we have ϵ ∈ GR. As ϵ ∈ GR, there exists g ∈ G
such that L = ḡ−1R F ḡR. Finally, as F = ḡRLḡ
−1
R and StabL(0, 1G) = StabL(1, 1G),
F = ḡRLḡ
−1
R stabilizes (1, g).
Corollary 2.13. Let A be an abelian group with S ⊆ A such that Γ = Haar(A,S)
is connected, and consequently Aut(Haar(A,S)) has B as a block system. Set F =
fixAut(Γ)(B). Then one of the following is true:
(1) the induced action of F on B1 is not faithful,
(2) the induced action of F on B0 is permutation inequivalent to the induced action of
F on B1, or
(3) Aut(Haar(A,S)) = ā−1R [Z2 ⋉Aut(Cay(A, a+ S))]āR for some a ∈ A.
Proof. We may assume without loss of generality that the action of F on B1 is faithful.
As Aut(Γ) is transitive as A is abelian, we have F is faithful on B0. With that assumption
made, we may then assume without loss of generality that the actions of F on B0 and B1
are permutation equivalent. Set L = {(i, j) 7→ (i, g(j)) : g ∈ FB0} so StabL(0, 0) =
StabL(1, 0). Applying Theorem 2.12, there is a ∈ A such that L = āRF ā−1R . By
[32, Lemma 2.2], āR(Γ) = Haar(A, a + S). By Lemma 1.4, Aut(Cay(A, a + S)) ∼= L,
so Aut(Haar(A, a+ S)) = Z2 ⋉Aut(Cay(A,−a+ S). The result follows as Aut(Γ) =
ā−1R Aut(Haar(A, a+ S))āR.
Remark 2.14. Suppose Haar(A,S) is connected and the induced action of F on B1 is
faithful. If a, b ∈ A with a ̸= b, then āR(Haar(A,S)) ̸= b̄R(Haar(A,S)) as otherwise
b̄−1R āR ∈ Aut(Haar(A,S)) and the induced action of fixAut(Cay(A,S))(B) is not faithful
on B0.
Remark 2.15. While Haar(A,S) ∼= Haar(A, a + S) for every a ∈ A, Cay(A,S) is
not necessarily isomorphic to Cay(A, a + S). Indeed, let n be an odd positive integer,
A = Zn, S = {±1}, and a = 1. Then Cay(A,S) is a cycle, while Cay(A, a + S) =
Cay(Zn, {0, 2}) is a directed cycle together with a loop at every vertex.
The following result is a combination of Theorems 2.5 and 2.8, and Corollary 2.13, and
summarizes the possibilities for Aut(Haar(A,S)) with A abelian.
Corollary 2.16. Let A be an abelian group, S ⊆ A, and Γ = Haar(A,S). Then one of
the following is true:
(1) Γ is disconnected, then there is a ∈ A and H < A such that Γ = ā−1R (Haar(A,
a+ S)) and Aut(Γ) ∼= ā−1R (SA/H ≀Aut(Haar(H, a+ S)))āR,
T. Dobson: On automorphisms of Haar graphs of abelian groups 13
(2) Γ is connected, and the action of fixAut(Γ)(B) is unfaithful on B1. There exists a
subgroup 1 < H ≤ A such that Γ ∼= Haar(A/H,S) ≀ K̄β where β = |H| and S is
interpreted as a set of cosets of H in A, and Aut(Γ) ∼= Aut(Haar(A/H,S)) ≀ Sβ .
Additionally, denoting the natural bipartition of Haar(A/H,S) by D, the action of
fixAut(Haar(A/H,S))(D) on D ∈ D is faithful,
(3) Aut(Γ) ∼= ā−1R Z2 ⋉Aut(Cay(A, a+ S))āR for some a ∈ A, or
(4) the action of fixAut(Γ)(B) on B1 is faithful but the actions on B0 and B1 are not
equivalent permutation groups.
We now give a group theoretic description of the graphs in (4) of Corollary 2.16.
Theorem 2.17. Let A be an abelian group and S ⊆ A such that Γ = Haar(A,S) is
connected and S is not a union of cosets of some subgroup of A. Let F = fixAut(Γ)(B),
H = FB0 , and L = StabH(b), where b ∈ B0. If the actions of F on B0 and B1 are in-
equivalent, then there exists σ ∈ Aut(H) which is an involution and maps L to a subgroup
R of H which is not conjugate in H to L.
Proof. The hypothesis implies that the action of F on B0 and B1 is faithful by Lemma 2.6.
As ι ∈ Aut(Γ), conjugation of F by ι induces automorphisms σ and δ which map FB0
to FB1 , and FB1 to FB0 , so we may view FB1 as being contained in H as the image of
δ. As ι has order 2, so does σ. Also, as the action of F on B0 and B1 are not equivalent,
by [8, Lemma 1.6B] σ(L) = R is not the stabilizer of a point in B0, so R is not conjugate
in H to L.
We next give a partial converse of the previous result. We begin with a more general
construction of bipartite graphs than that of the Haar graphs.
Definition 2.18. Let G be a group, let L,R ≤ G, and let D be a union of double cosets
of R and L in G, that is D = ∪iRdiL. Define a bipartite graph Γ = B(G,L,R;D) with
bipartition V (Γ) = G/L∪G/R (here G/L and G/R are the sets of left cosets of L and R
in G) and edge set E(Γ) = {{gL, gdR} : g ∈ G, d ∈ D}. We call Γ the bi-coset graph of
G with respect to L, R and D.
The bi-coset graphs B(G,L,R;D) were introduced in [14], where they were shown
to be well-defined bipartite graphs whose automorphism group contain a natural subgroup
isomorphic to G. Haar graphs are bi-coset graphs with L = R = 1. In [14, Lemma 2.6]
a sufficient condition to ensure that a bi-coset graph is vertex-transitive is given, and then
several more specific circumstances were given where this condition was satisfied. We will
need another such more specific circumstance.
Lemma 2.19. Let G be a group with L ≤ G and σ ∈ Aut(G) an involution such that
σ(L) = R. The bi-coset graph Γ = B(G,L,R;S) is vertex-transitive with σ ∈ Aut(Γ)
provided S = LDR with σ(D) = D−1 = D.
Proof. By [14, Lemma 2.6] we need only show that S−1 = σ(S). Then
σ(S) = σ(LDR) = σ(L)σ(D)σ(R) = RD−1L = R−1D−1L−1 = (LDR)−1 = S−1.
14 Art Discrete Appl. Math. 5 (2022) #P3.06
Definition 2.20. Let K be a group and H ≤ K. The largest normal subgroup of K
contained in H is called the core of H in K. If the core of H in K is trivial, then we say
H is core-free in K.
It is well known that the left action of a group K on a subgroup H ≤ K is faithful if
and only if H is core-free in K.
Theorem 2.21. Let A be an abelian group. Suppose AL ≤ K ≤ SA with L = StabK(a),
a ∈ A, σ ∈ Aut(K) an involution, and R = σ(L) ≤ K which is not conjugate in K to L
and satisfies R ∩ AL = 1. Then the bi-coset graph Γ = B(K,L,R;LR) ∼= Haar(A,S)
for some S ⊆ A, and there is a subgroup H ≤ Aut(Γ) such that H fixes B, H ∼= K, the
action of H on B0 is faithful, and the action of H on B0 = K/L is inequivalent to the
action of H on B1 = K/R.
Proof. By [14, Lemma 2.3] the left multiplication action of K on V (Γ) is contained in
Aut(Γ), fixes B, and is transitive on B0 and B1. Denote by H the corresponding subgroup
of Aut(Γ). Then HB0 is permutation equivalent to K ≤ SA as their stabilizers are the
same, namely L. It is clear that the left multiplication action of K on B0 is faithful as L
is core-free in K as L is the stabilizer of a point in K ≤ SA. Similarly, as σ ∈ Aut(K)
and σ(L) = R, R is core-free in K (as it is the isomorphic image of a core-free subgroup
of K) so the left multiplication action of K on K/R is faithful as well. Note that as
left multiplication of R by an element of K fixes R if and only if it is contained in R,
StabHB1 (R) = R. We see that the action of H on B1 is faithful, and the action of H on
B0 is inequivalent to the action of H on B1. Setting D = {1K} we see by Lemma 2.19
that σ ∈ Aut(Γ) as D−1 = D, and so Γ is vertex-transitive.
To see Γ is isomorphic to a Haar graph of A, let à ≤ Aut(Γ) be the subgroup of
Aut(Γ) induced by left multiplication by elements of AL in K. Of course, Ã is transitive
on B0 as ALL = K. Suppose ã1R = ã2R for some ã1, ã2 ∈ Ã. This occurs if and
only if there are a1, a2 ∈ A such that (a1)LR = (a2)LR, which occurs if and only if
(a−12 a1)LR = R. As AL ∩ R = 1, we see (a1)L = (a2)L. Then à is faithful on B1. As
à ∩ R = 1, ÃB1 is semiregular. By [42, Proposition 4.1] à has one orbit of size |A|, and
so is transitive. The result follows by [14, Lemma 2.5].
Theorem 2.21 is only a partial converse to Theorem 2.17 as it is possible that the graph
constructed in the result (or any graph constructed in a similar way) will have an auto-
morphism group larger than the group K in the statement. For example, if LR = K is
transitive, the graph B(K,L,R;LR) constructed will be a complete bipartite graph with
automorphism group K2 ≀ Sn with n = |A|. The next example shows that this can occur.
Example 2.22. Let K = S6, L = StabS6(x) ∼= S5 and σ be the outer automorphism of
S6 of order 2. Set R = σ(L). Then B(K,L,R;LR) ∼= K6,6.
Proof. We show that L is transitive on R. That is, we will show that σ(S5) is transitive,
where we view S5 as the stabilizer of the point 0 in the set Z6 on which we view S6
permuting. Now, the three cycle (3, 4, 5) is mapped by σ to a product of two disjoint 3-
cycles (see for example [20]). So σ(3, 4, 5) has two orbits of size 3. This implies that
σ(S5) is transitive, or σ(S5) has two orbits of size 3, in which case its maximum order is
|S3| · |S3| = 36. But S5 has order 120, so σ(S5) is transitive and the result follows.
Note that the graph B(K,L,R;LR) as constructed in the previous example does not
satisfy the hypothesis of Theorem 2.21 as R ∩AL ̸= 1.
T. Dobson: On automorphisms of Haar graphs of abelian groups 15
Definition 2.23. Let n ≥ 3 be an integer, q a prime-power, and set
L = PG(n− 1, q) = {rv⃗ : r ∈ Fq and v⃗ is in the vector space Fnq },
the set of lines (or projective points) in Fnq . Let H be the set of hyperplanes in Fnq . Define
a graph B(PG(n− 1, q)) to have vertex set L ∪H and edges {L,H} where L ⊆ H .
Note that the ‘B’ used in defining the above graphs has an entirely different meaning
from the ‘B’ used in the previous example. No confusion can arise though as the parameters
of the families are completely different. Also, it is known that B(2, 2) is isomorphic to the
Heawood graph. Our next example, which is also fairly well-known, shows it is a member
of a much larger family of graphs with many of the same properties.
Example 2.24. The graph B(PG(n − 1, q)) is isomorphic to a Haar graph of the cyclic
group of order (qn − 1)/(q − 1). Additionally, if n ≥ 3, F = fixAut(B(PG(n−1,q)))(B) in
its induced actions on B0 and B1 are inequivalent representations of PGL(n, q) with the
actions being on points and hyperplanes.
Proof. It is well-known that PGL(n, q) contains a regular cyclic subgroup of order
(qn − 1)/(q − 1) - see for example [24, Theorem 3] or [29, Theorem 1.1]. It is easy
to see that FB0 ∼= FB1 contains PGL(n, q) as a point contained in a hyperplane is mapped
to a point contained in a hyperplane by an element of PGL(n, q). That F is not larger
than PGL(n, q) basically follows from the Fundamental Theorem of Finite Geometry.
By [18, Lemma 4.2] we see B(PG(n − 1, q)) is isomorphic to a Haar graph, and, as
n ≥ 3, points and hyperplanes have different dimensions. It is then not hard to see that
the stabilizer in PGL(n, q) of a line does not stabilize any hyperplane. Thus the induced
actions of F on B0 and B1 are inequivalent by [8, Lemma 1.6B].
3 Applications to arc-transitive graphs
The study of s-arc-transitive graphs was initiated in a celebrated paper by Tutte [40]. There
has been strong and consistent interest in s-arc-transitive graphs for several decades. Per-
haps the most important tool in this area is Praeger’s Normal Quotient Lemma [36]. This
lemma shows how to reduce an s-arc-transitive graph Γ to an s-arc-transitive quotient of Γ
provided one can find N ◁ Aut(Γ) that has at least three orbits. If Aut(Γ) is quasiprimi-
tive, then one can study such groups and graphs using the O’Nan-Scott Theorem [8, Theo-
rem 4.1A] and Praeger’s quasiprimitive counterpart [37]. So other techniques are necessary
to deal with the case when Γ only has normal subgroups with exactly two orbits. In this
case, if Γ is disconnected, then there is an obvious reduction to s-arc-transitive graphs of
smaller order. If Γ is connected, then there are edges between the two orbits and so no
edges inside the orbits. Hence Γ is bipartite, and if the subgroup N of Aut(Γ) fixing the
parts of the bipartition set-wise contains a semiregular subgroup isomorphic to G, then Γ
is a Haar graph of G by [14, Lemma 2.4]. Thus the study of Haar graphs is essential to the
study of s-arc-transitive graphs.
Definition 3.1. Let s ≥ 1, and Γ a digraph. An s-arc of Γ is a sequence x0, x1, . . . , xs of
vertices of Γ such that (xi, xi+1) ∈ A(Γ), 0 ≤ i ≤ s − 1, and xi ̸= xi+2, 0 ≤ i ≤ s − 2.
The digraph Γ is s-arc-transitive if Aut(Γ) is transitive on the set of s-arcs of Γ.
As an arc-transitive graph without isolated vertices is vertex-transitive, we restrict our
attention to vertex-transitive Haar graphs.
16 Art Discrete Appl. Math. 5 (2022) #P3.06
Definition 3.2. Let Γ be a digraph, and s ≥ 1. A sequence of arcs a1, . . . , as ∈ A(Γ) is an
alternating s-arc if there exists vertices x0, . . . , xs ∈ V (Γ), xi ̸= xi+2, and for 1 ≤ m ≤ s
the arc am = (xm−1, xm) if m is odd while am = (xm, xm−1) if m is even. An alternating
s-arc-transitive digraph is a digraph whose automorphism group is transitive on the set of
alternating s-arcs. The vertices x0, . . . , xs are the vertex-sequence of the alternating s-arc
a1, . . . , as.
• • • • •
• • • • •
// // // //
// oo // oo
Figure 3: A 4-arc (top) versus an alternating 4-arc (bottom).
Clearly if s = 1, then an alternating s-arc is simply an s-arc. If s ≥ 2 then an alternating
s-arc can be obtained from an s-arc by reversing the direction of every other arc - see
Figure 3. Now choose an s-arc, and fix an orientation of this s-arc. An s-arc-transitive
graph is transitive on the set of all s-arcs in Γ with this fixed reorientation, so an s-arc-
transitive graph is trivially alternating s-arc-transitive. The next result gives a relationship
between alternating s-arc-transitive Cayley digraphs and s-arc-transitive Haar graphs.
Theorem 3.3. Let G be a group, S ⊆ G, and s ≥ 1. If Cay(G,S) is an alternating
s-arc-transitive digraph and Haar(G,S) is vertex-transitive, then Haar(G,S) is s-arc-
transitive. Conversely, if Aut(Haar(G,S)) = Z2 ⋉ Aut(Cay(G,S)) and Haar(G,S) is
s-arc-transitive, then Cay(G,S) is alternating s-arc-transitive.
Proof. Suppose Cay(G,S) is alternating s-arc-transitive and Haar(G,S) is vertex-transitive.
Let P1 = (i, x0), (i+1, x1), . . . , (i+s, xs) and P2 = (j, y0), (j+1, y1), . . . , (j+s, ys) be
two s-arcs in Haar(G,S). As Aut(Haar(G,S)) is transitive, replacing P1 or P2 by ι(P1)
or ι(P2), for appropriate ι ∈ Aut(Haar(G,S)), if necessary, we may assume without loss
of generality that i = j = 0. Then there exist t1, . . . , ts ∈ S such that for 1 ≤ m ≤ s
xm =
{
x0t1t
−1
2 . . . t
−1
m−1tm, if m is odd,
x0t1t
−1
2 . . . tm−1t
−1
m , if m is even.
Similarly, there exist u1, . . . , um ∈ S such that for 1 ≤ m ≤ s
ym =
{
y0u1u
−1
2 . . . u
−1
m−1um, if m is odd,
y0u1u
−1
2 . . . um−1u
−1
m , if m is even.
The arcs am = (xm, xmtm) if m is even and am = (xm−1t−1m , xm−1) if m is odd,
0 ≤ m ≤ s − 1, are then contained in A(Cay(G,S)), and Q1 = a0, . . . , as−1 is an
alternating s-arc in Cay(G,S). Similarly, the arcs bm = (ym, xyum) if m is even and
bm = (ym−1u
−1
m , ym−1) if m is odd, 0 ≤ m ≤ s−1, are then contained in A(Cay(G,S)),
and Q2 = b0, . . . , bs−1 is an alternating s-arc in Cay(G,S). As Cay(G,S) is alternat-
ing s-arc-transitive, there exists γ ∈ Aut(Cay(G,S)) such that γ(Q1) = Q2. Then
γ̂ ∈ Aut(Haar(G,S)), and γ̂(P1) = P2. The result follows.
T. Dobson: On automorphisms of Haar graphs of abelian groups 17
Conversely, suppose Aut(Haar(G,S)) = Z2⋉Aut(Cay(G,S)) and Haar(G,S) is s-
arc-transitive. Let P1 = a1, . . . , as and P2 = b1, . . . , bs be alternating s-arcs in Cay(G,S)
with vertex-sequences x0, x1, . . . , xs and y0, y1, . . . , ys, respectively. Then there exist
t1, . . . , ts ∈ S such that for 1 ≤ m ≤ s we have
xm =
{
x0t1t
−1
2 . . . t
−1
m−1tm, if m is odd,
x0t1t
−1
2 . . . tm−1t
−1
m , if m is even.
Similarly, there exist u1, . . . , um ∈ S such that for 1 ≤ m ≤ s
ym =
{
y0u1u
−1
2 . . . u
−1
m−1um, if m is odd,
y0u1u
−1
2 . . . um−1u
−1
m , if m is even.
Corresponding to these two alternating s-arcs, there are two s-arcs
(0, x0), (1, x1), . . . , (s, xs) and (0, y0), (1, y1), . . . , (s, ys) in Haar(G,S). As Haar(G,S)
is s-arc-transitive, there exists δ ∈ Aut(Haar(G,S)) such that
δ((0, x0), (1, x1), . . . , (s, xs)) = (0, y0), (1, y1), . . . , (s, ys).
Then δ(0, x0) = (0, y0) so δ ∈ fixAut(Haar(G,S))(B). As Aut(Haar(G,S)) =
Z2 ⋉ Aut(Cay(G,S)), δ = γ̂ for some γ ∈ Aut(Cay(G,S)). Then γ(xi) = yi and
so γ(P1) = P2. So Cay(G,S) is alternating s-arc-transitive.
We next consider alternating s-arc-transitivity when Cay(G,S) contains arcs and edges.
We will need a preliminary lemma.
Lemma 3.4. Let Γ be a digraph such that every vertex has in- and out-degree at least two,
and s ≥ 2. If Γ is alternating s-arc-transitive, then Γ is alternating (s− 1)-arc-transitive.
Proof. Let a1, . . . , as−1 and b1, . . . , bs−1 be alternating (s − 1)-arcs in Γ with vertex-
sequences x0, . . . , xs−1 and y0, . . . , ys−1. As xs−1 and ys−1 have in- and out-degree at
least two, a1, . . . , as−1 and b1 . . . , bs−1 can be extended to alternating s-arcs a1, . . . , as
and b1, . . . , bs. As Γ is alternating s-arc-transitive, there is γ ∈ Aut(Γ) such that
γ(a1, . . . , as) = b1, . . . , bs.
So γ(a1, . . . , as−1) = b1, . . . , bs−1 and Γ is alternating (s− 1)-arc-transitive.
Corollary 3.5. Let G be a group, S ⊆ G such that Aut(Haar(G,S)) =
Z2 ⋉Aut(Cay(G,S)). Haar(G,S) is s-arc-transitive if and only if
(1) S = S−1 and Cay(G,S) is s-arc-transitive, or
(2) S ∩ S−1 = ∅ and Cay(G,S) is alternating s-arc-transitive.
Proof. Suppose Haar(G,S) is s-arc-transitive. By Theorem 3.3, Cay(G,S) is alternat-
ing s-arc-transitive. Also, |S| ≥ 2 as Aut(Haar(G,S)) = Z2 ⋉ Aut(Cay(G,S)), so by
Lemma 3.4 we have that Cay(G,S) is arc-transitive. This implies Cay(G,S) cannot con-
tain both edges and arcs, so S = S−1 or S ∩ S−1 = ∅. If S ∩ S−1 = ∅, then (2) follows.
Otherwise, S = S−1 so Cay(G,S) is a graph, and is alternating s-arc-transitive if and only
if it is s-arc-transitive and (1) follows.
For the converse, we have already observed that an s-arc-transitive graph is alternating
s-arc-transitive, and so the result holds if S = S−1. If S∩S−1 = ∅, then the result follows
by Theorem 3.3.
18 Art Discrete Appl. Math. 5 (2022) #P3.06
Definition 3.6. Let Γ be a digraph. we say that Γ is a strict digraph if for every arc
(a, b) ∈ A(Γ), the arc (b, a) ̸∈ A(Γ).
Note that if G is a group and S ⊆ G such that S ∩ S−1 = ∅, then Cay(G,S) is a strict
digraph.
Definition 3.7. Let A be an abelian group. The group Z2 ⋉ A ∼= ⟨ι, ÂL⟩ is a generalized
dihedral group.
Definition 3.8. A transitive permutation group G ≤ Sn is quasiprimitive if every nontrivial
normal subgroup of G is transitive.
We now characterize s-arc-transitive Cayley graphs of generalized dihedral groups with
abelian normal subgroup of odd order.
Theorem 3.9. Let s ≥ 2 and Γ be an s-arc-transitive Cayley graph of a generalized
dihedral group G with a normal abelian subgroup A of odd order n and index 2 in G.
Then one of the following is true:
(1) Γ is disconnected,
(2) Aut(Γ) is quasiprimitive or primitive,
(3) Γ ∼= Kn,n,
(4) Γ is isomorphic to a Haar graph corresponding to an s-arc-transitive Cayley graph
of A,
(5) Γ is isomorphic to a Haar graph corresponding to an alternating s-arc-transitive
Cayley strict digraph of A, or
(6) Γ is isomorphic to a Haar graph of A and its corresponding Cayley digraph need
not be s-arc-transitive. In this case, B is a block system of Aut(Γ) and the action of
fixAut(Γ)(B) on B1 is faithful and inequivalent to the action of fixAut(Γ)(B) on B0.
Proof. We assume (1) - (5) do not hold, and show (6) holds. As Aut(Γ) is neither quasiprim-
itive or primitive, there exists 1 < N ◁Aut(Γ) that is not transitive. Let C be the nontrivial
block system of Aut(Γ) formed by the orbits of N . As Γ is connected, there is some edge
in Γ between C1 and C2 ∈ C with C1 ̸= C2. As Γ is edge-transitive, there can be no edges
inside any block of C.
Case 1: N has two orbits. As there are no edges inside any block of C, Γ is a connected
bipartite graph with bipartition C. So Aut(Γ)/C ∼= Z2 has order 2. As GL has order
2n = |G|, we see fixGL(C) has order n. As the only proper normal subgroups of G
of odd order are contained in A as A is of odd order, fixGL(C) contains a semiregular
subgroup with two orbits isomorphic to A. Then Γ is isomorphic to a Haar graph of A
by [14, Lemma 2.5]. So we may assume without loss of generality that Γ = Haar(A,S)
(and C = B).
If the action of fixAut(Γ)(B) is not faithful, then by Theorem 2.16 we have Γ ∼=
Haar(A/H,S) ≀ K̄|H| for some maximal subgroup 1 < H ≤ A. If |A/H| = 1, then
Γ ∼= Kn,n ∼= Haar(G,A), contradicting our assumption that (3) does not hold. If |A/H| ≥
2, then it is straightforward to show that no such wreath product is 2-arc-transitive as
|V (Haar(A/H,S))| ≥ 4. To see this, first observe that Z2 × A/H is a block system of
T. Dobson: On automorphisms of Haar graphs of abelian groups 19
Aut(Γ) as Γ ∼= Haar(A/H,S) ≀K̄|H| and H was chosen to be maximal. Then some 2-arcs
in Γ are of the form ((0, ah), (1, ah), (0, ah′)) for some a ∈ G and h, h′ ∈ H with h′ ̸= h,
and some 2-arcs in Γ are of the form (0, ah), (1, ah), (0, bh) where h ∈ H , and a, b ∈ G
with aH ̸= bH . As Z2×A/H is a block system of Aut(Γ), no automorphism of Γ can map
a 2-arc of the first kind, whose vertices are in two blocks of Z2×A/H , to a 2-arc of the sec-
ond kind, whose vertices are in three blocks of Z2×A/H . Thus the action of fixAut(Γ)(B)
is faithful. If the action of fixAut(Γ)(B) on B1 is equivalent to the action of fixAut(Γ)(B)
on B0, then by Theorem 2.16 we may assume Aut(Γ) = Z2 ⋉Aut(Cay(A,S)), in which
case (4) or (5) would occur by Corollary 3.5. Then (6) follows from Theorem 2.16.
Case 2: If N has at least three orbits, then, as Γ is connected, Γ is a cover of some s-arc-
transitive graph by the Praeger Normal Cover Lemma [36], so N is semiregular. Let M be
the largest subgroup of N that is normal in Aut(Γ) and is contained in ÂL. Let D be the
block system of Aut(Γ) formed by the orbits of M .
Suppose M = 1. As C is the set of left cosets of some subgroup of G, and as the
square of every element of G is contained in A, C consists of blocks of size 2. So N is a
semiregular group of order 2. As A has odd order, ÂL/D ∼= A is a semiregular subgroup
of order n = |A| permutating n blocks, and so is regular. Then ⟨ÂL,M⟩ is transitive,
ÂL ∩ M = 1, and as M has order 2, ⟨ÂL,M⟩ is abelian. We conclude that ⟨ÂL,M⟩ ∼=
ÂL × M is abelian. Thus Γ is an s-arc-transitive Cayley graph of an abelian group, and
as n is odd, we have Γ is isomorphic to a Cayley graph of Z2n. By [30, Theorem 1.1],
Γ = K2n, Γ = Kn,n or Γ is Kn,n with a 1-factor deleted. If Γ = Kn, then Aut(Γ) is
primitive, a contradiction. By hypothesis, Γ ̸= Kn,n. Finally, Kn,n with a 1-factor deleted
is 2-arc-transitive but not 3-arc-transitive and is isomorphic to Haar(A,A−{0}). So Kn,n
with a 1-factor deleted is the Haar graph corresponding to the 2-arc-transitive Cayley graph
Kn = Cay(Zn,Zn − {0}), also a contradiction.
Suppose M ̸= 1. Then M has at least as many orbits as N . If G/M is abelian, then
the commutator subgroup G′ of G is contained in M , and as G′ ∼= A, [G : G′] = 2, M has
at most two orbits, a contradiction. Thus G/M is nonabelian. Then A/M is semiregular in
G/M with two orbits, so Γ/D is isomorphic to a Haar graph of A/M by [14, Lemma 2.5].
Then Γ/D is connected, and so B/D is a block system of Aut(Γ/D), and so of Aut(Γ)/D.
Then B is a block system of Aut(Γ), and this case reduces to the one above with N =
fixAut(Γ)(B) having two orbits.
Note that Kn,n does not satisfy 3.9(4) as Kn,n = Haar(A,A) but Cay(A,A) has
loops, and even if the loops were deleted, Kn is 2-arc-transitive but not 3-arc-transitive
while Kn,n is 3-arc-transitive. So including Kn,n in the conclusion of the above theorem
is not superfluous.
Also, the Heawood graph is 4-arc-transitive by the Foster Census [16], while its corre-
sponding Cayley digraph Cay(Z7, {1, 2, 4}) is arc-transitive but not 2-arc-transitive. This
follows as by [1, Theorem 2] Aut(Cay(Z7, {1, 2, 4})) has automorphism group G = {x 7→
ax + b : a ∈ {1, 2, 4}, b ∈ Z7}, and it is arc-transitive. However, as G has order 21,
Cay(Z7, {1, 2, 4}) cannot be 2-arc-transitive as there are 42 2-arcs in Cay(Z7, {1, 2, 4}).
Finally, while we have shown that there are graphs satisfying Theorem 3.9(6) holds,
and some of the other possibilities obviously hold, it is not clear that for each of the six
possibilities, there are corresponding graphs.
20 Art Discrete Appl. Math. 5 (2022) #P3.06
ORCID iDs
Ted Dobson https://orcid.org/0000-0003-2013-4594
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