ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P2.07 / 287–304
https://doi.org/10.26493/1855-3974.2443.02e
(Also available at http://amc-journal.eu)
Generalised dihedral CI-groups
Ted Dobson *
University of Primorska, UP IAM, Muzejski trg 2, SI-6000 Koper, Slovenia, and
University of Primorska, UP FAMNIT, Glagoljaşka 8, SI-6000 Koper, Slovenia
Mikhail Muzychuk
Department of Mathematics, Ben-Gurion University of the Negev, Israel
Pablo Spiga
Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca,
Via Cozzi 55, 20125 Milano, Italy
Received 24 September 2020, accepted 16 August 2021, published online 27 May 2022
Abstract
In this paper, we find a strong new restriction on the structure of CI-groups. We show
that, if R is a generalised dihedral group and if R is a CI-group, then for every odd prime
p the Sylow p-subgroup of R has order p, or 9. Consequently, any CI-group with quotient
a generalised dihedral group has the same restriction, that for every odd prime p the Sylow
p-subgroup of the group has order p, or 9.
Keywords: CI-group, DCI-group, generalised dihedral, Cayley isomorphism.
Math. Subj. Class. (2020): 05E18, 05E30
1 Introduction
Let R be a finite group and let S be a subset of R. The Cayley digraph of R with con-
nection set S, denoted Cay(R,S), is the digraph with vertex set R and with (x, y) being
an arc if and only if xy−1 ∈ S. Now, Cay(R,S) is said to be a DCI-graph (here CI
stands for Cayley isomorphic while the D stands for directed), if whenever Cay(R,S) is
isomorphic to Cay(R, T ), there exists an automorphism φ of R with Sφ = T . Clearly,
*Corresponding author. This work is supported in part by the Slovenian Research Agency (research program
P1-0285 and research projects N1-0062, J1-9108, J1-1695, N1-0140, N1-0160, J1-2451, N1-0208).
E-mail addresses: ted.dobson@upr.si (Ted Dobson), muzychuk@bgu.ac.il (Mikhail Muzychuk),
pablo.spiga@unimib.it (Pablo Spiga)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
288 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
Cay(R,S) ∼= Cay(R,Sφ) for every φ ∈ Aut(R) and hence, loosely speaking, for a DCI-
graph Cay(R,S) deciding when a Cayley digraph over R is isomorphic to Cay(R,S) is
theoretically and algorithmically elementary, but computationally efficient only if Aut(R)
is small; that is, the solving set for Cay(R,S) is reduced to simply Aut(R) (for the defini-
tion of a solving set see for example [24, 26]). The group R is a DCI-group if Cay(R,S)
is a DCI-graph for every subset S of R. Moreover, R is a CI-group if Cay(R,S) is a
DCI-graph for every inverse-closed subset S of R. Thus every DCI-group is a CI-group.
After roughly 50 years of intense research, the classification of DCI- and CI-groups is
still open. The current state of the art in this problem is as follows. There exist two rather
short lists of candidates for DCI- and CI-groups and it is known that every DCI- and every
CI-group must be a member of the corresponding list, see for instance [20]. Showing that a
candidate on the lists of possible DCI- or CI-groups is actually a DCI- or CI-group, though,
takes a considerable amount of effort. Just to give an example, the recent paper of Feng
and Kovács [15] is a tour de force that shows that elementary abelian groups of rank 5 are
DCI-groups.
In this paper we find an unexpected new restriction on which generalised dihedral
groups are CI-groups, and significantly shorten the list of candidates for CI-groups.
Definition 1.1. Let A be an abelian group. The generalised dihedral group Dih(A) over
A is the group ⟨A, x | ax = a−1,∀a ∈ A⟩. A group is called generalised dihedral if it is
isomorphic to Dih(A) for some A. When A is cyclic, Dih(A) is called a dihedral group.
Our main result is the following.
Theorem 1.2. Let Dih(A) be a generalised dihedral group over the abelian group A. If
Dih(A) is a CI-group, then, for every odd prime p the Sylow p-subgroup of A has order p,
or 9. If Dih(A) is a DCI-group, then, in addition, the Sylow 3-subgroup has order 3.
Generalised dihedral groups are amongst the most abundant members in the list of pu-
tative CI-groups. The importance of Theorem 1.2 is the arithmetical condition on the order
of such groups, which greatly reduces even further the list of candidates for CI-groups.
We believe that every generalised dihedral group satisfying this numerical condition on its
order is a genuine CI-group. (This is in line with the partial result in [8].) Additionally, this
result further reduces to two other groups on the list, whose definitions we now give.
Definition 1.3. Let A be an abelian group such that every Sylow p-subgroup of A is el-
ementary abelian. Let n ∈ {2, 4, 8} be relatively prime to |A|. Set E(A,n) = A ⋊ ⟨g⟩,
where g has order n and ag = a−1, ∀a ∈ A.
Note that E(A, 2) = Dih(A). The groups E(A, 4) and E(A, 8) have centres Z1 and
Z2 of order 2 and 4, respectively, and E(A, 4)/Z1 ∼= E(A, 8)/Z2 ∼= Dih(A). Babai
and Frankl [2, Lemma 3.5] showed that a quotient of a (D)CI-group by a characteristic
subgroup is a (D)CI-group, while the first author and Joy Morris [7, Theorem 8] showed
that a quotient of a (D)CI-group is a (D)CI-group. Applying either result and Theorem 1.2
we have the following.
Corollary 1.4. If E(A, 4) or E(A, 8) is a CI-group, then, for every odd prime p the Sylow
p-subgroup of A has order p or 9. If E(A,n), n ∈ {2, 4, 8} is a DCI-group, then, in
addition, n ̸= 8 and the Sylow 3-subgroup of A has order 3.
T. Dobson et al.: Generalised dihedral CI-groups 289
Not much is known about which of the groups under consideration in this paper are
CI-groups. Let p be a prime. Babai [1, Theorem 4.4] showed D2p is a CI-group. The first
author [4, Theorem 22] extended this to some special values of square-free integers. With
Joy Morris, the first and third authors [8] showed that D6p is a CI-group, p ≥ 5. Also, Li,
Lu, and Pálfy showed E(p, 4) and E(p, 8) are CI-groups.
We have one other result of interest, for which we will need an additional definition.
Definition 1.5. Let G be a group, and S ⊆ G. A Haar graph of G with connection set S
has vertex set G× Z2 and edge set {{(g, 0), (sg, 1)} : g ∈ G and s ∈ S}.
So a Haar graph is a bipartite analogue of a Cayley graph. There is a corresponding iso-
morphism problem for Haar graphs, and if the group A is abelian, it is equivalent to the
isomorphism problem for Cayley graphs of generalised dihedral groups Dih(A) that are
bipartite (for nonabelian groups the problems are not equivalent, as for non-abelian groups
Haar graphs need not be transitive), see [17, Lemma 2.2]. If isomorphic bipartite Cayley
graphs of Dih(A) are isomorphic by group automorphisms of A, we say A is a BCI-group.
We will also show that Zk3 is not a BCI-group for every k ≥ 3, while it is known that Zk3 is
a CI-group for every 1 ≤ k ≤ 5 [32].
1.1 Some notation
Babai [1, Lemma 3.1] has proved a very useful criterion for determining when a finite
group is a DCI-group and, more generally, when Cay(R,S) is a DCI-graph.
Lemma 1.6. Let R be a finite group, and let S be a subset of R. Then, Cay(R,S) is a
DCI-graph if and only if Aut(Cay(R,S)) contains a unique conjugacy class of regular
subgroups isomorphic to R.
Let Ω be a finite set and let G be a permutation group on Ω. An orbital graph of G is a
digraph with vertex set Ω and with arc set a G-orbit (α, β)G = {(αg, βg) | g ∈ G}, where
(α, β) ∈ Ω × Ω. In particular, each orbital graph has for its arcs one orbit on the ordered
pairs of elements of Ω, under the action of G. Moreover, we say that the orbital graphs
(α, β)G and (β, α)G are paired. When (α, β)G = (β, α)G, we say that the orbital graph is
self-paired.
When G is transitive and ω0 ∈ Ω, there exists a natural one-to-one correspondence
between the orbits of G on Ω × Ω (a.k.a. orbitals or 2-orbits of G) and the orbits of the
stabiliser Gω0 on Ω (a.k.a. suborbits of G). Therefore, under this correspondence, we may
naturally define paired and self-paired suborbits.
Two subgroups of the symmetric group Sym(Ω) are called 2-equivalent if they have the
same orbitals. A subgroup of Sym(Ω) generated by all subgroups 2-equivalent to a given
G ≤ Sym(Ω) is called the 2-closure of G, denoted G(2).
The group G is said to be 2-closed if G = G(2). It is easy to verify that G(2) is a sub-
group of Sym(Ω) containing G and, in fact, G(2) is the largest (with respect to inclusion)
subgroup of Sym(Ω) preserving every orbital of G.
290 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
2 Construction and basic results
Let q be a power of an odd prime and let F be a field of cardinality q. We let
G :=

a x z0 b y
0 0 c
 | x, y, z ∈ F, a, b, c ∈ {−1, 1}, abc = 1
 ,
D :=

a ax ax2/20 1 x
0 0 a
 | x ∈ F, a ∈ {−1, 1}
 ,
H :=

a 0 x0 a y
0 0 1
 | x, y ∈ F, a ∈ {−1, 1}
 ,
K :=

1 x y0 a 0
0 0 a
 | x, y ∈ F, a ∈ {−1, 1}
 .
It is elementary to verify that G, D, H and K are subgroups of the special linear group
SL3(F). Moreover, D, H and K are subgroups of G, |G| = 4q3, |D| = 2q and |H| =
|K| = 2q2. We summarise in Proposition 2.1 some more facts.
Proposition 2.1. The group D is generalised dihedral over the abelian group (F,+) and,
H and K are generalised dihedral over the abelian group (F⊕ F,+). The core of D in G
is 1. Moreover,
DK = DH = G = HD = KD and D ∩H = 1 = D ∩K.
Proof. The first two assertions follow with easy matrix computations. Let
g :=
1 0 00 −1 0
0 0 −1
 ∈ G
and observe that
g−1
a ax ax2/20 1 x
0 0 a
 g =
a −ax −ax2/20 1 x
0 0 a
 .
As the characteristic of F is odd, from this it follows that
D ∩Dg =
〈−1 0 00 1 0
0 0 −1
〉 .
It is now easy to see that D is core-free in G.
It is readily seen from the definitions that D ∩H = 1 = D ∩K. Therefore, |DH| =
|D||H| = 4q3 and |DK| = |D||K| = 4q3. As DH and DK are subsets of G and
|G| = 4q3, we deduce DH = G = DK and hence also HD = G = KD.
T. Dobson et al.: Generalised dihedral CI-groups 291
We let D\G := {Dg | g ∈ G} be the set of right cosets of D in G. In view of Propo-
sition 2.1, G acts faithfully by right multiplication on D\G and H and K act regularly by
right multiplication on D\G.
Proposition 2.2. The subgroups H and K are normal in G and, therefore, are in distinct
G-conjugacy classes.
Proof. The normality of H and K in G can be checked by direct computations.
2.1 Schur notation
Since G = DH and D ∩ H = 1, for every g ∈ G, there exists a unique h ∈ H with
Dg = Dh. In this way, we obtain a bijection θ : D\G → H , where θ(Dg) = h ∈ H
satisfies Dg = Dh.
Using the method of Schur (see [33]), we may identify via θ the G-set D\G with H .
Moreover, we may define an action of G on H via the following rule: for every g ∈ G and
for every h ∈ H ,
hg = h′ if and only if Dhg = Dh′.
A classic observation of Schur yields that the action of G on D\G is permutation isomor-
phic to the action of G on H . In the rest of the paper, we use both points of view.
In the action of G on H , D is a stabiliser of the identity e ∈ H , i.e. Ge = D, and H
acts on itself via its right regular representation. Since H is normal in G, the action of the
point stabiliser Ge on H is permutation equivalent to the action of Ge via conjugation on
H (Proposition 20.2 [33]). More precisely, hg = g−1hg for any g ∈ Ge and h ∈ H .
In what follows, we represent the elements of H and D as pairs [a, x] and [a, w⃗], where
x ∈ F, w⃗ ∈ F2 and a ∈ {±1}. In particular, [a, x] represents the matrixa ax ax2/20 1 x
0 0 a

of D and, if w⃗ = (x, y), then [a, w⃗] represents the matrixa 0 x0 a y
0 0 1

of H . Under this identification, the product in D and H greatly simplifies. Indeed, for
every [a, x], [b, y] ∈ D and for every [a, v⃗], [b, w⃗] ∈ H , we have
[a, x][b, y] = [ab, bx+ y], (2.1)
[a, v⃗][b, w⃗] = [ab, bv⃗ + w⃗].
Using this identification, the action of D on H also becomes slightly easier. Indeed, for
every [a, v⃗] ∈ H (with v⃗ = (x, y)) and for every [b, z] ∈ D, we have
[a, (x, y)][b,z] = [a,
(
(1− a)z2/2− byz + x, (−1 + a)z + by
)
]. (2.2)
This equality can be verified observing thata 0 x0 a y
0 0 1
b bz bz2/20 1 z
0 0 b
=
b bz bz2/20 1 z
0 0 b
a 0 (1− a)z2/2− byz + x0 a (−1 + a)z + by
0 0 1
 .
292 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
2.2 One special case
Let A := ⟨e1, e2, e3⟩, where e1 := (1 2 3), e2 := (4 5 6), e1 := (7 8 9), let
x := (1 2)(4 5)(7 8) and let R := ⟨A, x⟩. Then R is a generalised dihedral group over
the elementary abelian 3-group A of order 33 = 27. Let
S := {x, e1x, e2x, e3x, e1e2x, e21e22x, e2e3x, e22e23x, e21e22e23x}
and define
Γ := Cay(R,S).
It can be verified with the computer algebra system Magma that Aut(Γ) has order
46656 = 26 · 36, acts transitively on the arcs of Γ and (most importantly) contains two
conjugacy classes of regular subgroups isomorphic to R and hence, via Babai’s lemma, R
is not a CI-group.
This example has another interesting property from the isomorphism problem point of
view. Observe that each element of S is an involution contained in R \A. This implies that
Γ is a bipartite graph, in which case Γ is isomorphic to a Haar graph, also called a bi-coset
graph. In our example above, as every element of the connection set is an involution, it is
a Haar graph of Z33 but as it is not a CI-graph of Dih(Z33), Z33 is not a BCI-group. This is
the first example the authors are aware of where a group is an abelian DCI-group but not a
BCI-group, as Z3p is a DCI-group [3]. Our next result shows Zk3 is not a BCI-group for any
k ≥ 3.
Lemma 2.3. Let R be an abelian group and let H ≤ R. If R is BCI-group, then R/H is
BCI-group.
Proof. For this result, it is most convenient to have the vertex sets of Haar graphs and
Cayley graphs of dihedral groups be the same. So, for an abelian group R, we will have
Dih(R) permuting the set R× Z2 (the vertex set of a Haar graph of R), where an element
s ∈ R is identified with the map st : R × Z2 → R × Z2 given by st(r, i) 7→ (r + s, i).
Define ι : R × Z2 → R × Z2 by ι(r, i) = (−r, i + 1). Then Dih(R) is canonically
isomorphic to G = ⟨ι, st : s ∈ R⟩. It is straightforward to show that ι ∈ Aut(Haar(R,S)),
and so we have G ≤ Aut(Haar(R,S)) for every S ⊆ R. By [28, Theorem 2], we have
Haar(R,S) ∼= Cay(Dih(R), T ), for some T ⊆ G, by the map ϕ which identifies (r, i) with
the unique element of G which maps (0, 0) to (r, i), r ∈ R, i ∈ Z2. Hence ϕ(r, i) = rtιi,
and T = {sι : s ∈ S} = S · ι.
If R is a BCI-group, then Haar(R,S) is a BCI graph. Let C = {R×{0}, R×{1}}, B
be the set of right cosets of H in Dih(R), and U = {sH : s ∈ S}. Then, as partitions of
R × Z2, B refines C. As C is a bipartition of Cay(Dih(R), S · ι), Cay(Dih(R/H), U · ι)
is bipartite with bipartition {{(rH, i) : r ∈ R} : i ∈ Z2} and so Cay(Dih(R/H), U · ι) =
Haar(R/H,U).
As Cay(Dih(R), S · ι) is a CI-graph of Dih(R), by the proof of [6, Theorem 8],
we see Cay(Dih(R/H), U · ι) is a CI-graph of Dih(R/H) and any Cayley graph of
Dih(R/H) isomorphic to Cay(Dih(R/H), U · ι) is isomorphic by a group automorphism
of Dih(R/H). But this means any two Haar graphs of R/H are isomorphic by a group
automorphism of Dih(R/H), and so R/H is a BCI-group.
Finally, Γ, as well as the graphs constructed in the next section, have the property
that the Sylow p-subgroups of their automorphism groups are not isomorphic to Sylow p-
subgroups of any 2-closed group of degree 33 or p2 (in the next section). For the example
T. Dobson et al.: Generalised dihedral CI-groups 293
above, the Sylow p-subgroups of the automorphism groups of Cayley digraphs of Z3p can
be obtained from [5, Theorem 1.1], and none have order 36 as a Sylow p-subgroup of
AGL(3, 3) is not 2-closed (for p2 in the next section, the Sylow p-subgroup has order p3,
but Sylow p-subgroups of the automorphism groups of Cayley digraphs of Z2p have order
p2 or pp+1 [10, Theorem 14]).
3 The permutation group G is 2-closed
In this section we prove the following.
Proposition 3.1. The group G in its action on H is 2-closed.
We start with some preliminary observations.
Lemma 3.2. The orbits of Ge on H have one of the following forms:
(1) St := {[1, (t, 0)]}, for every t ∈ F;
(2) Ct ∪ C−t, where Ct := {[1, (z, t)] | z ∈ F} and t ∈ F \ {0};
(3) Pt :=
{
[−1, (t+ z2, 2z)] | z ∈ F
}
with t ∈ F.
Proof. Let g := [a, (x, y)] ∈ H . If a = 1 and y = 0, then (2.2) yields
g[b,z] = [1, (x, 0)] = g
and hence the Ge-orbit containing g is simply {g}. Therefore we obtain the orbits in
Case (1).
Suppose then a = 1 and y ̸= 0. Now, 2.2 yields
g[1,z] = [1, (−yz + x, y)],
g[−1,z] = [1, (yz + x,−y)].
In particular, Cy = {g[1,z] | z ∈ F} and C−y = {g[−1,z] | z ∈ F} and we obtain the orbits
in Case (2).
Finally suppose a = −1. Now, (2.2) yields
g[b,z] = [1, (z2 − byz + x,−2z + by)].
In particular, if we choose z := by/2 and t = −y2/4 + x, then g and [−1, (t, 0)] are in the
same Ge-orbit. Therefore [−1, (x, y)]Ge = [−1, (t, 0)]Ge . Using again (2.2), we get
[−1, (t, 0)][b,−z] = [−1, (t+ z2, 2z)].
In particular, Pt = {g[b,z] | [b, z] ∈ Ge} and we obtain the orbits in Case (3).
We call the Ge-orbits in (1) singleton orbits, the Ge-orbits in (2) coset orbits and the
Ge-orbits in (3) parabolic orbits. Clearly, singleton orbits have cardinality 1, coset orbits
have cardinality 2q and parabolic orbits have cardinality q. Also, it follows from Lemma 3.2
that there are q singleton orbits, q−12 coset orbits and q parabolic orbits. Indeed,
q · 1 + q − 1
2
· 2q + q · q = 2q2 = |H|.
It is also clear from Lemma 3.2 that all non-singleton orbits are self-paired and the only
self-paired singleton orbit is S0.
Before continuing, we recall [14, Definitions 2.5.3 and 2.5.4] tailored to our needs.
294 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
Definition 3.3. We say that h ∈ H separates the pair (h1, h2) ∈ H × H , if (h, h1) and
(h, h2) belong to distinct G-orbitals, that is, hh−11 and hh
−1
2 are in distinct Ge-orbits.
We also say that a subset S ⊆ H separates G-orbitals if, for any two distinct elements
h1, h2 ∈ H \ S, there exists s ∈ S separating the pair (h1, h2).
Proposition 3.4. If q ≥ 5, then {e} ∪ P0 separates G-orbitals.
Proof. Set S := {e} ∪ P0. Let h1, h2 ∈ H \ S be two distinct elements. If h1 and h2
belong to distinct Ge-orbits, then e ∈ S separates (h1, h2). Therefore, we assume that h1
and h2 belong to the same Ge-orbit, say, O. Since h1 ̸= h2, O is not a singleton orbit and
hence O is either a coset or a parabolic orbit.
Assume first that O is a parabolic orbit, that is, O = Pt, for some t ∈ F. By Lemma 3.2,
for each i ∈ {1, 2}, there exists xi ∈ F with hi = [−1, (t + x2i , 2xi)]. As q = |F| ≥ 5, it
is easy to verify that there exists x ∈ F with x /∈ {x1, x2} and with x− x1 ̸= −(x− x2).
Now, let s := [−1, (x2, 2x)] ∈ P0 ⊆ S. From (2.1), we deduce
sh−1i = [1, (t+ x
2
i − x2, 2xi − 2x)].
As 2xi−2x ̸= 0, from Lemma 3.2, we obtain sh−1i ∈ C2(x−xi)∪C−2(x−xi). As x−x1 ̸=
−(x− x2), we deduce that sh−11 and sh
−1
2 are in distinct Ge-orbits and hence s separates
(h1, h2).
Assume now that O is a coset orbit, that is, O = Ct ∪ C−t, for some t ∈ F \ {0}. In
this case, for each i ∈ {1, 2}, there exist xi ∈ F and ai ∈ {±1} with hi = [1, (xi, ait)].
Let x ∈ F with
xt(a2 − a1) ̸= x2 − x1.
(The existence of x is clear when a1 ̸= a2 and it follows from the fact that h1 ̸= h2 when
a1 = a2.) Set s := [−1, (x2, 2x)] ∈ P0 ⊆ S. From (2.1), we have
sh−1i ∈ [−1, (x
2 − xi, 2x− ait)].
In particular, from Lemma 3.2, we have sh−1i ∈ Pti , for some ti ∈ F. Thus, (x2−xi, 2x−
ait) = (ti + y
2, 2y), for some y ∈ F. From this it follows that
ti = x
2 − xi −
(2x− ait)2
4
.
As xt(a2 − a1) ̸= x2 −x1, a simple computation yields t1 ̸= t2 and hence sh−11 and sh
−1
2
are in distinct Ge-orbits. Therefore, s separates (h1, h2).
Proof of Proposition 3.1. When q = 3, the proof follows with a computation with the
computer algebra system Magma. Therefore, for the rest of the proof we suppose q ≥
5. Let T be the 2-closure of G. As {e} ∪ P0 separates the G-orbitals, it follows from
[14, Theorem 2.5.7] that the action of Te on P0 is faithful, and hence so is the action of
Ge on P0. We denote by GP0e (respectively, T
P0
e ) the permutation group induced by Ge
(respectively, Te) on P0. In particular, Ge ∼= GP0e and Te ∼= TP0e .
We claim that
(Te)
P0 = (Ge)
P0 . (3.1)
Observe that from (3.1) the proof of Proposition 3.1 immediately follows. Indeed, Te ∼=
TP0e = G
P0
e
∼= Ge and hence Te = Ge. As H is a transitive subgroup of G, we deduce that
T. Dobson et al.: Generalised dihedral CI-groups 295
G = GeH = TeH = T and hence G is 2-closed. Therefore, to complete the proof, we
need only establish (3.1).
From Lemma 3.2, |P0| = q. Hence (Ge)P0 is a dihedral group of order 2q in its natural
action on q points.
For each t ∈ F∗ let Φt be the subgraph of Cay(H,Ct ∪ C−t) induced by P0. Let
(h1, h2) be an arc of Φt. As h1, h2 ∈ P0, there exist x1, x2 ∈ F with h1 = [−1, (x21, 2x1)]
and h2 = [−1, (x22, 2x2)]. Moreover, h2h−11 ∈ Ct ∪ C−t and hence, by (2.1), we obtain
h2h
−1
1 = [1, (x
2
2 − x21, 2x2 − 2x1)] ∈ Ct ∪ C−t,
that is, 2x2 − 2x1 ∈ {−t, t}. This shows that the mapping
P0 → F+
(x2, 2x) 7→ 2x
is an isomorphism between the graphs Φt and Cay(F+, {−t, t}). Therefore
(Ge)
P0 ≤ (Te)P0 ≤
⋂
t∈F∗
Aut(Φt) ∼=
⋂
t∈F∗
Aut(Cay(F+, {−t, t})) ∼= Dih(F+).
Since (Ge)P0 and Dih(F+) are dihedral groups of order 2q, we conclude that (Ge)P0 =
(Te)
P0 =
⋂
t∈F∗ Aut(Φt), proving 3.1.
4 Generating graph
Combining Proposition 3.1, Proposition 2.2, and Lemma 1.6, we have proven that Dih(Z2p)
is not a CI-group with respect to colour Cayley digraphs for odd primes p. In this section
we strengthen that result to Cayley graphs.
4.1 Schur rings
Let R be a finite group with identity element e. We denote the group algebra of R over the
field Q by QR. For Y ⊆ R, we define
Y :=
∑
y∈Y
y ∈ QR.
Elements of QR of this form will be called simple quantities, see [33]. A subalgebra A of
the group algebra QR is called a Schur ring over R if the following conditions are satisfied:
(1) there exists a basis of A as a Q-vector space consisting of simple quantities
T 0, . . . , T r;
(2) T0 = {e}, R =
⋃r
i=0 Ti and, for every i, j ∈ {0, . . . , r} with i ̸= j, Ti ∩ Tj = ∅;
(3) for each i ∈ {0, . . . , r}, there exists i′ such that Ti′ = {t−1 | t ∈ Ti}.
Now, T 0, . . . , T r are called the basic quantities of A. A subset S of R is said to be an
A- subset if S ∈ A, which is equivalent to S =
⋃
j∈J Tj , for some J ⊆ {0, . . . , r}.
Given two elements a :=
∑
x∈R axx and b :=
∑
y∈R byy in QR, the Schur-Hadamard
product a ◦ b is defined by
a ◦ b :=
∑
z∈R
azbzz.
296 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
It is an elementary exercise to observe that, if A is a Schur ring over R, then A is closed by
the Schur-Hadamard product.
The following statement is known as the Schur-Wielandt principle, see [33, Proposi-
tion 22.1].
Proposition 4.1. Let A be a Schur ring over R, let q ∈ Q and let x :=
∑
r∈R arr ∈ A.
Then
xq :=
∑
r∈R
ar=q
r ∈ A.
Let X be a permutation group containing a regular subgroup R. As in Section 2.1, we
may identify the domain of X with R. Let T0, . . . , Tr be the orbits of Xe with T0 = {e}.
A fundamental result of Schur [33, Theorem 24.1] shows that the Q-vector space spanned
by T 0, T 1, . . . , T r in QR is a Schur ring over R, which is called the transitivity module of
the permutation group X and is usually denoted by V (R,Ge). In particular, the V (R,Ge)-
subsets of the Schur ring V (R,Ge) are unions of Ge-orbits.
Let A := ⟨T 0, . . . , T r⟩ be a Schur ring over R (where T0, . . . , Tr are the basic quanti-
ties spanning A). The automorphism group of A is defined by
Aut(A) :=
r⋂
i=0
Aut(Cay(R, Ti)). (4.1)
Given a subset S of R, we denote by
⟨⟨S⟩⟩,
the smallest (with respect to inclusion) Schur ring containing S. Now, ⟨⟨S⟩⟩ is called the
Schur ring generated by S.
We conclude this brief introduction to Schur rings recalling [25, Theorem 2.4].
Proposition 4.2. Let S be a subset of R. Then Aut(⟨⟨S⟩⟩) = Aut(Cay(R,S)).
4.2 The group G is the automorphism group of a single (di)graph
It was shown above that the group G is 2-closed, i.e. it is the automorphism of a coloured di-
graph. In this section we give a Cayley digraph Cay(H,T ) having automorphism group G.
To build such a digraph it is sufficient to find a subset T ⊆ H such that
⟨⟨T ⟩⟩ = V (H,Ge) (Proposition 4.2). Such a set is constructed in Proposition 4.3. Note
that T is symmetric for q ≥ 7, so the digraph Cay(H,T ) is undirected. The cases of
q = 3, 5 are exceptional, because in those cases no inverse-closed subset of H has the
required property.
Proposition 4.3. Let q be prime, and
T :=

P0 ∪ P1 ∪ Px ∪ C1 ∪ C−1 where x ∈ F with x ̸∈ {0,±1,±2, 12} and x
6 ̸= 1,
when q > 7,
P0 ∪ P1 ∪ P3 ∪ C1 ∪ C−1 when q = 7,
S1 ∪ P0 when q = 5,
S1 ∪ P0 when q = 3.
Then ⟨⟨T ⟩⟩ = V (H,Ge). In particular, T is not a (D)CI-subset of H .
T. Dobson et al.: Generalised dihedral CI-groups 297
Proof. When q ≤ 7, the result follows by computations with the computer algebra system
Magma. Therefore for the rest of the proof we suppose q > 7.
According to Proposition 3.2 the basic sets of V (H,Ge) are of three types: Sa, Cb ∪
C−b, Pc with a, b, c ∈ F and b ̸= 0. Thus we have three types of basic quantities Sa,
Cb + C−b, Pc and
V (H,Ge) = ⟨Sa, Cb + C−b, Pc a, b, c ∈ F, b ̸= 0⟩.
Set
H1 := {[1, v⃗] | v⃗ ∈ F2},
H2 := {[1, (t, 0)] | t ∈ F}.
By (2.1), H1 and H2 are subgroups of H with |H2| = q, |H1| = q2 and, by Lemma 3.2,
H2 = ∪t∈FSt. In Table 4.2 we have reported the multiplication table among the basic
quantities of V (H,Ge): this will serve us well.
Sr Cs Pt
Sa Sa+r Cs Pt−a
Cb Cb
{
qCb+s if b+ s ̸= 0
qH2 if b+ s = 0
H \H1
Pc Pc+r H \H1 qS−c+t +H1 \H2
Table 1: Multiplication table for the basic quantities of V (H,Ge).
Fix a, b, c ∈ F with b, c ̸= 0 and let A be the smallest Schur ring of the group algebra
QH containing Pa, Cb + C−b, Sc. We claim that
A = V (H,Ge). (4.2)
Clearly, A ≤ V (H,Ge). From Table 4.2, for every k ∈ {0, . . . , q−1}, we have Sck = Sck
and hence Sck ∈ A. As c ̸= 0, Si ∈ A, for each i ∈ {0, . . . , q − 1}. Now, as Pa ∈ A,
from Table 4.2, we have Pa · Si = Pa+i ∈ A for any i ∈ {0, . . . , q − 1}. The equality
(Cb+C−b)
2 = 2qH2+ qC2b+ qC−2b implies C2b+C−2b ∈ A. Now arguing inductively
we deduce Ck + C−k ∈ A, for all k ∈ {1, . . . , q − 1}. Thus (4.2) follows.
Let x ∈ F with x ̸∈ {0,±1,±2, 12} and x
6 ̸= 1, let T := P0 ∪ P1 ∪ Px ∪ C1 ∪ C−1
and let T := ⟨⟨T ⟩⟩ (the existence of x is guaranteed by the fact that q > 7). We claim that
H2, H1, C2 + C−2, S1 + S−1 + Sx + S−x + S1−x + Sx−1 ∈ T . (4.3)
Using Table 4.2 for squaring T , we obtain (after rearranging the terms):
T 2 =3qS0 + qS1 + qS−1 + qSx + qS−x + qS1−x + qSx−1
+ 9H1 \H2 + 12H \H1 + qC2 + qC−2 + 2qH2.
298 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
From the assumptions on x, the elements −1, 1,−x, x,−(x−1), x−1 are pairwise distinct.
Therefore
T 2 ◦ Sb =

5qS0, b = 0,
3qSb, if b ∈ {±1,±x,±(x− 1)},
2qSb, if b ̸∈ {0,±1,±x,±(x− 1)},
T 2 ◦ Cb =
{
(q + 9)Cb, if b ∈ {±2},
9Cb, if b ̸∈ {0,±2},
T 2 ◦ Pb = 12Pb, if b ∈ F.
Since the numbers 6, 9, q + 9, 2q, 3q, 5q are also pairwise distinct (because q ̸= 3), an
application of the Schur-Wielandt principle yields
(T 2)3q = S1 + S−1 + Sx + S−x + S1−x + Sx−1 ∈ T ,
(T 2)12 = H \H1 ∈ T ,
(T 2)2q = H2 − (S0 + S1 + S−1 + Sx + S−x + S1−x + Sx−1) ∈ T ,
(T 2)q+9 = C2 + C−2 ∈ T .
From this, (4.3) immediately follows.
We claim that
S1 + S−1 ∈ T . (4.4)
Let
TH2 := T ∩QH2
and observe that TH2 is a Schur ring over the cyclic group H2 ∼= Zq of prime order q. It is
well known that every Schur ring over Zq is determined by a subgroup M ≤ Aut(Zq) ∼=
Z∗q such that, every basic set of the corresponding Schur ring is an M -orbit. Let M be such
a subgroup for TH2 . From (4.3), the simple quantity S1 + S−1 + Sx + S−x + S1−x +
Sx−1 belongs to TH2 and hence {±1,±x,±(1 − x)} is a TH2 -subset of cardinality 6.
It follows that |M | divides six and M ⊆ {±1,±x,±(1 − x)}. If |M | ∈ {3, 6}, then
{±1,±x,±(1− x)} is a subgroup of Z∗q , contrary to the assumption x6 ̸= 1. Therefore
either M = {1} or |M | = {±1}. (4.5)
In both cases, {−1, 1} is a union of M -orbits. Therefore, S1+S−1 ∈ TH2 . From this, (4.4)
follows immediately.
We are now ready to conclude the proof. Clearly, T ∈ V (H,Ge) and hence T ⊆
V (H,Ge). From (4.3), H1 ∈ T and, from (4.4), S1 + S−1 ∈ T . Therefore H1 ◦ T =
C1 + C−1 ∈ T and (T −H1) ◦ T = P0 + P1 + Px ∈ T . Therefore(
(P0 + P1 + Px)(S1 + S−1)
)
◦ (P0 + P1 + Px) ∈ T .
As (P0 + P1 + Px)(S1 + S−1) = P1 + P2 + Px+1 + P−1 + P0 + Px−1, we deduce(
(P0 + P1 + Px)(S1 + S−1)
)
◦ (P0 + P1 + Px) = P0 + P1
T. Dobson et al.: Generalised dihedral CI-groups 299
and hence P0 + P1 ∈ T . Therefore, Px = (P0 + P1 + Px)− (P0 + P1) ∈ T . As
(P0 + P1)Px = qSx + qSx−1 + 2(H \H1),
from the Schur-Wielandt principle, we obtain Sx+Sx−1 ∈ T . Therefore Sx+Sx−1 ∈ TH2
and hence {x, x − 1} is a TH2 -subset. Thus {x, x − 1} is an M -orbit. Recall (4.5). If
M = {−1, 1}, then x− 1 = −1 · x = −x, contrary to the assumption x ̸= 1/2. Therefore
M = {1} and TH2 = QH2. Thus Si ∈ T , for each i ∈ Zq . Thus S1, Px, C1 + C−1 ∈ T
and (4.2) implies V (H,Ge) ⊆ T .
5 Proof of Theorem 1.2
Proof of Theorem 1.2. The list of candidate CI-groups is on page 323 in [20]. From here,
we see that, if R is in this list and if R = Dih(A) is generalised dihedral, then for every
odd prime p the Sylow p-subgroup of R is either elementary abelian or cyclic of order 9.
Assume that the Sylow p-subgroup (p is an odd prime) of A is elementary abelian of
rank at least 2. Let P ≤ A be a subgroup isomorphic to Z2p and let x ∈ R\A. Then ⟨P, x⟩ ∼=
Dih(Z2p). By Proposition 4.3, Dih(Z2p) contains a non-DCI subset. Therefore Dih(Z2p) is
a non-DCI-group. Since subgroups of a (D)CI-group are also (D)CI, we conclude that R is
a not a DCI-group as well. The non-DCI set T constructed in Proposition 4.3 is symmetric
for p ≥ 7. Hence Dih(Z2p) and, therefore, R are non-CI groups when p ≥ 7. If p = 5, then
the group Dih(Z2p) contains a non-CI subset, namely: P0 ∪ S1 ∪ S−1 (this was checked by
Magma1). Combining these arguments we conclude that if Dih(A) is a CI-group, then its
Sylow p-subgroup is cyclic if p ≥ 5. If p = 3, then the Sylow 3-subgroup is either cyclic
of order 9 or elementary abelian. The example in Section 2.2 shows that the rank of an
elementary abelian group is bounded by 2.
We now give the updated list of CI-groups. It is a combination of the list in [20],
together with our results here and [12, Corollary 13] (note [12, Corollary 13] contains an
error, and should list Q8 on line (1c), not on line (1b)). We need to define one more group:
Definition 5.1. Let M be a group of order relatively prime to 3, and exp(M) be the largest
order of any element of M . Set E(M, 3) = M ⋊ϕZ3, where ϕ(g) = gℓ, and ℓ is an integer
satisfying ℓ3 ≡ 1 (mod exp(M)) and gcd(ℓ(ℓ− 1), exp(M)) = 1.
Theorem 5.2. Let G, M , and K be CI-groups with respect to graphs such that M and K
are abelian, all Sylow subgroups of M are elementary abelian, and all Sylow subgroups of
K are elementary abelian of order 9 or cyclic of prime order.
(1) If G does not contain elements of order 8 or 9, then G = H1 ×H2 ×H3, where the
orders of H1, H2, and H3 are pairwise relatively prime, and
(a) H1 is an abelian group, and each Sylow p-subgroup of H1 is isomorphic to Zkp
for k < 2p+ 3 or Z4;
(b) H2 is isomorphic to one of the groups E(K, 2), E(M, 3), E(K, 4), A4, or 1;
(c) H3 is isomorphic to one of the groups D10, Q8, or 1.
1The automorphism group of the corresponding Cayley graph is 4 times bigger than G but the subgroups H
and K are non-conjugate inside it.
300 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
(2) If G contains elements of order 8, then G ∼= E(K, 8) or Z8.
(3) If G contains elements of order 9, then G is one of the groups Z9 ⋊ Z2, Z9 ⋊ Z4,
Z22 ⋊ Z9, or Zn2 × Z9, with n ≤ 5.
Remark 5.3. The rank bound of an elementary abelian group used in part (1)(a) is due to
[29].
Other than positive results already mentioned, the abelian groups known to be CI-
groups are Z2n [22], Z4n [23] with n an odd square-free integer, Zq × Z2p [18],
Zq × Z3p [31], and Zq × Z4p [19] with q and p and distinct primes, and Z32 × Zp [9]. Addi-
tional results are given in [4, Theorem 16] and [11] with technical restrictions on the orders
of the groups. A similar result with technical restrictions on M is given in [4, Theorem
22] for some E(M, 3). Also, E(Zp, 4) and E(Zp, 8) were shown to be CI-groups in [21],
and Q8 × Zp in [30]. Finally, Holt and Royle have determined all CI-groups of order at
most 47 [16]. Applying Theorem 5.2 to determine possible CI-groups, and then checking
the positive results above to see that all possible CI-groups are known to be CI-groups, we
extend the census of CI-groups up to groups of order at most 59. The isomorphism problem
for circulant digraphs was independently solved in [13] and [26] (in both cases a polyno-
mial time algorithm for solving the isomorphism problem was given). A polynomial time
algorithm for finding the automorphism group of circulant digraph was provided in [27].
Finally, we remark that the groups E(M, 3) and E(M, 8) are not DCI-groups.
Appendix A An alternative approach
In this section we give an alternative approach to the proof of Theorem 1.2. We do not
give all of the details - just the basic idea. In principle, this section is independent from the
previous sections, but for convenience we deduce the main result from our previous work.
For each g ∈ GL3(F), let g⊤ denote the transpose of the matrix g and let gι := (g−1)⊤.
It is easy to verify that ι : GL3(F) → GL3(F) is an automorphism. Let
s =
0 0 10 1 0
1 0 0

and let α be the automorphism of GL3(F) defined by
gα := s−1gιs = s−1(g−1)⊤s, (A.1)
for every g ∈ GL3(F).
We now define α̂ ∈ Sym(H) by
[a, (x, y)]α̂ = [a, (y2/2− x, ay)], (A.2)
for every [a, (x, y)] ∈ H .
Lemma A.1. Let α and α̂ be as in (A.1) and (A.2). We have
(1) Gα = G and Dα = D;
(2) K = Hα and H = Kα;
T. Dobson et al.: Generalised dihedral CI-groups 301
(3) for every h ∈ H , (Dh)α = Dhα̂;
(4) for every x ∈ F and for every t ∈ F∗, Sα̂x = S−x, Cα̂t = Ct, P α̂x = P−x.
Proof. The proof follows from straightforward computations. For every a ∈ {−1, 1} and
x ∈ F, we havea ax ax2/20 1 x
0 0 a
α =
0 0 10 1 0
1 0 0


a ax ax2/20 1 x
0 0 a
−1

⊤ 0 0 10 1 0
1 0 0

=
0 0 10 1 0
1 0 0
a −x a(−x)2/20 1 a(−x)
0 0 a
⊤ 0 0 10 1 0
1 0 0

=
0 0 10 1 0
1 0 0
 a 0 0−x 1 0
a(−x)2/2 a(−x) a
0 0 10 1 0
1 0 0

=
a a(−x) a(−x)2/20 1 −x
0 0 a
 ∈ D.
This shows Dα = D. The computations for proving G = Gα, K = Hα and H = Kα
are similar.
Let h := [a, (x, y)] ∈ H . A direct computation shows that
hα =
a 0 x0 a y
0 0 1
α =
1 −ay −ax0 a 0
0 0 a

and hence
hα(hα̂)−1 =
1 −ay −ax0 a 0
0 0 a
a 0 y2/2− x0 a ay
0 0 1
−1
=
1 −ay −ax0 a 0
0 0 a
a 0 −ay2/2 + ax0 a −y
0 0 1

=
a −y ay2/20 1 −ay
0 0 a
 ∈ D.
Therefore
(Dh)α = Dαhα = Dhα = Dhα̂
and part (3) follows. Now, part (4) follows immediately from Lemma 3.2 and part (3).
Lemma A.2. Let x ∈ F with x ̸∈ {0,±1,±2, 12} and x
6 ̸= 1, and let
T := P0 ∪ P1 ∪ Px ∪ C1 ∪ C−1,
T ′ := P0 ∪ P−1 ∪ P−x ∪ C1 ∪ C−1.
302 Ars Math. Contemp. 22 (2022) #P2.07 / 287–304
Then Cay(H,T ) and Cay(H,T ′) are isomorphic but not Cayley isomorphic. In particular,
H is not a CI-group.
Proof. We view G as a permutation group on D\G, which we may identify with H via the
Schur notation.
It follows from Lemma A.1(1) and (3) that α̂ normalizes G. Therefore, α̂ permutes
the orbitals of G. Since α̂ fixes e = [1, (0, 0)], α̂ permutes the suborbits of G and, from
Lemma A.1(4), we have Cay(H,T α̂) = Cay(H,T ′). Hence Cay(H,T )α̂ = Cay(H,T ′)
and Cay(H,T ) ∼= Cay(H,T ′).
Assume that there exists β ∈ Aut(H) with Cay(H,T )β = Cay(H,T ′). Then α̂β−1
is an automorphism of Cay(H,T ). It follows from Propositions 4.2 and 4.3 that α̂β−1 ∈
Aut(Cay(H,T )) = G. Therefore α̂ ∈ Gβ. Since G and β normalize H , so does α.
However, this contradicts Lemma A.1(2).
On the previous proof, one could prove directly that there exists no automorphism β of
H with T β = T ′; however, this requires some detailed computations, in the same spirit as
the computations in Section 4.2.
ORCID iDs
Mikhail Muzychuk https://orcid.org/0000-0002-6346-8976
Pablo Spiga https://orcid.org/0000-0002-0157-7405
Ted Dobson https://orcid.org/0000-0003-2013-4594
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