Also available at http://amc.imfm.si
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 6 (2013) 351–364
CI-groups with respect to ternary
relational structures: new examples
Edward Dobson ∗
Department of Mathematics and Statistics
Mississipi State University
PO Drawer MA, Mississipi State, MS 39762 USA
and
UP IAM, University of Primorska
Muzejski trg 2, 6000 Koper, Slovenia
Pablo Spiga
Dipartimento di Matematica Pura e Applicata
Università degli Studi di Milano-Bicocca
Via Cozzi 53, 20126 Milano, Italy
Received 22 February 2012, accepted 25 October 2012, published online 14 January 2013
Abstract
We find a sufficient condition to establish that certain abelian groups are not CI-groups
with respect to ternary relational structures, and then show that the groups Z3×Z22, Z7×Z32,
and Z5 ×Z42 satisfy this condition. Then we completely determine which groups Z32 ×Zp,
p a prime, are CI-groups with respect to color binary and ternary relational structures.
Finally, we show that Z52 is not a CI-group with respect to ternary relational structures.
Keywords: CI-group, ternary relation.
Math. Subj. Class.: 05C15, 05C10
1 Introduction
In recent years, there has been considerable interest in which groups G have the property
that any two Cayley graphs of G are isomorphic if and only if they are isomorphic by a
group automorphism of G. Such a group is a called a CI-group with respect to graphs,
and this problem is often referred to as the Cayley isomorphism problem. The interested
∗Project sponsored by the National Security Agency under Grant Number H98230-11-1-0179.
E-mail addresses: dobson@math.msstate.edu (Edward Dobson), pablo.spiga@unimib.it (Pablo Spiga)
Copyright c© 2013 DMFA Slovenije
352 Ars Math. Contemp. 6 (2013) 351–364
reader is referred to [11] for a survey on CI-groups with respect to graphs. Of course,
the Cayley isomorphism problem can and has been considered for other types of objects
(see, for example, [9, 14, 16] for work on this problem on codes and on designs). Before
proceeding we give the relevant definitions. (There are several equivalent definitions of
combinatorial object [1, 15], here we follow [13].)
Definition 1.1. A k-ary relational structure is an ordered pair X = (V,E), with V a set
and E a subset of V k. Furthermore, a color k-ary relational structure is an ordered pair
X = (V, (E1, . . . , Ec)), with V a set and E1, . . . , Ec pairwise disjoint subsets of V k. If
k = 2, 3, or 4, we simply say that X is a (color) binary, ternary, or quaternary relational
structure. A combinatorial object is a pair X = (V,E), with V a set and E a subset of⋃∞
i=1 V
i.
The following two definitions are due to Babai [1].
Definition 1.2. For a group G, define gL : G → G by gL(h) = gh, and let GL = {gL :
g ∈ G}. Then GL is a permutation group on G, called the left regular representation of G.
We will say that a (color) k-ary relational structure X is a Cayley (color) k-ary relational
structure ofG ifGL ≤ Aut(X) (note that this implies V = G). In general, a combinatorial
object X will be called a Cayley object of G if GL ≤ Aut(X).
Definition 1.3. For a class C of Cayley objects of G, we say that G is a CI-group with
respect to C if whenever X,Y ∈ C, then X and Y are isomorphic if and only if they are
isomorphic by a group automorphism of G.
It is clear that if G is a CI-group with respect to color k-ary relational structures, then
G is a CI-group with respect to k-ary relational structures.
Perhaps the most significant result in this area is a well-known theorem of Pálfy [15]
which states that a groupG of order n is a CI-group with respect to every class of combina-
torial objects if and only if n = 4 or gcd(n, ϕ(n)) = 1, where ϕ is the Euler phi function.
In fact, in proving this result, Pálfy showed that if a group G is not a CI-group with respect
to some class of combinatorial objects, then G is not a CI-group with respect to quater-
nary relational structures. As much work has been done on the case of binary relational
structures (i.e., digraphs), until recently there was a “gap” in our knowledge of the Cayley
isomorphism problem for k-ary relational structures with k = 3. As additional motivation
to study this problem, we remark that a group G that is a CI-group with respect to ternary
relational structures is necessarily a CI-group with respect to binary relational structures,
see [5, page 227].
Although Babai [1] showed in 1977 that the dihedral group of order 2p is a CI-group
with respect to ternary relational structures, no additional work was done on this problem
until the first author considered the problem in 2003 [5]. Indeed, in [5] a relatively short
list of groups is given and it is proved that every CI-group with respect to ternary relational
structures lies in this list (although not every group in this list is necessarily a CI-group
with respect to ternary relational structures). Additionally, several groups in the list were
shown to be CI-groups with respect to ternary relational structures. Recently, the second
author [17] has shown that two groups given in [5] are not CI-groups with respect to ternary
relational structures, namely Z3 n Q8 and Z3 × Q8. In this paper, we give a sufficient
condition to ensure that certain abelian groups are not CI-groups with respect to ternary
relational structures (Theorem 2.1), and then show that Z22 × Z3, Z32 × Z7, and Z42 × Z5
E. Dobson and P. Spiga: CI-groups with respect to ternary relational structures. . . 353
satisfy this condition in Corollary 2.4 (and so are not CI-groups with respect to ternary
relational structures). We then show that Z32 × Z5 is a CI-group with respect to ternary
relational structures. As the first author has shown [6] that Z32 × Zp is a CI-group with
respect to ternary relational structures provided that p ≥ 11, we then have a complete
determination of which groups Z32 × Zp, p a prime, are CI-groups with respect to ternary
relational structures.
Theorem A. The group Z32 × Zp is a CI-group with respect to color ternary relational
structures if and only if p /∈ {3, 7}.
We will show that both Z32×Z3 and Z32×Z7 are CI-groups with respect to color binary
relational structures. As it is already known that Z42 is a CI-group with respect to binary
relational structures [11], we have the following result.
Corollary A. The group Z32 × Zp is a CI-group with respect to color binary relational
structures for all primes p.
We are then left in the situation of knowing whether or not any subgroup of Z32 ×Zp is
a CI-group with respect to color binary or ternary relational structures, with the exception
of Z22×Z7 with respect to color ternary relational structures (as Z22×Z7 is a CI-group with
respect to color binary relational structures [10]). We show that Z22×Z7 is a CI-group with
respect to color ternary relational structures (which generalizes a special case of the main
result of [10]) and we prove the following.
Corollary B. The group Z22 × Zp is a CI-group with respect to color ternary relational
structures if and only if p 6= 3.
Finally, using Magma [2] and GAP [8], we show that Z52 is not a CI-group with respect
to ternary relational structures. We conclude this introductory section by recalling the
following.
Definition 1.4. For g, h in G, we denote the commutator g−1h−1gh of g and h by [g, h].
2 The main ingredient and Theorem A
We start by proving the main ingredient for our proof of Theorem A.
Theorem 2.1. Let G be an abelian group and p an odd prime. Assume that there exists
an automorphism α of G of order p fixing only the zero element of G. Then Zp ×G is not
a CI-group with respect to color ternary relational structures. Moreover, if there exists a
ternary relational structureZ onG with Aut(Z) = 〈GL, α〉, then Zp×G is not a CI-group
with respect to ternary relational structures.
Proof. Since α fixes only the zero element of G, we have |G| ≡ 1 (mod p) and so
gcd(p, |G|) = 1.
For each g ∈ G, define ĝ : Zp × G → Zp × G by ĝ(i, j) = (i, j + g). Additionally,
define τ, γ, ᾱ : Zp × G → Zp × G by τ(i, j) = (i + 1, j), γ(i, j) = (i, αi(j)), and
ᾱ(i, j) = (i, α(j)). Then (Zp ×G)L = 〈τ, ĝ : g ∈ G〉.
Clearly, 〈GL, α〉 = GL o 〈α〉 is a subgroup of Sym(G) (where GL acts on G by left
multiplication and α acts as an automorphism). Note that the stabilizer of 0 in 〈GL, α〉
is 〈α〉. As α fixes only 0, we conclude that for every g ∈ G with g 6= 0, the point-wise
354 Ars Math. Contemp. 6 (2013) 351–364
stabilizer of 0 and g in 〈GL, α〉 is 1. Therefore, by [18, Theorem 5.12], there exists a color
Cayley ternary relational structure Z of G such that Aut(Z) = 〈GL, α〉. If there exists
also a ternary relational structure with automorphism group 〈GL, α〉, then we let Z be one
such ternary relational structure.
Let
U = {((0Zp , g), (0Zp , h)) : (0G, g, h) ∈ E(Z)}, and
S = {([ĝ, γ](1, 0G), [ĝ, γ](2, 0G)) : g ∈ G} ∪ U
and define a (color) ternary relational structure X by
V (X) = Zp ×G and E(X) = {k(0Zp×G, s1, s2) : (s1, s2) ∈ S, k ∈ (Zp ×G)L}.
If Z is a color ternary relational structure, then we assign to the edge k(0Zp×G, s1, s2) the
color of the edge (0G, g, h) in Z if (s1, s2) ∈ U and (s1, s2) = ((0Zp , g), (0Zp , h)), and
otherwise we assign a fixed color distinct from those used in Z. By definition ofX we have
(Zp×G)L ≤ Aut(X) and so X is a (color) Cayley ternary relational structure of Zp×G.
We claim that ᾱ ∈ Aut(X). As ᾱ is an automorphism of Zp × G, we see that ᾱ ∈
Aut(X) if and only if ᾱ(S) = S and ᾱ preserves colors (if X is a color ternary relational
structure). By definition of Z and U , we have ᾱ(U) = U and ᾱ preserves colors (if X is a
color ternary relational structure). So, it suffices to consider the case s ∈ S − U , i.e., s =
([ĝ, γ](1, 0), [ĝ, γ](2, 0)) for some g ∈ G. Note that now we need not consider colors as all
the edges in S−U are of the same color. Then ᾱĝ(i, j) = (i, α(j) +α(g)) = α̂(g)ᾱ(i, j).
Thus ᾱĝ = α̂(g)ᾱ. Similarly, ᾱĝ−1 = α̂(g)
−1
ᾱ. Clearly ᾱ commutes with γ, and so
ᾱ[ĝ, γ] = [α̂(g), γ]ᾱ. As ᾱ fixes (1, 0) and (2, 0), we see that
ᾱ(s) = ᾱ([ĝ, γ](1, 0), [ĝ, γ](2, 0)) = (ᾱ[ĝ, γ](1, 0), ᾱ[ĝ, γ](2, 0))
= ([α̂(g), γ]ᾱ(1, 0), [α̂(g), γ]ᾱ(2, 0))
= ([α̂(g), γ](1, 0), [α̂(g), γ](2, 0)) ∈ (S − U).
Thus ᾱ(S) = S, ᾱ preserves colors (if X is a color ternary relational structure) and ᾱ ∈
Aut(X).
We claim that γ−1(Zp × G)Lγ is a subgroup of Aut(X). We set τ ′ = γ−1τγ and
g′ = γ−1ĝγ, for g ∈ G. Note that τ ′ = τᾱ−1. As ᾱ ∈ Aut(X), we have τ ′ ∈ Aut(X).
Therefore it remains to prove that 〈g′ : g ∈ G〉 is a subgroup of Aut(X). Let e ∈ E(X)
and g ∈ G. Then e = k((0, 0), s), where s ∈ S and k = τa l̂, for some a ∈ Zp, l ∈ G.
We have to prove that g′(e) ∈ E(X) and has the same color as e (if X is a color ternary
relational structure).
Assume that s ∈ U . As g′(i, j) = (i, j + α−i(g)), by definition of U , we have
g′[k((0, 0), s)] ∈ E(X) and has the same color of e (ifX is a color ternary relational struc-
ture). So, it remains to consider the case s ∈ S − U , i.e., s = ([x̂, γ](1, 0), [x̂, γ](2, 0)) for
some x ∈ G. As before, we need not concern ourselves with colors because all the edges
in S − U are of the same color.
Set m = kα̂−a(g). Since ᾱĝ = α̂(g)ᾱ and ᾱ, γ commute, we get ᾱg′ = (α(g))′ᾱ.
Also observe that as G is abelian, g′ commutes with ĥ for every g, h ∈ G. Hence
E. Dobson and P. Spiga: CI-groups with respect to ternary relational structures. . . 355
g′k = γ−1ĝγτa l̂ = γ−1ĝτaγᾱa l̂ = γ−1τaĝᾱaγl̂
= τaγ−1ᾱ−aĝᾱaγl̂ = τa(α−a(g))′ l̂ = τa l̂(α−a(g))′
= kα̂−a(g)α̂−a(g)
−1
γ−1α̂−a(g)γ = m[α̂−a(g), γ]
and
g′[k((0, 0), s)] = g′k((0, 0), [x̂, γ](1, 0), [x̂, γ](2, 0))
= m[α̂−a(g), γ]((0, 0), [x̂, γ](1, 0), [x̂, γ](2, 0))
= m((0, 0), [α̂−a(g), γ][x̂, γ](1, 0), [α̂−a(g), γ][x̂, γ](2, 0))
= m((0, 0), [ ̂α−a(g)x, γ](1, 0), [ ̂α−a(g)x, γ](2, 0)) ∈ E(X).
This proves that g′ ∈ Aut(X). Since g is an arbitrary element of G, we have γ−1GLγ ⊆
Aut(X). As claimed, γ−1(Zp × G)Lγ is a regular subgroup of Aut(X) conjugate in
Sym(Zp ×G) to (Zp ×G)L.
We now see that Y = γ(X) is a Cayley (color) ternary relational structure of Zp ×
G as Aut(Y ) = γAut(X)γ−1. We will next show that Y 6= X . Assume by way of
contradiction that Y = X . As γ(0, g) = (0, g) for every g ∈ G, the permutation γ
must map edges of U to themselves, so that γ(S − U) = S − U . We will show that
γ(S − U) 6= S − U . Note that we need not concern ourselves with colors as all the edges
derived from S − U via translations of (Zp ×G)L have the same color. Observing that
([ĝ, γ](1, 0), [ĝ, γ](2, 0)) = (ĝ−1γ−1ĝγ(1, 0), ĝ−1γ−1ĝγ(2, 0))
= (ĝ−1γ−1ĝ(1, 0), ĝ−1γ−1ĝ(2, 0))
= (ĝ−1γ−1(1, g), ĝ−1γ−1(2, g))
= (ĝ−1(1, α−1(g), ĝ−1(2, α−2(g))
= ((1, α−1(g)− g), (2, α−2(g)− g)),
we see that γ(S−U) = {((1, g−α(g)), (2, g−α2(g))) : g ∈ G}. Moreover, as S−U =
{(1, α−1(g)− g), (2, α−2(g)− g) : g ∈ G}, we conclude that for each g ∈ G, there exists
hg ∈ G such that
g − α(g) = α−1(hg)− hg and g − α2(g) = α−2(hg)− hg.
Setting ι : G→ G to be the identity permutation, we may rewrite the above equations as
(ι− α)(g) = (α−1 − ι)(hg) and (ι− α2)(g) = (α−2 − ι)(hg).
Computing in the endomorphism ring of the abelian group G, we see that (α−2 − ι) =
(α−1 + ι)(α−1 − ι). Applying the endomorphism (α−1 + ι) to the first equation above,
we then have
(α−1 + ι)(ι− α)(g) = (α−1 + ι)(α−1 − ι)(hg) = (α−2 − ι)(hg) = (ι− α2)(g).
356 Ars Math. Contemp. 6 (2013) 351–364
Hence (α−1 + ι)(ι− α) = ι− α2, and so
0 = (α−1 + ι)(ι− α)− (ι− α2) = ((α−1 + ι)− (ι+ α))(ι− α) = (α−1 − α)(ι− α),
(here 0 is the endomorphism of G that maps each element of G to 0). As α fixes only 0,
the endomorphism ι − α is invertible, and so we see that α−1 − α = 0, and α = α−1.
However, this implies that p = |α| = 2, a contradiction. Thus γ(S − U) 6= S − U and so
Y 6= X .
We set T = γ(S), so that ((0, 0), t) ∈ E(Y ) for every t ∈ T , where if X is a color
ternary relational structure we assume that γ preserves colors. Now suppose that there
exists β ∈ Aut(Zp×G) such that β(X) = Y . Since gcd(p, |G|) = 1, we obtain that Zp×
1G and 1Zp×G are characteristic subgroups of Zp×G. Therefore β(i, j) = (β1(i), β2(j)),
where β1 ∈ Aut(Zp) and β2 ∈ Aut(G).
As β fixes (0, 0), we must have β(S) = T . Observe that every element of S and of T
is of the form ((0, g), (0, h)) or ((1, g), (2, h)), for some g, h ∈ G. In particular, we must
have β1(1) = 1 and hence β1 = 1. As ᾱ ∈ Aut(X) and X 6= Y , we have β2 6∈ 〈α〉. Now
observe that β(U) = U . Thus β2 ∈ Aut(Z) = 〈GL, α〉. We conclude that β2 ∈ 〈α〉, a
contradiction. Thus X,Y are not isomorphic by a group automorphism of Zp×G, and the
result follows.
The following two lemmas, which in our opinion are of independent interest, will be
used (together with Theorem 2.1) in the proof of Corollary 2.4.
Lemma 2.2. Let G be a transitive permutation group on Ω. If x ∈ Ω and StabG(x) in its
action on Ω − {x} is the automorphism group of a k-ary relational structure with vertex
set Ω− {x}, then G is the automorphism group of a (k + 1)-ary relational structure.
Proof. Let Y be a k-ary relational structure with vertex set Ω − {x} and automorphism
group StabG(x) in its action on Ω − {x}. Let W = {(x, v1, . . . , vk) : (v1, . . . , vk) ∈
E(Y )}, and define a (k + 1)-ary relational structure X by V (X) = Ω and E(X) =
{g(w) : w ∈ W and g ∈ G}. We claim that Aut(X) = G. First, observe that StabG(x)
maps W to W . Also, if e ∈ E(X) and e = (x, v1, . . . , vk) for some v1, . . . , vk ∈ Ω,
then there exists (x, u1, . . . , uk) ∈ W and g ∈ G with g(x, u1, . . . , uk) = (x, v1, . . . , vk).
We conclude that g(x) = x and g(u1, . . . , uk) = (v1, . . . , vk). Hence g ∈ StabG(x) and
(v1, . . . , vk) ∈ E(Y ). Then W is the set of all edges of X with first coordinate x.
By construction, G ≤ Aut(X). For the reverse inclusion, let h ∈ Aut(X). As G is
transitive, there exists g ∈ G such that g−1h ∈ StabAut(X)(x). Note that as g ∈ G, the
element g−1h ∈ G if and only if h ∈ G. We may thus assume without loss of generality
that h(x) = x. Then h stabilizes set-wise the set of all edges of X with first coordinate x,
and so h(W ) = W and h induces an automorphism of Y . As Aut(Y ) = StabG(x) ≤ G,
the result follows.
Lemma 2.3. Let m ≥ 2 be an integer and ρ ∈ Sym(Zms) be a semiregular element
of order m with s orbits. Then there exists a digraph Γ with vertex set Zms and with
Aut(Γ) = 〈ρ〉.
Proof. For each i ∈ Zs, set
ρi = (0, 1, . . . ,m−1) · · · (im, im+1, . . . , im+m−1) and Vi = {im+j : j ∈ Zm}.
E. Dobson and P. Spiga: CI-groups with respect to ternary relational structures. . . 357
We inductively define a sequence of graphs Γ0, . . . ,Γs−1 = Γ such that the subgraph of Γ
induced by Z(i+1)m is Γi, the indegree in Γ of a vertex in Vi is i+ 1, and Aut(Γi) = 〈ρi〉,
for each i ∈ Zs.
We set Γ0 to be the directed cycle of length m with edges {(j, j + 1) : j ∈ Zm} and
with automorphism group 〈ρ0〉. Inductively assume that Γs−2, with the above properties,
has been constructed. We construct Γs−1 as follows. First, the subgraph of Γs−1 induced
by Z(s−1)m is Γs−2. Then we place the directedm cycle {((s−1)m+j, (s−1)m+j+1) :
j ∈ Zm} whose automorphism group is 〈((s−1)m, (s−1)m+1, . . . , (s−1)m+m−1)〉
on the vertices in Vs−1. Additionally, we declare the vertex (s− 1)m to be outadjacent to
(s− 2)m and to every vertex that (s− 2)m is outadjacent to that is not contained in Vs−2.
Finally, we add to Γs−1 every image of one of these edges under an element of 〈ρs−1〉.
By construction, ρs−1 is an automorphism of Γs−1 and the subgraph of Γs−1 induced
by Z(s−1)m is Γs−2. Then each vertex in Γs−1 ∩ Vi has indegree i+ 1 for 0 ≤ i ≤ s− 2,
while it is easy to see that each vertex of Vs−1 has indegree s. Finally, if δ ∈ Aut(Γs−1),
then δ maps vertices of indegree i + 1 to vertices of indegree i + 1, and so δ fixes set-
wise Vi, for every i ∈ Zs. Additionally, the action induced by 〈δ〉 on Vs−1 is necessarily
〈((s−1)m, (s−1)m+1, . . . , (s−1)m+m−1)〉 as this is the automorphism group of the
subgraph of Γs−1 induced by Vs−1. Moreover, arguing by induction, we may assume that
the action induced by δ on V (Γs−1)−Vs−1 is given by an element of 〈ρs−2〉. If δ 6∈ 〈ρs−1〉,
then Aut(Γs−1) has order at least m2, and there is some element of Aut(Γs−1) that is the
identity on V (Γs−2) but not on Vs−1 and vice versa. This however is not possible as each
vertex of Vs−2 is inadjacent to exactly one vertex of Vs−1. Then Aut(Γs−1) = 〈ρs−1〉 and
the result follows.
Corollary 2.4. None of the groups Z3 × Z22, Z7 × Z32, or Z5 × Z42 are CI-groups with
respect to ternary relational structures.
Proof. Observe that Z22 has an automorphism α3 of order 3 that fixes 0 and acts regularly
on the remaining 3 elements, and similarly, Z32 has an automorphism α7 of order 7 that
fixes 0 and acts regularly on the remaining 7 elements. As a regular cyclic group is the
automorphism group of a directed cycle, we see that 〈(Z3×Z22)L, α3〉 and 〈(Z7×Z32)L, α7〉
are the automorphism groups of ternary relational structures by Lemma 2.2. The result then
follows by Theorem 2.1.
Now Z42 has an automorphism α5 of order 5 that fixes 0 and acts semiregularly on the
remaining 15 points. Then 〈α5〉 in its action on Z42 − {0} is the automorphism group of
a binary relational structure by Lemma 2.3. By Lemma 2.2, there exists a ternary rela-
tional structure with automorphism group 〈(Z5 × Z42)L, α5〉. The result then follows by
Theorem 2.1.
Before proceeding, we will need terms and notation concerning complete block sys-
tems.
Let G ≤ Sym(n) be a transitive permutation group (acting on Zn, say). A subset
B ⊆ Zn is a block for G if g(B) = B or g(B) ∩ B = ∅ for every g ∈ G. Clearly Zn and
its singleton subsets are always blocks for G, and are called trivial blocks. If B is a block,
then g(B) is a block for every g ∈ G, and the set B = {g(B) : g ∈ G} is called a complete
block system for G, and we say that G admits B. A complete block system is nontrivial if
its blocks are nontrivial. Observe that a complete block system is a partition of Zn, and any
two blocks have the same size. If G admits B as a complete block system, then each g ∈ G
358 Ars Math. Contemp. 6 (2013) 351–364
induces a permutation of B, which we denote by g/B. We set G/B = {g/B : g ∈ G}.
The kernel of the action of G on B, denoted by fixG(B), is then the subgroup of G which
fixes each block of B set-wise. That is, fixG(B) = {g ∈ G : g(B) = B for all B ∈ B}.
For fixed B ∈ B, we denote the set-wise stabilizer of B in G by StabG(B). That is
StabG(B) = {g ∈ G : g(B) = B}. Note that fixG(B) = ∩B∈BStabG(B). Finally, for
g ∈ StabG(B), we denote by g|B the action induced by g on B ∈ B.
Note that Corollary 2.4, together with the fact that Z32 × Zp, p ≥ 11, is a CI-group
with respect to color ternary relational structures [6], settles the question of which groups
Z32 × Zp are CI-groups with respect to color ternary relational structures except for p = 5.
Our next goal is to show that Z32 ×Z5 is a CI-group with respect to color ternary relational
structures. From a computational point of view, the number of points is too large to enable
a computer to determine the answer without some additional information. Lemma 6.1
in [6] is the only result that uses the hypothesis p ≥ 11. For convenience, we report [6,
Lemma 6.1].
Lemma 2.5. Let p ≥ 11 be a prime and write H = Z32 × Zp. For every φ ∈ Sym(H),
there exists δ ∈ 〈HL, φ−1HLφ〉 such that 〈HL, δ−1φ−1HLφδ〉 admits a complete block
system consisting of 8 blocks of size p.
In particular, to prove that Z32×Z5 is a CI-group with respect to color ternary relational
structures, it suffices to prove that Lemma 2.5 holds true also for the prime p = 5. We begin
with some intermediate results which accidentally will also help us to prove that Z32 × Z7
is a CI-group with respect to color binary relational structures. (Here we denote by Alt(X)
the alternating group on the set X and by Alt(n) the alternating group on {1, . . . , n}.)
Lemma 2.6. Let p be an arbitrary divisor of n with p 6= 1 and let P1 and P2 be partitions
of Zn where each block in P1 and P2 has size p. Then there exists φ ∈ Alt(Zn) such that
φ(P1) = P2.
Proof. Let P1 = {∆1, . . . ,∆n/p} and P2 = {Ω1, . . . ,Ωn/p}. As Alt(n) is (n − 2)-
transitive, there exists φ ∈ Alt(n) such that φ(∆i) = Ωi, for i ∈ {1, . . . , n/p − 1}. As
both P1 and P2 are partitions, we see that φ(∆n/p) = Ωn/p as well.
Lemma 2.7. Let p be a prime, let G = Z32 × Zp and let δ ∈ Sym(G). Suppose that
〈GL, δ−1GLδ〉 admits a complete block system C with p blocks of size 8 such that Alt(C) ≤
Stab〈GL,δ−1GLδ〉(C)|C , where C ∈ C. Then there exists γ ∈ 〈GL, δ−1GLδ〉 such that
〈GL, γ−1δ−1GLδγ〉 admits a complete block system B with 4p blocks of size 2.
Proof. Write H = 〈GL, δ−1GLδ〉, N = fixH(C) and M = [N,N ]. Clearly both GL and
δ−1GLδ are regular, and so both fixGL(C) and fixδ−1GLδ(C) are semiregular of order 8.
Moreover, as fixGL(C)|C and fixδ−1GLδ(C)|C have exponent 2, we see that they are both
consist of even permutations and hence they are contained in Alt(C), for each C ∈ C.
From the previous paragraph, as Alt(8) is simple and 1 6= N |C / Stab〈G,δ−1Gδ〉(C)|C ,
we have Alt(C) = M |C , for every C ∈ C. In particular, M is isomorphic to a subgroup of
Alt(8)p.
Denote by M(C) the pointwise stabilizer of C ∈ C. Define an equivalence relation ≡
on C by C ≡ C ′ if and only if M(C) = M(C′). Clearly, ≡ is an H-invariant equivalence
relation because M /H . As |C| = p, we see that ≡ is either the identity or the universal
relation. From this, we infer that either M ∼= Alt(8) (when ≡ is the identity relation) or
M ∼= Alt(8)p (when ≡ is the universal relation). Observe further that, when M ∼= Alt(8),
E. Dobson and P. Spiga: CI-groups with respect to ternary relational structures. . . 359
since Alt(8) has only one permutation representation of degree 8 [3, Theorem 5.3], the
groupM induces equivalent actions on C and on C ′, for every C and C ′ in C. In particular,
in both cases, given a subgroup I of GL and J of δ−1GLδ both of order 2, there exists
γ ∈M with I = γ−1Jγ.
Write K = 〈GL, γ−1δ−1GLδγ〉. Clearly, I is centralized by GL and by γ−1δ−1GLδγ
because I ≤ GL and I ≤ γ−1δ−1GLδγ. So I is centralized by K. As I /K, the orbits of
I form a complete block system for K with 4p blocks of size 2.
The proof of the following result is similar to the proof of [6, Lemma 6.1], and gener-
alizes it.
Lemma 2.8. Let H be an abelian group of order `p, where ` < p and p is prime. Let
φ ∈ Sym(H). Then there exists δ ∈ 〈HL, φ−1HLφ〉 such that 〈HL, δ−1φ−1HLφδ〉 admits
a complete block system with blocks of size p.
Proof. Let ρ ∈ H be of order p. Then HL admits a complete block system B of `
blocks of size p formed by the orbits of 〈ρ〉. Note that as ` < p, a Sylow p-subgroup
of Sym(H) has order p`. In particular, 〈ρ|B : B ∈ B〉 is a Sylow p-subgroup of Sym(H)
isomorphic to Z`p, an elementary abelian p-group of order p`. Let P and P1 be Sylow
p-subgroups of 〈HL, φ−1HLφ〉 containing ρ and φ−1ρφ, respectively. Then there exists
δ ∈ 〈HL, φ−1HLφ〉 such that δ−1P1δ = P . Now, every element of HL normalizes 〈ρ〉,
and so normalizes 〈ρ|B : B ∈ B〉. This then implies that HL normalizes P because
P = 〈ρ|B : B ∈ B〉 ∩ 〈HL, φ−1HLφ〉.
Let B′ be the complete block system of δ−1φ−1HLφδ formed by the orbits of the
cyclic group δ−1φ−1〈ρ〉φδ. Arguing as above, we see that δ−1φ−1HLφδ normalizes
M = 〈(δ−1φ−1ρφδ)|B′ : B′ ∈ B′〉 ∩ 〈HL, δ−1φ−1HLφδ〉. However, M is the Sylow
p-subgroup of 〈HL, δ−1φ−1HLφδ〉 containing δ−1φ−1〈ρ〉δφ, which is P . Thus we have
P / 〈HL, δ−1φ−1HLφδ〉, and the orbits of P form the required complete block system.
Lemma 2.9. Let p ≥ 5, H = Z32 × Zp, and φ ∈ Sym(H). Then either there exists
δ ∈ 〈HL, φ−1HLφ〉 such that 〈HL, δ−1φ−1HLφδ〉 admits a complete block system with
blocks of size p or 〈HL, φ−1HLφ〉 admits a complete block system B with blocks of size 8
and fix〈HL,φ−1HLφ〉(B)|B is isomorphic to a primitive subgroup of AGL(3, 2), for B ∈ B.
Proof. Set K = 〈HL, φ−1HLφ〉. As H has a cyclic Sylow p-subgroup, we have by [4,
Theorem 3.5A] that K is doubly-transitive or imprimitive. If K is doubly-transitive, then
by [12, Theorem 1.1] we have Alt(H) ≤ K. Now Lemma 2.6 reduces this case to the
imprimitive case. Thus we may assume thatK is imprimitive with a complete block system
C.
Suppose that the blocks of C have size `p, where ` = 2 or 4. Notice that p > `. As H is
abelian, fixHL(C) is a semiregular group of order `p and fixφ−1HLφ(C) is also a semiregular
group of order `p. Then, for C ∈ C, both fixHL(C)|C and fixφ−1HLφ(C)|C are regular
groups of order `p. Let C ∈ C. By Lemma 2.8, there exists δ ∈ 〈fixHL(C),fixφ−1HLφ(C)〉
such that 〈fixHL(C),fixδ−1φ−1HLφδ(C)〉|C admits a complete block system BC consisting
of blocks of size p. Let C ′ ∈ C with C ′ 6= C. Arguing as above, there exists δ′ ∈
〈fixHL(C),fixδ−1φ−1HLφδ(C)〉 such that 〈fixHL(C),fixδ′−1δ−1φ−1HLφδδ′(C)〉|C′ admits a
complete block system BC′ consisting of blocks of size p. Note that the restriction δ′|C is
in 〈fixHL(C),fixδ−1φ−1HLφδ(C)〉|C and so 〈fixHL(C),fixδ′−1δ−1φ−1HLφδδ′(C)〉|C admits
BC as a complete block system. Repeating this argument for every block in C, we find
360 Ars Math. Contemp. 6 (2013) 351–364
that there exists δ ∈ 〈fixHL(C),fixφ−1HLφ(C)〉 such that 〈fixHL(C),fixδ−1φ−1HLφδ(C)〉|C
admits a complete block system BC consisting of blocks of size p. Let B = ∪CBC . We
claim that B is a complete block system for 〈HL, δ−1φ−1HLφδ〉, which will complete the
argument in this case.
Let ρ ∈ HL be of order p. By construction, ρ ∈ fixHL(B). AsH is abelian, fixHL(C)|C
is abelian, for every C ∈ C. Then BC is formed by the orbits of some subgroup of
fixHL(C)|C of order p, and as 〈ρ〉|C is the unique subgroup of fixHL(C)|C of order p, we
obtain that BC is formed by the orbits of 〈ρ〉|C . Then B is formed by the orbits of 〈ρ〉 /HL
and B is a complete block system for HL. An analogous argument for δ−1φ−1〈ρ〉φδ gives
that B is a complete block system for δ−1φ−1HLφδ. Then B is a complete block system
for 〈HL, δ−1φ−1HLφδ〉 with blocks of size p, as required.
Suppose that the blocks of C have size 8. NowHL/C and φ−1HLφ/C are cyclic of order
p, and as Zp is a CI-group [1, Theorem 2.3], replacing φ−1HLφ by a suitable conjugate,
we may assume that 〈HL, φ−1HLφ〉/C = HL/C. Then K/C is regular and StabK(C) =
fixK(C), for every C ∈ C.
Suppose that StabK(C)|C is imprimitive, for C ∈ C. By [4, Exercise 1.5.10], the group
K admits a complete block system D with blocks of size 2 or 4. Then K/D has degree 2p
or 4p and, by Lemma 2.8, there exists δ ∈ K such that 〈HL, δ−1φ−1HLφδ〉/D admits a
complete block system B′ with blocks of size p. In particular, B′ induces a complete block
system B′′ for 〈HL, δ−1φ−1HLφδ〉 with blocks of size 2p or 4p, and we conclude by the
case previously considered applied with C = B′′. Suppose that StabK(C)|C is primitive,
forC ∈ C. If StabK(C)|C ≥ Alt(C), then the result follows by Lemma 2.7, and so we may
assume this is not the case. By [12, Theorem 1.1], we see that StabK(C)|C ≤ AGL(3, 2).
The result now follows with B = C.
Corollary 2.10. LetH = Z32×Z5 and φ ∈ Sym(H). Then there exists δ ∈ 〈HL, φ−1HLφ〉
such that 〈HL, δ−1φ−1HLφδ〉 admits a complete block system with blocks of size 5.
Proof. Set K = 〈HL, φ−1HLφ〉. By Lemma 2.9, we may assume that K admits a com-
plete block system B with blocks of size 8 and with StabK(B)|B ≤ AGL(3, 2), forB ∈ B.
As |AGL(3, 2)| = 8 · 7 · 6 · 4, we see that a Sylow 5-subgroup of K has order 5. Let 〈ρ〉
be the subgroup of HL of order 5. So 〈ρ〉 is a Sylow 5-subgroup of K. Then φ−1〈ρ〉φ
is also a Sylow 5-subgroup of K, and by a Sylow theorem there exists δ ∈ K such
that δ−1φ−1〈ρ〉φδ = 〈ρ〉. We then see that 〈HL, δ−1φ−1HLφδ〉 has a unique Sylow 5-
subgroup, whose orbits form the required complete block system B.
We are finally ready to prove Theorem A.
Proof of Theorem A. If p is odd, then the paragraph following the proof of Corollary 2.4
shows that it suffices to prove that Lemma 2.5 holds for the prime p = 5. This is done in
Corollary 2.10. If p = 2, then the result can be verified using GAP or Magma.
3 Proof of Corollaries A and B
Before proceeding to our next result we will need the following definitions.
Definition 3.1. LetG be a permutation group on Ω and k ≥ 1. A permutation σ ∈ Sym(Ω)
lies in the k-closure G(k) of G if for every k-tuple t ∈ Ωk there exists gt ∈ G (depending
on t) such that σ(t) = gt(t). We say that G is k-closed if the permutations lying in the
E. Dobson and P. Spiga: CI-groups with respect to ternary relational structures. . . 361
k-closure of G are the elements of G, that is, G(k) = G. The group G is k-closed if and
only if there exists a color k-ary relational structure X on Ω with G = Aut(X), see [18].
Definition 3.2. For color digraphs Γ1 and Γ2, we define the wreath product of Γ1 and
Γ2, denoted Γ1 o Γ2, to be the color digraph with vertex set V (Γ1) × V (Γ2) and edge
set E1 ∪ E2, where E1 = {((x1, y1), (x1, y2)) : x1 ∈ V (Γ1), (y1, y2) ∈ E(Γ2)} and
E2 = {((x1, y1), (x2, y2)) : (x1, x2) ∈ E(Γ1), y1, y2 ∈ V (Γ2)}.
The edge ((x1, y1), (x1, y2)) ∈ E1 is colored with the same color as (y1, y2) in Γ2 and
the edge ((x1, y1), (x2, y2)) ∈ E2 is colored with the same color as (x1, x2) in Γ1.
Definition 3.3. Let G ≤ Sym(X) and let H ≤ Sym(Y ). We define the wreath product
of G and H , denoted by G o H , to be the semidirect product G n HX , where HX is the
direct product of |X| copies of H (labeled by the elements of X) and where G acts on
HX as a group of automorphisms by permuting the coordinates according to its action on
X . The group G o H has a natural faithful action on X × Y , where for (x, y) ∈ X × Y
the element g ∈ G acts via (x, y) 7→ (g(x), y) and the element (hz)z∈X ∈ HX acts
via (x, y) 7→ (x, hx(y)). We refer the reader to [4, page 46] for more details on this
construction.
The following very useful result (see [1, Lemma 3.1]) characterizes CI-groups with
respect to a class of combinatorial objects.
Lemma 3.4. LetH be a group and letK be a class of combinatorial objects. The following
are equivalent.
1. H is a CI-group with respect to K,
2. whenever X is a Cayley object of H in K and φ ∈ Sym(H) such that φ−1HLφ ≤
Aut(X), then HL and φ−1HLφ are conjugate in Aut(X).
Proof of Corollary A. From Theorem A, it suffices to show that Z32 × Z3 and Z32 × Z7 are
CI-groups with respect to color binary relational structures. As the transitive permutation
groups of degree 24 are readily available in GAP or Magma, it can be shown using a
computer that Z32 × Z3 is a CI-group with respect to color binary relational structures. It
remains to consider H = Z32 × Z7.
Fix φ ∈ Sym(H) and set K = 〈HL, φ−1HLφ〉. Assume that there exists δ ∈ K such
that 〈HL, δ−1φ−1HLφδ〉 admits a complete block system with blocks of size 7. Now, it
follows by [6] (see the two paragraphs following the proof of Corollary 2.4) that HL and
δ−1φ−1HLφδ are conjugate in 〈HL, δ−1φ−1HLφδ〉(3). Since 〈HL, δ−1φ−1HLφδ〉(3) ≤
〈HL, δ−1φ−1HLφδ〉(2), the corollary follows from Lemma 3.4 (and from Definition 3.1).
Assume that there exists no δ ∈ K such that 〈HL, δ−1φ−1HLφδ〉 admits a complete
block system with blocks of size 7. By Lemma 2.9, the group K admits a complete block
system C with blocks of size 8 and fixK(C)|C is isomorphic to a primitive subgroup of
AGL(3, 2), for C ∈ C. Suppose that 7 and |fixK(C)| are relatively prime. So, a Sylow 7-
subgroup of K has order 7. We are now in the position to apply the argument in the proof
of Corollary 2.10. Let 〈ρ〉 be the subgroup of HL of order 7. Then φ−1〈ρ〉φ is a Sylow 7-
subgroup of K, and by a Sylow theorem there exists δ ∈ K such that δ−1φ−1〈ρ〉φδ = 〈ρ〉.
We then see that 〈HL, δ−1φ−1HLφδ〉 has a unique Sylow 7-subgroup, whose orbits form
a complete block system with blocks of size 7, contradicting our hypothesis on K. We thus
assume that 7 divides |fixK(C)| and so fixK(C) acts doubly-transitively on C, for C ∈ C.
362 Ars Math. Contemp. 6 (2013) 351–364
Fix C ∈ C and let L be the point-wise stabilizer of C in fixK(C). Assume that
L 6= 1. Now, we compute K(2) and we deduce that HL and φ−1HLφ are conjugate in
K(2), from which the corollary will follow from Lemma 3.4. As L/fixK(C), we have
L|C′ / fixK(C)|C′ , for every C ′ ∈ C. As a nontrivial normal subgroup of a primitive
group is transitive [19, Theorem 8.8], either L|C′ is transitive or L|C′ = 1. Let Γ be
a Cayley color digraph on H with K(2) = Aut(Γ). Let C = {Ci : i ∈ Z7} where
Ci = {(x1, x2, x3, i) : x1, x2, x3 ∈ Z2}, and assume without loss of generality that
C = C0. Suppose that there is an edge of color κ from some vertex of Ci to some vertex
of Cj , where i 6= j. Then there is an edge of color κ from some vertex of C0 to some
vertex of Cj−i. Additionally, j − i generates Z7, so there is a smallest integer s such
that L|Cs(j−i) = 1 while L|C(s+1)(j−i) is transitive. As there is an edge of color κ from
some vertex of Cs(j−i) to some vertex of C(s+1)(j−i), we conclude that there is an edge
of color κ from every vertex of Cs(j−i) to every vertex of C(s+1)(j−i). This implies that
there is an edge of color κ from every vertex of Ci to every vertex of Cj , and then Γ is
the wreath product of a Cayley color digraph Γ1 on Z7 and a Cayley color digraph Γ2 on
Z32. Since fixK(C) is doubly-transitive on C, we have Aut(Γ2) ∼= Sym(8). Therefore
K(2) = Aut(Γ1) o Aut(Γ2) ∼= Aut(Γ1) o Sym(8). By [7, Corollary 6.8] and Lemma 3.4
HL and φ−1HLφ are conjugate inK(2). We henceforth assume that L = 1, that is, fixK(C)
acts faithfully on C, for each C ∈ C.
Define an equivalence relation on H by h ≡ k if and only if it holds StabfixK(C)(h) =
StabfixK(C)(k). The equivalence classes of ≡ form a complete block system D for K. As
fixK(C)|C is primitive and not regular, each equivalence class of ≡ contains at most one
element from each block of C. We conclude that D either consists of 8 blocks of size 7 or
each block is a singleton. Since we are assuming that K has no block system with blocks
of size 7, we see that each block of D is a singleton.
Fix C and D in C with C 6= D and h ∈ C. Now, StabfixK(C)(h) is isomorphic to a
subgroup of GL(3, 2) and acts with no fixed points on D. From [4, Appendix B]), we see
that AGL(3, 2) is the only doubly-transitive permutation group of degree 8 whose point
stabilizer admits a fixed-point-free action of degree 8. Therefore fixK(C) ∼= AGL(3, 2).
Additionally, StabfixK(C)(h)|D is transitive on D.
Suppose that Γ is a color digraph with K(2) = Aut(Γ) and suppose that there is an
edge of color κ from h to ` ∈ E, with E ∈ C and E 6= D. Then StabfixK(C)(h)|E is
transitive, and so there is an edge of color κ from h to every vertex of E. As fixK(C) is
transitive on both C and E, we see that there is an edge of color κ from every vertex of C
to every vertex of D. We conclude that Γ is a wreath product of two color digraphs Γ1 and
Γ2, where Γ1 is a Cayley color digraph on Z7 and Γ2 is either complete or the complement
of a complete graph, and K(2) = Aut(Γ1) o Sym(8). The result then follows by the same
arguments as above.
Proof of Corollary B. From Corollary 2.4 and Theorem A, it suffices to show that Z22×Z7
is a CI-group with respect to color ternary relational structures. As the transitive permuta-
tion groups of degree 28 are readily available in GAP or Magma, it can be shown using a
computer that Z22×Z7 is a CI-group with respect to color ternary relational structures. (We
note that a detailed analysis similar to the proof of Corollary A for the group Z32 × Z7 also
gives a proof of this theorem.)
E. Dobson and P. Spiga: CI-groups with respect to ternary relational structures. . . 363
4 Concluding remarks
In the rest of this paper, we discuss the relevance of Theorem A to the study of CI-groups
with respect to ternary relational structures. Using the software packages [2] and [8], we
have determined that Z52 is not a CI-group with respect to ternary relational structures. Here
we report an example witnessing this fact: the group G has order 2048, V and W are two
nonconjugate elementary abelian regular subgroups of G, and X = ({1, . . . , 32}, E) is a
ternary relational structure with G = Aut(X). The group V is generated by
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32),
(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13, 15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32),
(1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32),
(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32),
(1,17)(2,18)(3,19)(4,20)(5,21)(6,22)(7,23)(8,24)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32),
the group W is generated by
(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32),
(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13, 15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31),
(1,5)(2,6)(3,7)(4,8)(9,14)(10,13)(11,16)(12,15)(17,22)(18,21)(19,24)(20, 23)(25,29)(26,30)(27,31)(28,32),
(1,9)(2,10)(3,11)(4,12)(5,14)(6,13)(7,16)(8,15)(17,27)(18,28)(19,25)(20,26)(21,32)(22,31)(23,30)(24,29),
(1,17)(2,18)(3,20)(4,19)(5,22)(6,21)(7,23)(8,24)(9,27)(10,28)(11,26)(12,25)(13,32)(14,31)(15,29)(16,30),
the group G is generated by
V,W, (25,26)(27,28)(29,30)(31,32),(1,11)(2,12)(3,9)(4,10)(5,13)(6, 14)(7,15)(8,16)(17,19)(18,20)(25,27)(26,28),
the set E is defined by
{g((1, 3, 9)), g((1, 5, 25)) : g ∈ G}.
Definition 4.1. For a cyclic group M = 〈g〉 of order m and a cyclic group 〈z〉 of order 2d,
d ≥ 1, we denote by D(m, 2d) the group 〈z〉nM with gz = g−1.
Combining Theorem A with [5, Theorem 9], [5, Lemma 6], the construction given
in [17] and the previous paragraph, we have the following result which lists every group
that can be a CI-group with respect to ternary relational structures (although not every
group on the list needs to be a CI-group with respect to ternary relational structures).
Theorem 4.2. If G is a CI-group with respect to ternary relational structures, then all Sy-
low subgroups of G are of prime order or isomorphic to Z4, Zd2, 1 ≤ d ≤ 4, or Q8. More-
over, G = U ×V , where gcd(|U |, |V |) = 1, U is cyclic of order n, with gcd(n, ϕ(n)) = 1,
and V is one of the following:
1. Zd2, 1 ≤ d ≤ 4, D(m, 2), or D(m, 4), where m is odd and gcd(nm,ϕ(nm)) = 1,
2. Z4, Q8.
Furthermore,
364 Ars Math. Contemp. 6 (2013) 351–364
(a) if V = Z4, Q8, or D(m, 4) and p | n is prime, then 4 6 | (p− 1),
(b) if V = Zd2, d ≥ 2, or Q8, then 3 6 | n,
(c) if V = Zd2, d ≥ 3, then 7 6 | n,
(d) if V = Z42, then 5 6 | n.
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