ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 22 (2022) #P3.08 / 489–504
https://doi.org/10.26493/1855-3974.2147.da8
(Also available at http://amc-journal.eu)
Two families of pseudo metacirculants*
Li Cui , Jin-Xin Zhou †
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P. R. China
Received 12 October 2019, accepted 31 December 2021, published online 9 June 2022
Abstract
A split weak metacirculant which is not metacirculant is simply called a pseudo metacir-
culant. In this paper, two infinite families of pseudo metacirculants are constructed.
Keywords: Metacirculant, metacyclic, split weak metacirculant.
Math. Subj. Class. (2020): 20B25, 05C25
1 Introduction
Metacirculant graphs were introduced by Alspach and Parsons [1]. In 2008 Marušič and
Šparl [11] gave an equivalent definition of metacirculant graphs as follows. Let m ≥ 1 and
n ≥ 2 be integers. A graph Γ of order mn is called [11] an (m,n)-metacirculant graph
(in short (m,n)-metacirculant) if it has an automorphism σ of order n such that ⟨σ⟩ is
semiregular on the vertex set of Γ, and an automorphism τ normalizing ⟨σ⟩ and cyclically
permuting the m orbits of ⟨σ⟩ such that τ has a cycle of size m in its cycle decomposition.
A graph is called a metacirculant if it is an (m,n)-metacirculant for some m and n.
It follows from the definition above that a metacirculant Γ has a vertex-transitive au-
tomorphism group ⟨σ, τ⟩ which is metacyclic. If we, instead, require that the graph has a
vertex-transitive metacyclic group of automorphisms, then we obtained the so-called weak
metacirculants, which were introduced by Marušič and Šparl [11] in 2008. In [10] Li et
al. initiated the study of relationship between metacirculants and weak metacirculants, and
they divided the weak metacirculants into the following two subclasses: A weak metacircu-
lant which has a vertex-transitive split metacyclic automorphism group is called split weak
metacirculant. Otherwise, a weak metacirculant Γ is called a non-split weak metacirculant
if its full automorphism group does not contain any split metacyclic subgroup which is
vertex-transitive. In [10, Lemma 2.2] it was proved that every metacirculant is a split weak
*This work was partially supported by the National Natural Science Foundation of China (12071023,
1211101360).
†Corresponding author.
E-mail addresses: 16118417@bjtu.edu.cn (Li Cui), jxzhou@bjtu.edu.cn (Jin-Xin Zhou)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
490 Ars Math. Contemp. 22 (2022) #P3.08 / 489–504
metacirculant, but it was unknown whether the converse of this statement is true. In [14]
Sanming Zhou and the second author asked the following question:
Question 1.1 ([14, Question A]). Is it true that any split weak metacirculant is a metacir-
culant?
If the graph under consideration is of prime-power order, it was shown by the authors
[5, 14] that the answer to the above question is positive. However, in [6] we show that
there do exist infinitely many split weak metacirculants which are not metacirculants. For
convenience, we shall say that a split metacirculant is a pseudo metacirculant if it is not
metacirculant. To the best of our knowledge, up to now the only known pseudo metacircu-
lants are the graphs constructed in [6]. So it might be interesting to find some other families
of pseudo metacirculants. Furthermore, it seems that the existence of pseudo metacircu-
lants is closely related to the orders of graphs. Motivated by this, it is natural to consider
the following problem.
Problem 1.2. Characterize those integers n for which there is a pseudo metacirculant of
order n.
There are some partial answers to Problem 1.2. By [5, 14], there do not exist a pseudo
metacirculant with a prime-power order. The construction of pseudo metacirculants in [6]
shows that for two any primes p, q with q | p − 1, there exists a pseudo metacirculant of
order pmq for each m ≥ 3. In this paper, two new infinite families of pseudo metacirculants
are constructed. Our construction implies that for any primes p, q, if either p⌊
m
2 ⌋+1 | q − 1
or p = 2 and 4 | q − 1, then there exists a pseudo metacirculant of order pmq with m ≥ 3.
Our research also shows that the three families of pseudo metacirculants constructed
in [6] and this paper are crucial for solving Problem 1.2, and we shall use them to give a
complete solution of Problem 1.2 in our subsequent paper [4].
2 Preliminaries
2.1 Definitions and notation
For a positive integer n, we denote by Cn the cyclic group of order n, by Zn the ring of
integers modulo n, by Z∗n the multiplicative group of Zn consisting of numbers coprime to
n, and by D2n the dihedral group of order 2n. For two groups M and N , N : M denotes
a semidirect product of N by M . Given a group G, denote by 1, Aut(G), and Z(G) the
identity element, full automorphism group and center of G, respectively. Denote by o(x)
the order of an element x of G. For a subgroup H of G, denote by CG(H) the centralizer
of H in G. A group G is called metacyclic if it contains a normal cyclic subgroup N such
that G/N is cyclic. In other words, a metacyclic group G is an extension of a cyclic group
N ∼= Cn by a cyclic group G/N ∼= Cm, written as G ∼= Cn.Cm. If this extension is split,
namely G ∼= Cn : Cm, then G is called a split metacyclic group.
Let G be a permutation group on a set Ω and α ∈ Ω. Denote by Gα the stabilizer of
α in G, that is, the subgroup of G fixing the point α. We say that G is semiregular on Ω
if Gα = 1 for every α ∈ Ω and regular if G is transitive and semiregular. For any subset
∆ of Ω, use G∆ and G(∆) to denote the subgroups of G fixing ∆ setwise and pointwise,
respectively. A block of imprimitivity of G on Ω is a subset ∆ of Ω with 1 < |∆| < |Ω|
such that for any g ∈ G, either ∆g = ∆ or ∆g ∩∆ = ∅. In this case the blocks ∆g, g ∈ G
form a G-invariant partition of Ω.
L. Cui and J.-X. Zhou: Two families of pseudo metacirculants 491
All graphs in this paper are finite, simple and undirected. For a graph Γ, we denote
its vertex set and edge set by V (Γ) and E(Γ), respectively. Given two adjacent vertices
u, v of Γ, denote by {u, v} the edge between u and v. Denote by Γ(v) the neighbourhood
of v, and by Γ[B] the subgraph of Γ induced by a subset B of V (Γ). An s-cycle in Γ,
denoted by Cs, is an (s + 1)-tuple of pairwise distinct vertices (v0, v1, . . . , vs) such that
{vi−1, vi} ∈ E(Γ) for 1 ≤ i ≤ s and {vs, v0} ∈ E(Γ). Denote by Kn the complete graph
of order n, and by Kn,n the complete bipartite graph with biparts of cardinality n. The full
automorphism group of Γ is denoted by Aut(Γ).
2.2 Quotient graph
Let Γ be a connected vertex-transitive graph, and let G ≤ Aut(Γ) be vertex-transitive on
Γ. A partition B of V (Γ) is said to be G-invariant if for any B ∈ B and g ∈ G we have
Bg ∈ B. For a G-invariant partition B of V (Γ), the quotient graph ΓB is defined as the
graph with vertex set B such that, for any two different vertices B,C ∈ B, B is adjacent to
C if and only if there exist u ∈ B and v ∈ C which are adjacent in Γ. Let N be a normal
subgroup of G. Then the set B of orbits of N on V (Γ) is a G-invariant partition of V (Γ).
In this case, the symbol ΓB will be replaced by ΓN , and the original graph Γ is said to be a
cover of ΓN if Γ and ΓN have the same valency.
2.3 Cayley graph
Given a finite group G and an inverse closed subset S ⊆ G \ {1}, the Cayley graph
Cay(G,S) on G with respect to S is a graph with vertex set G and edge set {{g, sg} |
g ∈ G, s ∈ S}. For any g ∈ G, R(g) is the permutation of G defined by R(g) : x 7→ xg
for x ∈ G. Set R(G) := {R(g) | g ∈ G}. It is well-known that R(G) is a subgroup of
Aut(Cay(G,S)). A Cayley graph Cay(G,S) is said to be normal if R(G) is normal in
Aut(Cay(G,S)). This concept was introduced by Xu in [13], and for more results about
normal Cayley graphs, we refer the reader to [7].
The following proposition determines the normalizer of R(G) in the full automorphism
group of Cay(G,S).
Proposition 2.1 ([8, Lemma 2.1]). Let Γ = Cay(G,S) be a Cayley graph on G with
respect to S. Then NAut(Γ)(R(G)) = R(G) : Aut(G,S), where Aut(G,S) is the group
of automorphisms of G fixing the set S setwise.
2.4 Coset graph
Let G be a group and for a subgroup H of G, let Ω = [G : H] = {Hx | x ∈ G}, the
set of right cosets of H in G. For g ∈ G, define RH(g) : Hx 7→ Hxg, x ∈ G, and set
RH(G) = {RH(g) | g ∈ G}. The map g 7→ RH(g), g ∈ G, is a homomorphism from G
to SΩ and it is called the coset action of G relative to H . The kernel of the coset action is
HG = ∩g∈GHg , the largest normal subgroup of G contained in H , and G/HG ∼= RH(G).
It is well-known that any transitive action of G on Ω is equivalent to the coset action of G
relative the subgroup Gα for any given α ∈ Ω. If HG = 1, we say that H is core-free in G.
Let D be a union of several double-cosets of the form HgH with g /∈ H such that
D = D−1. The coset graph Γ = Cos(G,H,D) of G with respect to H and D is defined as
the graph with vertex set V (Γ) = [G : H], and edge set E(Γ) = {{Hg,Hdg} | g ∈ G, d ∈
D}. It is easy to see that Γ is well defined and has valency |D|/|H|, and Γ is connected if
492 Ars Math. Contemp. 22 (2022) #P3.08 / 489–504
and only if D generates G. Further, RH(G) ≤ Aut(Γ), and hence Γ is vertex-transitive.
Let Aut(G,H,D) = {α ∈ Aut(G) | Hα = H,Dα = D}. For any α ∈
Aut(G,H,D), define αH : Hx 7→ Hxα, x ∈ G, and consider the action of Aut(G,H,D)
on [G : H] induced by α 7→ αH . It follows that Aut(G,H,D)/L ∼= Aut(G,H,D)H ,
where Aut(G,H,D)H = {αH | α ∈ Aut(G,H,D)} and L is the kernel of the action.
Furthermore, it is easy to see that Aut(G,H,D)H ≤ Aut(Γ) and Aut(G,H,D)H fixes
the vertex H . For h ∈ H , let σ(h) be the inner automorphism of G induced by h, that is,
σ(h) : g 7→ h−1gh, g ∈ G. One may show that σ(H) = {σ(h) | h ∈ H} is a subgroup of
Aut(G,H,D) and hence RH(H) = {RH(h) | h ∈ H} is a subgroup of Aut(G,H,D)H .
The following proposition determines the normalizer of RH(G) in the full automor-
phism group of Cos(G,H,D).
Proposition 2.2 ([12, Lemma 2.10]). Let G be a finite group, H a core-free subgroup of G
and D a union of several double-cosets HgH such that H ⊈ D. Let Γ = Cos(G,H,D)
and A = Aut(Γ). Then RH(G) ∼= G, Aut(G,H,D)H ∼= Aut(G,H,D), RH(H) ∼=
σ(H), and NA(RH(G)) = RH(G)Aut(G,H,D)H with RH(G) ∩ Aut(G,H,D)H =
RH(H).
Below, we prove a technical lemma.
Lemma 2.3. Let G be a finite group and let H be a core-free subgroup of G. Let S be a
set of non-identity elements of G such that S is self-inverse, and let T be any self-inverse
subset of S. Let D = HSH and let C = HTH . Set
Γ = Cos(G,H,D) and Σ = Cos(G,H,C).
If the vertex-stabilizer Aut(Γ)H fixes Σ(H) = {Hd | d ∈ C} setwise, then Aut(Γ) ≤
Aut(Σ).
Proof. Let A = Aut(Γ). Suppose that AH fixes Σ(H) = {Hd | d ∈ C} setwise. Then for
any g ∈ G, we have AHg = (AH)RH(g), and so AHg fixes the following set setwise:
{Hd | d ∈ C}RH(g) = {Hdg | d ∈ C} = Σ(Hg).
Take x ∈ A and take any edge e = {Hg,Hdg} of Σ. To show A ≤ Aut(Σ), it suffices
to show that ex ∈ E(Σ). Since G acts transitively on V (Γ) by right multiplication, there
exists g′ ∈ G such that (Hg)x = Hgg′, and then (Hg)xRH((g′)−1) = Hg. It follows
that (Hdg)xRH((g
′)−1) ∈ Σ(Hg) and so (Hdg)x ∈ (Σ(Hg))RH((g′)) = Σ(Hgg′) =
Σ((Hg)x). Hence, we have ex ∈ E(Σ), and consequently, A ≤ Aut(Σ).
2.5 Circulants
A circulant of order n is a Cayley graph over a cyclic group of order n. The following
proposition gives a classification of arc-transitive circulants. Before stating this result, we
introduce several concepts.
If a graph Γ has n > 1 connected components, each of which is isomorphic to a graph
Σ, then we shall write Γ = nΣ. The lexicographic (or wreath) product of graphs Γ1 and Γ2
is a graph Γ1◦Γ2 with vertex set V (Γ1)×V (Γ2) such that {(x1, x2), (y1, y2)} ∈ E(Γ1◦Γ2)
if and only if either x1 = y1 and {x2, y2} ∈ E(Γ2), or {x1, y1} ∈ E(Γ1). If Γ2 is of
order m with V (Γ2) = {y1, y2, . . . , ym}, then we have a natural embedding of mΓ1 in
L. Cui and J.-X. Zhou: Two families of pseudo metacirculants 493
Γ1 ◦ Γ2, where, for 1 ≤ i ≤ m, the ith copy of Γ1 is the subgraph induced by the subset
of vertices of Γ1 ◦ Γ2. The deleted lexicographic product of graphs Γ1 and Γ2, denoted by
Γ1 ◦ Γ2 −mΓ1, is the graph obtained by deleting from Γ1 ◦ Γ2 the edges of mΓ1.
Proposition 2.4 ([9, Theorem 1]). Let Γ be a connected arc-transitive circulant of order
n. Then one of the following holds:
(a) Γ = Kn;
(b) Γ is a normal circulant;
(c) Γ = Σ ◦ K̄d, where n = md and Σ is a connected arc-transitive circulant of order
m;
(d) Γ = Σ ◦ K̄d − dΣ, where n = md, d > 3, gcd(d,m) = 1 and Σ is a connected
arc-transitive circulant of order m.
3 Pseudo metacirculants–Family A
Consruction A. Let p be a prime such that 4 | p− 1 and let n ≥ 2 be an integer. If p > 5,
then let r ∈ Z∗p be such that r2 ≡ −1 (mod p), and if p = 5, then let r = 2. Let
G2,n,p,r = ⟨a, b, c | a2
n
= bp = c4 = 1, ab = ba, c−1ac = a−1, c−1bc = br⟩.
Let
Γ2,n,p,r = Cos(G2,n,p,r, H,H{(ab)±1, (abc)±1, (bc)±1, c±1}H),
where H = ⟨a2n−1c2⟩.
We first prove a lemma.
Lemma 3.1. Let G = G2,n,p,r, H = ⟨a2
n−1
c2⟩, and let D = H{(ab)±1, (abc)±1,
(bc)±1, c±1}H . If n = 2 and p > 5, then Aut(G,H,D) contains exactly one involu-
tion.
Proof. Since H ∼= C2, by Proposition 2.2, Aut(G,H,D) has an involution. Take an
involution α ∈ Aut(G,H,D). Since ⟨b⟩ is a normal Sylow p-subgroup of G, ⟨b⟩ is char-
acteristic in G, implying that bα ∈ ⟨b⟩. Since α has order 2, one has bα = b or b−1. By a
direct computation, we see that CG(b) = ⟨a⟩ × ⟨b⟩. Then
aα ∈ CG(bα) = CG(b) = ⟨a⟩ × ⟨b⟩.
It follows that aα ∈ ⟨a⟩. Since α has order 2 and n = 2, one has aα = a or a−1, and hence
(a2)α = a2.
Assume that cα = aibjck for some i ∈ Z2n , j ∈ Zp and k ∈ Z∗4. Considering the
image of the equality c−1bc = br under α, we obtain that (aibjck)−1(b±1)aibjck = b±r,
and hence br
k
= br. It follow that k = 1 in Z4, and so cα = aibjc. Moreover, (c2)α =
bj(1−r)c2. From Hα = H we obtain that (a2c2)α = a2c2. As (a2)α = a2, one has
(c2)α = c2. Thus, j = 0 in Zp, and so cα = aic.
Now by a direct computation, we have
ac, a−1c, ba2c, b−1a2c, a−1bc, a−1b−1c /∈ D. (3.1)
494 Ars Math. Contemp. 22 (2022) #P3.08 / 489–504
Remember that Dα = D. Since c ∈ D, cα = aic implies that aic ∈ D. So the only
possibility is either cα = c or cα = a2c. If the latter happens, then (bc)α = ba2c or
b−1a2c, and since bc ∈ D, either ba2c or b−1a2c belongs to D, contrary to Equation (3.1).
Thus, cα = c.
Recall that aα = a−1 or a. For the former, we have (abc)α = a−1bc or a−1b−1c,
and since abc ∈ D, either a−1bc or a−1b−1c belongs to D. This is again impossible by
Equation (3.1). Thus, aα = a, and hence we have that
α : a 7→ a, b 7→ b−1, c 7→ c.
This implies that Aut(G,H,D) has exactly one involution.
Below we shall determine the full automorphism group of Γ2,n,p,r.
Lemma 3.2. Let Γ = Γ2,n,p,r and let G = G2,n,p,r. Then Aut(Γ) = RH(G) ∼= G.
Proof. Let A = Aut(Γ). It is easy to see that H is a non-normal subgroup of G, and
so H is core-free in G. It follows that G acts faithfully and transitively on V (Γ) by right
multiplication, and so we may view G as a transitive subgroup of A. If n = 2 and p = 5,
then by Magma [3], we obtain that Aut(Γ) = G. In what follows, we shall always assume
that either p > 5 or n > 2.
Noting that ab = ba, we have ⟨ab⟩ ∼= C2np. Clearly, ⟨ab⟩ ⊴ G, so ⟨ab⟩ is semiregular
on V (Γ). Since |V (Γ)| = |G : H| = 2n+1p, ⟨ab⟩ has two orbits on V (Γ) which are listed
as follows:
V0 = {Haibj | i ∈ Z2n , j ∈ Zp} and V1 = {Haibjc | i ∈ Z2n , j ∈ Zp}.
The kernel of G acting on {V0, V1} is ⟨ab⟩ : ⟨c2⟩. We can also easily obtain the following
two observations:
∀i ∈ Z2n , j ∈ Zp, Haibjc2 = Hc2ai(c2bjc2) = Ha2
n−1+ib−j ,
∀i1, i2 ∈ Z2n , j1, j2 ∈ Zp, k ∈ Z2 Hai1bj1ck = Hai2bj2ck ⇔
{
i1 ≡ i2 (mod 2n),
j1 ≡ j2 (mod p).
(3.2)
Set ∆1 = {Hd | d ∈ H{(ab)±1}H} and ∆2 = {Hd | d ∈ H{(abc)±1, (bc)±1, c±1}H}.
Then Γ(H) = ∆1 ∪∆2. Furthermore, an easy computation shows that
∆1 = {Hab,H(ab)−1, Hab−1, Ha−1b},
∆2 = {Habc,Ha1+2
n−1
brc,Hab−1c,Ha1+2
n−1
b−rc,
Hbc,Ha2
n−1
brc,Hb−1c,Ha2
n−1
b−rc,Hc,Ha2
n−1
c}.
L. Cui and J.-X. Zhou: Two families of pseudo metacirculants 495
q
Habc
qHc q
Hab−1c
qHab−1 qHa2n−1c qHab
q
Ha1+2
n−1
brc
qHa−1b−1 qHa−1b q
Ha1+2
n−1
b−rc
q
Ha2
n−1
brc
q
Hbc
q
Hb−1c
q
Ha2
n−1
b−rc
Figure 1: The subgraph induced by Γ(H) when p > 5 and n > 2.
So for any i ∈ Z2n , j ∈ Zp, we have
Γ(Haibj) = {Hai+1bj+1, Hai−1bj−1, Hai+1bj−1, Hai−1bj+1, Ha1−ib1−jrc,
Ha2
n−1+1−ibr−jrc,Ha1−ib−1−jrc,Ha1+2
n−1−ib−r−jrc,
Ha−ib1−jrc,Ha2
n−1−ibr−jrc,Ha−ib−1−jrc,Ha2
n−1−ib−r−jrc,
Ha−ib−jrc,Ha2
n−1−ib−jrc},
Γ(Haibjc) = {Hai+1bj+1c,Hai−1bj−1c,Hai+1bj−1c,Hai−1bj+1c,Ha2
n−1+1−ibjr−1,
Ha1−ibjr−r, Ha2
n−1+1−ib1+jr, Ha1−ibr+jr, Ha2
n−1−ibjr−1,
Ha−ibjr−r, Ha2
n−1−ib1+jr, Ha−ibr+jr, Ha2
n−1−ibjr, Ha−ibjr}.
We shall finish the proof by the following four steps.
Step 1: Let Σ = Cos(G,H,H{(bc)±1, c±1}H) and let M = ⟨bc, c,H⟩. Then A ≤
Aut(Σ). In particular, the orbit HM = {Hg | g ∈ M} of M on V (Γ) containing H is a
block of imprimitivity of A on V (Γ).
By direct computations, we may depict the subgraph induced by Γ(H) as in
Figures 1 – 3. From these three figures one may see that the vertex-stabilizer AH fixes the
following set setwise:
Σ(H) = {Hd | d ∈ H{(bc)±1, c±1}H}
= {Hbc,Ha2
n−1
brc,Hb−1c,Ha2
n−1
b−rc,Hc,Ha2
n−1
c}.
From Lemma 2.3 it follows that A ≤ Aut(Σ).
Since M = ⟨bc, c,H⟩ = ⟨a2n−1⟩ × ⟨b, c⟩ ∼= C2 × (Cp : C4), the coset graph
∆ = Cos(M,H,H{(bc)±1, c±1}H})
is just a component of Σ, and since A is transitive on V (Σ) = V (Γ), the orbit of M con-
taining H is a block of imprimitivity of A on V (Γ).
496 Ars Math. Contemp. 22 (2022) #P3.08 / 489–504
q
Ha−1b−1
qHc q
Ha−1b
qHa1+2n−1brc q



C
C
C
C
C
Habc qCCC
C
C





Hab−1c qHa1+2n−1b−rc
qHab−1 qHa2n−1c qHab
q
Ha2
n−1
brc
q
Hbc
q
Hb−1c
q
Ha2
n−1
b−rc
Figure 2: The subgraph induced by Γ(H) when p > 5 and n = 2.
qHa2n−1b2c q
Ha1+2
n−1
b−2c
qHa−1b q
Hab−1c
qHc q
Habc
qHa−1b−1 q
Ha1+2
n−1
b2c
qHa2n−1b−2c
qHbc qHab−1 qHa2n−1c qHab qHb−1c
Figure 3: The subgraph induced by Γ(H) when p = 5, r = 2 and n > 2.
Step 2: Set N = ⟨a2n−1⟩ × ⟨b⟩. Then each orbit of N is a block of imprimitivity of A
on V (Γ).
Let O be the orbit of M on V (Γ) containing H . Then O = {Hg | g ∈ M} = V (∆).
By Step 1, O is a block of imprimitivity of A on V (Γ). Let S = {Og | g ∈ G}. Then
S is an imprimitivity block system of A on V (Γ). Let K be the kernel of A acting on S.
Note that N ⊴ G. Since N ≤ M , N fixes every block Og in S. It follows that N ≤ K.
Note that the subgraph Σ[Og] of Σ is just a component which is isomorphic to ∆. It is
easy to see that N acts on each Og semiregularly with two orbits {Hng | n ∈ N} and
{Htcg | t ∈ N}. As
Σ(Hg) = {Hbcg,Ha2
n−1
brcg,Hb−1cg,Ha2
n−1
b−rcg,Hcg,Ha2
n−1
cg},
the component Σ[Og] is a bipartite graph with the two orbits of N on Og as its two parts.
This implies that every orbit of N on V (Γ) is a block of imprimitivity of A on V (Γ).
Step 3: Take two adjacent orbits B0, B1 of N on V (Γ) such that B0 ⊆ V0 and B1 ⊆ V1.
Then we have A(B0) = A(B1). In particular, AH fixes ∆1 = {Hab,H(ab)−1, Hab−1,
Ha−1b} = {Hd | d ∈ H{(ab)±1}H} setwise. Since G acts transitively on V (Γ), we may
let B0 = B = {Hn | n ∈ N}. Since N ⊴G, one has B = NH ≤ G. Recall that
Γ(H) = {Hab,H(ab)−1, Hab−1, Ha−1b,
Habc,Ha1+2
n−1
brc,Hab−1c,Ha1+2
n−1
b−rc,
Hbc,Ha2
n−1
brc,Hb−1c,Ha2
n−1
b−rc,Hc,Ha2
n−1
c}.
L. Cui and J.-X. Zhou: Two families of pseudo metacirculants 497
Each orbit of N on V (Γ) is also an independent subset of Γ. Consider the orbits Ba,Ba−1, Bc
and Bac of N . Since B = NH , one has
Hab,Hab−1 ∈ Ba,
Ha−1b,Ha−1b−1 ∈ Ba−1,
Habc,Ha1+2
n−1
brc,Hab−1c,Ha1+2
n−1
b−rc ∈ Bac,
Hbc,Ha2
n−1
brc,Hb−1c,Ha2
n−1
b−rc,Hc,Ha2
n−1
c ∈ Bc.
So Ba,Ba−1, Bc and Bac are all orbits of N adjacent to B. Furthermore, it is easy to
check that if n > 2, then Ba,Ba−1, Bc and Bac are four pair-wise different orbits of
N , and if n = 2, then Ba = Ba−1, Bc and Bac are three pair-wise different orbits of
N . Clearly, Bc,Bac ⊆ V1. So B1 = Bc or Bac. Note that Γ[B,Bc] has valency 6 and
Γ[B ∪Bac] has valency 4.
If n > 2, then both Γ[B ∪Ba] and Γ[B ∪Ba−1] have valency 2. This implies that AB
also fixes Bc and Bac. If n = 2, then Γ[B∪Ba] has valency 4, and from Figure 2 one may
see that AH fixes {Hab,Ha−1b−1, Hab−1, Ha−1b} setwise and so AH fixes Ba. Again,
we have AB also fixes each of Bc and Bac.
Thus, we always have that AB also fixes each of Bc and Bac. Clearly, the subgraph
Γ[B∪B1] is bipartite, where B1 is either Bc or Bac. Let K be the subgroup of Aut(Γ[B∪
B1]) fixing B setwise. Then B and B1 are two orbits of K. Let K(B) be the kernel of K
acting on B. If K(B) does not fix every vertex of B1, then each orbit of K(B) on B1 has
length 2p, p or 2. Take two vertices u, v of B1 such that u, v are in the same orbit of K(B).
Then u, v will share the common neighborhood in Γ[B ∪ B1]. Without loss of generality,
we may assume that H is one of their common neighbors.
If B1 = Bac, then u, v ∈ {Habc,Ha1+2
n−1
brc,Hab−1c,Ha1+2
n−1
b−rc}. Note that
Γ(Habc) ∩B = {Ha2
n−1
br−1, H,Ha2
n−1
b1+r, Hb2r},
Γ(Ha1+2
n−1
brc) ∩B = {Hb−2, Ha2
n−1
b−1−r, H,Ha2
n−1
br−1},
Γ(Hab−1c) ∩B = {Ha2
n−1
b−r−1, Hb−2r, Ha2
n−1
b1−r, H},
Γ(Ha1+2
n−1
b−rc) ∩B = {H,Ha2
n−1
b1−r, Hb2, Ha2
n−1
br+1}.
It is easy to see no two of the above four sets are the same, and so no two vertices
in {Habc,Ha1+2n−1brc,Hab−1c,Ha1+2n−1b−rc} share the common neighborhood in
Γ[B,Bac], a contradiction.
If B1 = Bc, then u, v ∈ {Hbc,Ha2
n−1
brc,Hb−1c,Ha2
n−1
b−rc,Hc,Ha2
n−1
c}.
Note that
Γ(Hbc) ∩B = {Ha2
n−1
br−1, H,Ha2
n−1
b1+r, Hb2r, Ha2
n−1
br, Hbr},
Γ(Ha2
n−1
brc) ∩B = {Hb−2, Ha−2
n−1
b−1−r, H,Ha−2
n−1
br−1, Hb−1, Ha−2
n−1
b−1},
Γ(Hb−1c) ∩B = {Ha2
n−1
b−r−1, Hb−2r, Ha2
n−1
b1−r, H,Ha2
n−1
b−r, Hb−r},
Γ(Ha2
n−1
b−rc) ∩B = {H,Ha−2
n−1
b1−r, Hb2, Ha−2
n−1
br+1, Hb,Ha−2
n−1
b},
Γ(Hc) ∩B = {Ha2
n−1
b−1, Hb−r, Ha2
n−1
b,Hbr, Ha2
n−1
, H},
Γ(Ha2
n−1
c) ∩B = {Hb−1, Ha2
n−1
b−r, Hb,Ha−2
n−1
br, H,Ha−2
n−1
}.
498 Ars Math. Contemp. 22 (2022) #P3.08 / 489–504
It is easily checked that the above six sets are pair-wise different, and no two vertices in
{Hbc,Ha2n−1brc,Hb−1c,Ha2n−1b−rc,Hc,Ha2n−1c} share the common neighborhood
in Γ[B,Bc], a contradiction. Thus, K(B) also fixes every vertex of B1. Consequently, we
have A(B0) = A(B) = A(B1) with B1 = Bc or Bac.
Step 4: A = G.
By Step 3, AH fixes ∆1 setwise and so fixes ∆2 setwise. By Lemma 2.3, we have
A ≤ Aut(Λ), where Λ = Cos(G,H,D2) with D2 = H{(abc)±1, (bc)±1, c±1}H . It is
easy to see that Λ is a connected bipartite graph with V0 and V1 as its two parts. Let K be
the kernel of A acting on {V0, V1}. Again, by Step 3, we obtain that K acts faithfully on
V0. It is easy to see that Γ[V0] ∼= Θ = Cay(⟨ab⟩, {ab, (ab)−1, ab−1, a−1b}). By [2, Corol-
lary 1.3], Θ is a normal Cayley graph on ⟨ab⟩. Then Aut(⟨ab⟩, {ab, (ab)−1, ab−1, a−1b}) ∼=
C2 × C2 is regular on {ab, (ab)−1, ab−1, a−1b}. So, |K| ≤ 4|V0|. It follows that |A| ≤
4|V (Γ)| and hence |A : G| ≤ 2. Consequently, we have G ⊴ A. By Proposition 2.2, we
have AH = Aut(G,H,D) ≤ Aut(⟨ab⟩, {ab, (ab)−1, ab−1, a−1b}). If n > 2, then from
Figures 1 and 3, we can see that AH is intransitive on the set of four neighbors of H con-
tained in V0. It follows that AH ∼= C2 and hence A = G. If n = 2 and p > 5, then by
Lemma 3.1, we must have Aut(G,H,D) ∼= C2, implying that A = G.
Corollary 3.3. The graph Γ2,n,p,r is non-Cayley.
Proof. Let Γ = Γ2,n,p,r, and let A = Aut(Γ). Suppose on the contrary that Γ is a Cay-
ley graph. Then A has a regular subgroup, say T , and then A = T : AH . Since A is
metacyclic, every Sylow 2-subgroup of T must be cyclic. It follows that every Sylow 2-
subgroup of A has a cyclic maximal subgroup. However, this is impossible because from
the Construction B we know that every Sylow 2-subgroup of A is isomorphic to C2n : C4,
a contradiction.
Theorem 3.4. The graph Γ2,n,p,r is a pseudo metacirculant.
Proof. Let Γ = Γ2,n,p,r, and let G = G2,n,p,r. Note that G acts faithfully and transitively
on V (Γ) by right multiplication. Since G = ⟨ab⟩ : ⟨c⟩ ∼= C2n·p : C4, Γ is a split weak
metacirculant.
Suppose that Γ is also a metacirculant. Then by the definition of metacirculant, Γ has
two automorphisms σ, τ satisfying the following conditions:
(1) ⟨σ⟩ is semiregular and has m orbits on V (Γ),
(2) τ normalizes ⟨σ⟩ and cyclically permutes the m orbits of ⟨σ⟩,
(3) τ has a cycle of size m in its cycle decomposition.
By Lemma 3.2, we have Aut(Γ) = G. By Corollary 3.3, Γ is a non-Cayley graph, and
then we have G = ⟨σ, τ⟩. Thus, τm ̸= 1 and hence ⟨τm⟩ ∼= C2. Since G is transitive on
V (Γ), we may assume that ⟨τm⟩ = GH = ⟨a2
n−1
c2⟩. Then there would exist an element
x of G of order 4 such that x2 = τm = a2
n−1
c2. By a direct computation, we have
CG(a
2n−1c2) = ⟨a, c⟩. Then x ∈ ⟨a, c⟩ and so x = aicj for some integers i, j. However,
x2 = c2j due to c−1ac = a−1. A contradiction occurs. Thus, Γ is not a metacirculant.
L. Cui and J.-X. Zhou: Two families of pseudo metacirculants 499
4 Pseudo metacirculants—Family B
Consruction B. Let m > n > 1 be positive integers and let p, q be primes such that
pm | q − 1. Let r ∈ Z∗q be of order pm, and let
Gp,q,m,n,r = ⟨a, b, c | ap
n
= bq = cp
m
= 1, ab = ba, ac = ca, c−1bc = br⟩.
Let
Γp,q,m,n,r = Cos(Gp,q,m,n,r, H,H{(ab)±1, c±1}H),
where H = ⟨cpm−1apn−1⟩.
We shall first give some basic properties of G.
Lemma 4.1. Let G = Gp,q,m,n,r, and let D = H{(ab)±1, c±1}H . Then the following
hold.
(1) |H| = p, H is non-normal in G and CG(H) = ⟨a, c⟩.
(2) CG(b) = ⟨a, b⟩.
(3) For any g ∈ G, if ⟨g⟩⊴G, then g ∈ ⟨a, b⟩.
(4) D = (
⋃
k∈Zp H(ab
rkp
m−1
)) ∪ (
⋃
k∈Zp H(ab
rkp
m−1
)−1) ∪Hc ∪Hc−1.
(5) Aut(G,H,D) ∼= Cp.
Proof. Note that G = ⟨a⟩ × (⟨b⟩ : ⟨c⟩) ∼= Zpn × (Zq : Zpm). Let P = ⟨a, c⟩. Clearly,
P = ⟨a⟩ × ⟨c⟩ ∼= Cpn × Cpm , so H = ⟨cp
m−1
ap
n−1⟩ has order p. If H ⊴ G, then b
centralizes cp
m−1
ap
n−1
and then centralizes cp
m−1
since a ∈ Z(G). This is impossible
because c−p
m−1
bcp
m−1
= br
pm−1 ̸= b. Thus, H is non-normal in G. It follows that
⟨a, c⟩ ≤ CG(H) < G. Observing that |G : ⟨a, c⟩| = q, we have CG(H) = ⟨a, c⟩.
Therefore, item (1) holds.
For (2) , it is easy to see that ⟨a, b⟩ ≤ CG(b) and G = ⟨a, b⟩ : ⟨c⟩. Recall that
c−1bc = br with r an element of Z∗q of order pm. This implies that CG(b) = ⟨a, b⟩.
For (3) , recall that G/⟨a⟩ ∼= ⟨b⟩ : ⟨c⟩ ∼= Zq : Zpm , and ⟨b⟩ is self-centralized in
⟨b⟩ : ⟨c⟩. This implies that ⟨b⟩ is the unique non-trivial normal cyclic subgroup of ⟨b⟩ : ⟨c⟩.
For any g ∈ G, if ⟨g⟩ ⊴ G, then ⟨g⟩⟨a⟩/⟨a⟩ is normal in G/⟨a⟩, and then ⟨g⟩⟨a⟩/⟨a⟩ ≤
⟨b⟩⟨a⟩/⟨a⟩. So g ∈ ⟨a, b⟩, as desired.
For (4) , we have D = H{(ab)±1, c±1}H = H{ab, (ab)−1}H ∪H{c, c−1}H . Since
c centralizes H , one has H{c, c−1}H = Hc ∪Hc−1. Clearly,
H{ab, (ab)−1}H = HabH ∪H(ab)−1H.
Then HabH =
⋃
k∈Zp Hab(c
pm−1ap
n−1
)k =
⋃
k∈Zp Habc
kpm−1akp
n−1
. Since c−1bc =
br, one has bckp
m−1
= ckp
m−1
br
kpm−1
. As a ∈ Z(G), it follows that
HabH =
⋃
k∈Zp
Habckp
m−1
akp
n−1
=
⋃
k∈Zp
Habr
kpm−1
.
500 Ars Math. Contemp. 22 (2022) #P3.08 / 489–504
Similarly, H(ab)−1H =
⋃
k∈Zp H(ab
rkp
m−1
)−1. (4) is proved.
Finally, we shall prove (5) . Take α ∈ Aut(G,H,D). Then Hα = H and Dα = D.
Observe that ⟨b⟩ is a normal Sylow q-subgroup of G and ⟨a⟩ is just the center of G. It
follows that bα ∈ ⟨b⟩ and aα ∈ ⟨a⟩. Since Hα = H , one has cα ∈ CG(H) = ⟨a, c⟩.
Assume that
aα = ai, bα = bj , cα = asct, with i ∈ Z∗pn , j ∈ Z∗q , s ∈ Zpn , t ∈ Z∗pm .
Considering the image of the equality c−1bc = br under α, we obtain that
(asct)−1bj(asct) = bjr,
and hence bjr
t
= bjr. It follows that rt ≡ r (mod q). Since r is an element of Z∗pm of
order pm and t ∈ Z∗pm , the equality rt ≡ r (mod q) implies that t = 1 in Zpm , and so
cα = asc. Consequently, asc = cα ∈ Dα = D. By a direct computation, we have aℓc /∈ D
for any ℓ ̸= 0 (in Zpn ). So we must have cα = c.
Since ab ∈ D, one has aibj = (ab)α ∈ D. Since j ∈ Z∗q , one has aibj /∈ Hc ∪Hc−1
because (Hc ∪Hc−1) ⊆ ⟨a, c⟩. Then by (4) we have
aibj ∈ (
⋃
k∈Zp
H(abr
kpm−1
)) ∪ (
⋃
k∈Zp
H(abr
kpm−1
)−1).
Note that ⟨a, b⟩ ∩H = 1. It follows that (ab)α = aibj ∈ {(abrkp
m−1
)±1 | k ∈ Zp}, and so
aibj = abr
kpm−1
or (abr
kpm−1
)−1 for some k ∈ Zp. Since ⟨a, b⟩ = ⟨a⟩× ⟨b⟩ ∼= Zpn ×Zq ,
one has
(a, b)α = (ai, bj) = (a, br
kpm−1
) or (a−1, b−r
kpm−1
).
Since Hα = H , one has cp
m−1
(ai)p
n−1
= (cp
m−1
ap
n−1
)α ∈ H , and hence aα = a. It
follows that (a, b)α = (ai, bj) = (a, br
kpm−1
). Hence |Aut(G,H,D)| ≤ p. Note that
H ≤ Aut(G,H,D). Then Aut(G,H,D) ∼= Cp.
Next we shall determine the full automorphism group of Γp,q,n,m,r.
Lemma 4.2. Let Γ = Γp,q,m,n,r and let G = Gp,q,m,n,r. Then Aut(Γ) = RH(G) ∼= G.
Moreover, Γ is a non-Cayley graph.
Proof. We shall first prove three claims.
Claim 1. Let Λ = Cos(G,H,H{ab, (ab)−1}H). Then Aut(Γ) ≤ Aut(Λ).
Proof of Claim 1. By Lemma 4.1(4), the neighborhood of H in Γ is
Γ(H) = {H(abr
kpm−1
)±1, Hc±1 | k ∈ Zp},
and the neighborhood of H is Λ is
Λ(H) = {H(abr
kpm−1
)±1 | k ∈ Zp}.
L. Cui and J.-X. Zhou: Two families of pseudo metacirculants 501
By direct computations, we see that for any k ∈ Zp,
{H,Ha2b1+r
kpm−1
| k ∈ Zp} ⊆ Γ(Habr
kpm−1
) ∩ Γ(Hab),
{H,H(a2b1+r
kpm−1
)−1 | k ∈ Zp} ⊆ Γ(H(abr
kpm−1
)−1) ∩ Γ(H(ab)−1),
and moreover, for any Hx ∈ {H(abrkp
m−1
)±1 | k ∈ Zp} and Hcℓ ∈ {Hc,Hc−1}, we
have
Γ(Hx) ∩ Γ(Hcℓ) = {H}.
It then follows that the vertex-stabilizer Aut(Γ)H fixes the following set setwise:
Λ(H) = {H(abr
kpm−1
)±1 | k ∈ Zp}.
By Lemma 2.3, we have Aut(Γ) ≤ Aut(Λ).
Claim 2. Let Vi = {Hajbkci | j ∈ Zpn , k ∈ Zq} with i ∈ Zpm−1 . Then the following
hold.
(1) Vi is a block of imprimitivity of Aut(Γ) on V (Γ).
(2) The edges of Γ between Vi and Vi+1 form a perfect matching, where the subscripts
are modulo pm−1.
(3) Let B = {V0, V1, . . . , Vpm−1−1}. Then the quotient graph ΓB is isomorphic to
Cpm−1 .
Proof of Claim 2. By Claim 1, we have Aut(Γ) ≤ Aut(Λ). Recall that
Λ = Cos(G,H,H{ab, (ab)−1}H).
Then Λ is disconnected, and the coset graph
∆ = Cos(⟨ab,H⟩, H,H{ab, (ab)−1}H)
is a component of Λ. Consequently, V0 = V (∆) = {Hg | g ∈ ⟨ab⟩} is a block of
imprimitivity of Aut(Γ) on V (Λ) = V (Γ). Clearly, each Vi is an orbit of ⟨ab⟩ on V (Γ).
Since ⟨ab⟩ ⊴ G and G is transitive on V (Γ), Vi is a block of imprimitivity of Aut(Γ) on
V (Γ).
For (2), observing that V1 ∩ Γ(H) = {Hc}, the edges between V0 and V1 form a
perfect matching. As c cyclically permutates the orbits V0, V1, . . . , Vpm−1−1 of ⟨ab⟩, for
every i ∈ Zpm−1 , the subgraph of Γ induces by the edges between Vi and Vi+1 form a
perfect matching, where the subscripts are modulo pm−1.
For (3), noting that Γ has valency 2p + 2 while ∆ has valency 2p, the quotient graph
ΓB must be a cycle of length pm−1.
Claim 3. Let ∆ = Cos(⟨ab,H⟩, H,H{ab, (ab)−1H}). Then |Aut(∆)| = 2pn+1q.
Proof of Claim 3. It is easy to see that ∆ is isomorphic to the following Cayley graph
Θ = Cay(⟨ab⟩, {abr
kpm−1
, (abr
kpm−1
)−1 | k ∈ Zp}).
502 Ars Math. Contemp. 22 (2022) #P3.08 / 489–504
Let M = ⟨a, b⟩(∼= Zpnq) and S = {abr
kpm−1
, (abr
kpm−1
)−1 | k ∈ Zp}. The maps
α : a 7→ a, b 7→ brp
m−1
and :
¯
a 7→ a−1, b 7→ b−1 induce two automorphisms of M which
fix S setwise. So, ⟨α, β⟩ ≤ Aut(M,S). It is easy to see that ⟨α, β⟩ acts transitively on S.
Hence Θ is a connected arc-transitive Cayley graph on M . Suppose that Θ is not normal.
It is obvious that Θ ≇ Kpnq . By Proposition 2.4, there exists a connected arc-transitive
circulant Σ of order t such that one of the following happens:
(i) Θ ∼= Σ ◦ K̄d with pnq = td,
(ii) Θ ∼= Σ ◦ K̄d − dΣ, where pnq = td, d > 3 and gcd(d, t) = 1.
Suppose that (i) happens. Let k be the valency of Σ. Then Σ[K̄d] has valency kd. Noting
that Θ has valency 2p, Θ ∼= Σ ◦ K̄d implies that kd = 2p. As pnq = td, one has d = p,
and hence k = 2. It follows that Σ is a cycle of length t. Also, Θ ∼= Σ ◦ K̄p implies
that there exist t independent subsets, say D0, D1, D2, . . . , Dt−1 of V (Θ) of cardinality p
such that the subgraph induced by Di ∪Di+1 is isomorphic to Kp,p, where the subscripts
are modulo t. Furthermore, these t subsets are also blocks of imprimitivity of Aut(Θ)
on V (Θ). Assume that D0 contains the identity of M . Since M acts on V (Θ) by right
multiplication, D0 will be a subgroup of M of order p, and then D0 = ⟨ap
n−1⟩ and then
each Di is a coset of D0 in M . Recall that the only two blocks adjacent to D0 are D1
and D2, and that any two adjacent blocks induce a subgraph isomorphic to Kp,p. Then
D1∪Dt−1 = S. We may assume that ab ∈ D1. Then D1 = D0ab and Dt−1 = D0(ab)−1.
This is contrary to the fact that S = {abrkp
m−1
, (abr
kpm−1
)−1 | k ∈ Zp}.
Suppose now that (ii) holds. Observing that the valency of Σ◦K̄d−dΣ is a multiple of
d− 1, one has d− 1 | 2p, and hence d− 1 = p due to d > 3. As pnq = td and d = p+ 1,
one has p+ 1 | pnq, implying p+ 1 = q. This is contrary to the fact that p, q are odd.
Thus, Θ is a normal Cayley graph on M . By Proposition 2.1, we have Aut(Θ) =
R(M) : Aut(M,S). As M is cyclic, Aut(M,S) is abelian, and so Aut(M,S) acts regu-
larly on S. It follows that |Aut(M,S)| = 2p, and so |Aut(Θ)| = 2pn+1q, as claimed.
Proof of Lemma 4.2, continued: Now we are ready to finish the proof. Let A = Aut(Γ).
By Claim 2(1), B = {V0, V1, . . . , Vpm−1−1} is a system of blocks of A. Let K be the
kernel of A acting on B. By Claim 2(3), we have A/K ≤ Aut(ΓB) ∼= D2pm−1 . By
Claim 2(2), one may see that K acts faithfully on V0. So K can be viewed as a subgroup of
the full automorphism group of the subgraph ∆ of Γ induced by V0. By Claim 3, we have
|Aut(∆)| = 2pn+1q, and so |K| ≤ 2pn+1q.
From Lemma 4.1(1) we know that H is non-normal in G, and so G acts faithfully
on V (Γ) by right multiplication. Therefore, we may identify G with RH(G), and then
G is a vertex-transitive subgroup of A. Then GK/K ∼= Cpm−1 which is normal in
A/K ≤ D2pm−1 . In particular, GK ⊴ A. Furthermore, |GK/K| = pm−1 implies that
|GK| = pm−1|K| ≤ 2pm+nq. So |GK : G| ≤ 2, and hence G is the unique Hall {p, q}-
subgroup of GK. Thus, G is characteristic in GK, and so normal in A since GK ⊴A. By
Proposition 2.2, the stabilizer of H in A is AH = Aut(G,H,D). By Lemma 4.1(5), we
have Aut(G,H,D) ∼= Cp. This implies that G = A.
Finally, suppose that Γ is a Cayley graph. Then A has a regular subgroup, say T , and
then A = T : AH . Since A = G is metacyclic, every Sylow p-subgroup of T must
be cyclic because AH ∼= Cp. It follows that every Sylow p-subgroup of A has a cyclic
L. Cui and J.-X. Zhou: Two families of pseudo metacirculants 503
maximal subgroup. However, this is impossible because from the Construction B we know
that every Sylow p-subgroup of A is isomorphic to Cpn : Cpm with m > n > 1.
Theorem 4.3. The graph Γp,q,m,n,r is a pseudo metacirculant.
Proof. Let Γ = Γp,q,n,m,r, and let G = Gp,q,n,m,r. Note that G acts faithfully and transi-
tively on V (Γ) by right multiplication. Since G = ⟨ab⟩ : ⟨c⟩ ∼= Cpnq : Cpm , Γ is a split
weak metacirculant.
Suppose that Γ is also a metacirculant. Then by the definition of metacirculant, Γ has
two automorphisms σ, τ satisfying the following conditions:
(1) ⟨σ⟩ is semiregular and has t orbits on V (Γ),
(2) τ normalizes ⟨σ⟩ and cyclically permutes the t orbits of ⟨σ⟩,
(3) τ has a cycle of size t in its cycle decomposition.
By Lemma 4.2, we have A = G and Γ is a non-Cayley graph. Since |G| = |V (Γ)|p, we
must have G = ⟨σ, τ⟩. Thus, τ t ̸= 1 and hence ⟨τ t⟩ ∼= Cp. Since G is transitive on V (Γ),
we may assume that ⟨τ t⟩ = GH = H = ⟨ap
n−1
cp
m−1⟩. By Lemma 4.1(3), we have σ ∈
⟨a, b⟩, and so o(σ) | pnq. Consequently, pm | o(τ). Let x ∈ ⟨τ⟩ be of order pm such that
xp
m−1
= τ t = ap
n−1
cp
m−1
. Then x ∈ CG(H), and by Lemma 4.1(1), CG(H) = ⟨a, c⟩.
So x = aicj for some integers i, j. However, xp
m−1
= aip
m−1
cjp
m−1
= cjp
m−1
due to
m > n. A contradiction occurs. Thus, Γ is not a metacirculant.
ORCID iDs
Li Cui https://orcid.org/0000-0002-7470-6511
Jin-Xin Zhou https://orcid.org/0000-0002-8353-896X
References
[1] B. Alspach and T. D. Parsons, A construction for vertex-transitive graphs, Can. J. Math. 34
(1982), 307–318, doi:10.4153/cjm-1982-020-8.
[2] Y.-G. Baik, Y. Feng, H.-S. Sim and M. Xu, On the normality of Cayley graphs of abelian
groups, Algebra Colloq. 5 (1998), 297–304.
[3] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I: The user language, J.
Symb. Comput. 24 (1997), 235–265, doi:10.1006/jsco.1996.0125.
[4] L. Cui and J.-X. Zhou, Split metacyclic groups and pseudo metacirculants, submitted.
[5] L. Cui and J.-X. Zhou, Absolutely split metacyclic groups and weak metacirculants, J. Algebra
Appl. 18 (2019), 13, doi:10.1142/s0219498819501172, id/No 1950117.
[6] L. Cui and J.-X. Zhou, A construction of pseudo metacirculants, Discrete Math. 343 (2020), 8,
doi:10.1016/j.disc.2020.111830, id/No 111830.
[7] Y.-Q. Feng, Z.-P. Lu and M.-Y. Xu, Automorphism groups of Cayley digraphs, in: Applications
of Group Theory to Combinatorics, Taylor & Francis Group, London, pp. 13–25, 2008, doi:
10.1201/9780203885765-5.
[8] C. Godsil, On the full automorphism group of a graph, Combinatorica 1 (1981), 243–256,
doi:10.1007/bf02579330.
504 Ars Math. Contemp. 22 (2022) #P3.08 / 489–504
[9] I. Kovács, Classifying arc-transitive circulants, J. Algebr. Comb. 20 (2004), 353–358, doi:10.
1023/b:jaco.0000048519.27295.3b.
[10] C. H. Li, S. J. Song and D. J. Wang, A characterization of metacirculants, J. Comb. Theory, Ser.
A 120 (2013), 39–48, doi:10.1016/j.jcta.2012.06.010.
[11] D. Marušič and P. Šparl, On quartic half-arc-transitive metacirculants, J. Algebr. Comb. 28
(2008), 365–395, doi:10.1007/s10801-007-0107-y.
[12] Y. Wang, Y.-Q. Feng and J.-X. Zhou, Cayley digraphs of 2-genetic groups of odd prime-power
order, J. Comb. Theory, Ser. A 143 (2016), 88–106, doi:10.1016/j.jcta.2016.05.001.
[13] M. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182
(1998), 309–319, doi:10.1016/s0012-365x(97)00152-0.
[14] J.-X. Zhou and S. Zhou, Weak metacirculants of odd prime power order, J. Comb. Theory, Ser.
A 155 (2018), 225–243, doi:10.1016/j.jcta.2017.11.007.