Chapter 5
Quotients of vertex-transitive graphs
This chapter will study some operations on vertex-transitive graphs.
5.1
Quotient graphs
Let Γ = (V, E) be a digraph. For a partition B of V , the quotient ΓB is defined as the
digraph with vertex set B such that, for any two vertices B, C ∈ B , B is connected to
C if and only if there exist u ∈ B and v ∈ C which are adjacent in Γ.
It is clear that a quotient of a connected graph is connected.
Proposition 5.1 A graph is connected if and only if it is a quotient of a cycle.
Let Γ = (V, E) be a vertex-transitive graph. Assume that G ≤ Aut Γ is transitive on
V . If B is a G-invariant partition of V , then the quotient ΓB is called an imprimitive
quotient (relative to G). In the case, Γ is called an extension of ΓB ; if in addition Γ and
ΓB have the same valency then Γ is called an imprimitive cover of ΓB .
Proposition 5.2 Let Γ = Cn be a cycle of size n. Then Γ has a non-trivial imprimitive
quotient if and only if n is not a prime. Each of imprimitive quotients of Γ is a cycle or
a single edge graph K2 .
Example 5.3 Let Γ = 6K2 , 6 disjoint copies of K2 .
43
44
Quotients of vertex-transitive graphs
There is a group X ≤ Aut Γ such that X ∼
= Z6 × Z2 is transitive on V . There exists
an X -partition B with two blocks of size 6, such that ΓB ∼
= K2 .
There exists a group Y ≤ Aut Γ such that Y ∼
= D12 is transitive on V , and there
0
exists an X -partition B with 6 blocks of size 2 such that ΓB0 ∼
= C6 .
Let Z = A4 , acting on {1, 2, 3, 4}. Let g = (12)(34). Then it is easily shown that
∼
Γ = Cay(Z, {g}). Thus Z acts on V regularly. Let H < Z be isomorphic to Z3 , and let
B 00 be the set of right cosets of H in Z . Then ΓB00 ∼
= K4 .
Therefore, the disconnected graph Γ has connected imprimitive quotient graphs isomorphic to K2 , K3 , K4 , C6 .
Exercise 5.4 Let Γ = 6K2 . Find all imprimitive quotients of Γ.
5.2
Imprimitive quotients of Cayley graphs
Let Γ be Petersen graph. Then Γ can be represented as a coset graph of A5 . Let G ∼
= A5
act on Ω = {1, 2, 3, 4, 5}. Let H = h(123), (23)(45)i, and let g = (24)(35). Then
Γ∼
= Γ(G, H, HgH).
It is known that Γ is not a Cayley graph. However, we will see that Γ is an imprimitive
quotient of a Cayley graph, for example, the Cayley graph Cay(G, {g}).
Theorem 5.5 Each vertex-transitive graph is an imprimitive quotient of a Cayley graph.
Proof. Let Γ = (V, E) be a vertex-transitive graph, and assume that G ≤ Aut Γ is
transitive on V . Then there exist a subgroup H < G and a subset S ⊂ G such that
Γ = Γ(G, H, HSH).
Let R be a subset of G such that S ⊆ R ⊆ HSH . Let
Σ = Cay(G, R).
Let B = [G : H]. We claim that
Γ∼
= ΣB .
Draft typeset on June 8, 2001.
Groups & Graphs
45
First we notice that Γ and ΣB have the same vertex set [G : H]. We need to prove
that for any Hx, Hy ∈ [G : H],
Hx is connected in Γ to Hy if and only if Hx is connected in ΣB to Hy .
Assume that Hx is connected in Γ to Hy . By definition, yx−1 ∈ HSH , so that
−1
yx−1 = h1 sh2 for some h1 , h2 ∈ H and some s ∈ S . Thus (h−1
= s ∈ S ⊆ R,
1 y)(h2 x)
−1
so the vertex h2 x is connected in Σ to the vertex h1 y . As h2 x ∈ Hx and h−1
1 y ∈ Hy ,
by definition, Hx is connected in ΣB to Hy .
Assume now that Hx is connected in ΣB to Hy . Then there exists some h1 x ∈ Hx
which is connected in Σ to some h2y ∈ Hy . By definition, (h2 y)(h1x)−1 ∈ R ⊆ HSH .
It follows that yx−1 ∈ HSH , and hence Hx is connected in Γ to Hy .
Therefore, Γ ∼
= ΣB , as desired.
2
Exercise 5.6 (1) Prove that Petersen graph is an imprimitive quotient of a connected
arc-transitive Cayley graph.
(2) Prove that the complete graph K6 is an imprimitive quotient of the icosahedron.
Draft typeset on June 8, 2001.
46
5.3
Quotients of vertex-transitive graphs
Normal quotients
Let Γ = (V, E), and assume that G ≤ Aut Γ is transitive on V . Let N be a normal
subgroup of G which is intransitive on V , and let B be the set of N -orbits in V . Then
B is a partition of V . Denote by ΓN the quotient graph ΓB .
Lemma 5.7 Using notation defined above,
(1) B is a G-invariant partition;
(2) for any B, C ∈ B , the vertex B is connected in ΓN to the vertex C if and only if
each u ∈ B is connected in Γ to some v ∈ C .
Proof. Suppose that, for some B ∈ B and some g ∈ G, B g 6= B . Then there exist
some u ∈ B and some C ∈ B such that ug ∈ C 6= B . Hence C = (ug )N = (uN )g = B g ,
and so B is G-invariant.
Assume that the vertex B ∈ B is connected in ΓN to the vertex C ∈ B . Then some
u ∈ B is connected in Γ to v ∈ C . Since N is transitive on B , for any u0 ∈ B , there
exists g ∈ N such that ug = u0 . Since g fixes C (setwise), we have v g ∈ C . Thus u0 is
2
connected to v g .
The quotient ΓN is called a normal quotient induced by N . In particular, a normal
quotient is an imprimitive quotient. The property given in Lemma 5.7 (2) is not shared by
imprimitive quotient, for example, 6K2 is A4 -vertex-transitive, and has an imprimitive
quotient isomorphic to K4 .
It is known that imprimitive quotients of arc-transitive graphs are arc-transitive. For
normal quotients, we have a stronger result, given in the next section.
Draft typeset on June 8, 2001.
Groups & Graphs
5.4
47
Normal quotients of s-arc-transitive graphs
Recall that a graph Γ = (V, E) is said to be (G, s)-arc-transitive if G ≤ Aut Γ is transitive
on the set of all s-arcs of Γ. For example, complete graphs and hypercubes are 2-arctransitive but not 3-arc-transitive; while regular complete bipartite graphs and odd graphs
are 3-arc-transitive but not 4-arc-transitive.
Exercise 5.8 Let Γ be a Cayley graph of an abelian group. Prove that Γ is not 4-arctransitive unless Γ is a cycle.
A graph Γ is said to be G-locally-primitive if G ≤ Aut Γ acts transitive on V Γ and
Gα acts primitively on Γ(α).
Lemma 5.9 Let Γ be a G-locally-primitive graph. Then for any intransitive normal
subgroup N of G, either N has exactly two orbits in V Γ and ΓN ∼
= K2 , or Γ is a cover
of ΓN , ΓN is (G/N)-locally-primitive, and G/N ≤ Aut ΓN .
Proof. Let B be the set of N -orbits in V Γ, and let K be the kernel of G acting on
B . Then N ≤ K , and K G. Thus Kα Gα , where α ∈ V Γ. Since Γ is (G, 2)-arctransitive, Gα acts 2-transitively on the neighborhood Γ(α). So Kα acts on Γ(α) either
trivially or transitively. Suppose that Kα 6= 1. It then follows that Kα acts non-trivially
on Γ(α), so that Kα acts on Γ(α) transitively. Thus the valency of ΓN is equal to 1,
and so ΓN ∼
= K2 , which is a contradiction since N has at least 3 orbits in V . Therefore,
Kα = 1, that is, K acts semiregularly on V . In particular, K = N , that is, N is the
kernel of G acting on V . Hence G/N may be identified with a subgroup of Aut ΓN , and
so ΓN is (G/N)-locally-primitive.
2
Perhaps the most important property of taking normal quotients of graphs is that the
s-arc-transitivity of a graph is inherited by normal quotients.
Theorem 5.10 (Praeger 1992) Let Γ be a connected (G, s)-arc-transitive graph for some
s ≥ 2. Assume that N is a normal subgroup of G which has at least three orbits in V .
Then the normal quotient ΓN is a (G/N, s)-arc transitive graph.
Proof. Let B0 , B1, . . . , Bs and C0 , C1, . . . , Cs be two s-arcs of the normal quotient
graph ΓN . Then for each i with 0 ≤ i ≤ s − 1, Bi is adjacent to Bi+1 and Ci is adjacent
Draft typeset on June 8, 2001.
48
Quotients of vertex-transitive graphs
to Ci+1 , in ΓN . By Lemma 5.7, there exist u0 , u1, . . . , us and v0, v1, . . . , vs such that for
each i ∈ {0, 1, . . . , s − 1}, ui ∈ Bi is adjacent to ui+1 ∈ Bi+1 , and vi ∈ Ci adjacent to
vi+1 ∈ Ci+1 . Since Γ is (G, s)-arc-transitive, there exists g ∈ G such that ugi = vi for all
i ∈ {0, 1, . . . , s}. Since B is G-invariant, we have Big = Ci , and hence g maps the s-arc:
B0 , B1, . . . , Bs to the s-arc: C0, C1 , . . . , Cs . Thus G induces a transitive action on the
set of s-arcs of ΓN . By Lemma 5.9, G/N may be identified with a subgroup of Aut ΓN .
2
Exercise 5.11 Let Γ = Cay(G, S) be connected and undirected. Assume that Aut(G, S)
is 2-transitive on S . Prove
(i) Γ is 2-arc-transitive,
(ii) all elements of S have order 2,
(iii) if in addition G is abelian, then G is an elementary abelian 2-group.
Theorem 5.10 suggests to study ‘minimal’ s-arc-transitive graphs, that is, s-arctransitive graphs which have no non-trivial normal quotients.
Let Γ be connected and undirected. Suppose that Γ is (G, s)-arc-transitive for some
G ≤ Aut Γ, and assume further that Γ is a minimal s-arc-transitive graph with respect
to G. Then either
(1) each non-trivial normal subgroup of G is transitive on V Γ, that is, G is quasiprimitive on V Γ; or
(2) some non-trivial normal subgroup of G has exactly two orbits in V Γ, that is, G is
bi-quasiprimitive on V Γ.
Draft typeset on June 8, 2001.
Groups & Graphs
5.5
49
Quasiprimitive 2-arc-transitive graphs
Let G be a quasiprimitive permutation group on Ω, and let N = soc(G), the socle of G.
Let M1, . . . , Mk be all minimal normal subgroups of G. It is easily shown that
N = M1 × · · · × Mk ,
for some k ≥ 1.
Suppose that k ≥ 2. Then since Mi is transitive, we have that Mi is nonabelian. Let
L = M2 × · · · × Mk . Then N = M1 × L. Since M1 is transitive on Ω, for any β ∈ Ω,
there exists x ∈ M1 such that β = αx . Therefore,
Lβ = Lαx = x−1 Lα x = Lα ,
so that Lα fixes all points of Ω. Hence Lα = 1. As L is transitive on Ω, we obtain that
L is regular. Since each Mi is transitive, it follows that L = M2 and k = 2. Similarly,
M1 is regular on Ω. Therefore, N = M1 × M2 such that Mi is nonabelian and regular
on Ω; in particular, |M1| = |M2 | = |Ω|.
Since N is transitive on Ω, it follows that |Nα | = |M1 |. An element of Nα may be
written as xx0 , where x ∈ M1 and x0 ∈ M2 . Suppose that for some x ∈ M1 , there exist
two different elements x0, y 0 ∈ M2 such that xx0, xy 0 ∈ Nα . Then M2 has a non-identity
element x0(y 0)−1 = xx0(xy 0)−1 ∈ Nα , which is a contradiction since M2 is regular on Ω.
Further, since |Nα | = |M1 |, for each element x ∈ M1 , there exists exactly one element
x0 ∈ M2 such that xx0 ∈ Nα . Let σ be a map from M1 to M2 defined:
x → x0 ,
where x ∈ M1 and xx0 ∈ Nα .
Then it is easily shown that σ is an isomorphism from M1 to M2 . In particular, M1 ∼
= M2 .
Thus we have
Lemma 5.12 Let G be a quasiprimitive permutation group on Ω. Then either soc(G)
is a minimal normal subgroup of G, or soc(G) = M1 × M2 such that M1 ∼
= M2 , and the
Mi are nonabelian and regular on Ω.
The next lemma characterizes minimal normal subgroups of a group.
Draft typeset on June 8, 2001.
50
Quotients of vertex-transitive graphs
Lemma 5.13 If M is a minimal normal subgroup of a group G, then M = T1 × · · · × Tk
such that T1 ∼
= ··· ∼
= Tk is a simple group.
In particular, if G is a quasiprimitive permutation group, then soc(G) = T1 ×· · ·×Tl ∼
=T,
∼
∼
where l ≥ 1 and T1 = · · · = Tl is a simple group. Quasiprimitive permutation groups G
are categorized into 8 types by O’Nan-Scott’s theorem (Praeger 1992), where N = soc(G):
• N is abelian (HA),
• N is non-abelian,
– N = M1 × M2 ,
∗ Mi is simple (HS),
∗ Mi is non-simple (HC),
– N is a minimal normal subgroup of G,
∗ N is regular (TW),
∗ N is non-regular,
· N is simple (AS),
· N is non-simple,
• Nα 6∼
= T r (PA),
• Nα ∼
= Tr,
- r = 1 (SD),
- r > 1 (CD).
A further analysis leads to the next theorem.
Theorem 5.14 (Praeger 1992) Let Γ be a connected undirected graph, and assume
further that G ≤ Aut Γ is quasiprimitive on V Γ and that Γ is (G, 2)-arc-transitive.
Then G is of type HA, TW, AS or PA.
Draft typeset on June 8, 2001.