On the point-stabiliser in a transitive permutation group

Transcription

1 On the point-stabiliser in a transitive permutation group Primož Potočnik 1 Department of Mathematics University of Auckland Private Bag Auckland, New Zealand potocnik@math.auckland.ac.nz and Steve Wilson Department of Mathematics and Statistics Northern Arizona University Box 5717 Flagstaff, AZ , USA stephen.wilson@nau.edu Proposed running head: Point-stabiliser The correspondence should be addressed to: Primož Potočnik Institute of Mathematics, Physics, and Mechanics Jadranska 19 SI-1000 Ljubljana Slovenia Preferably via primoz.potocnik@fmf.uni-lj.si 1 This author gratefully acknowledges support by the US Department of State and the Fulbright Scholar Programme, and thanks Northern Arizona University for hospitality during his visit in Spring

2 Abstract If V is a (possibly infinite) set, G a permutation group on V, v V, and Ω is an orbit of the stabiliser G v, let G Ω v denote the permutation group induced by the action of G v on Ω, and let N be the normaliser of G in Sym(V ). In this article, we discuss a relationship between the structures of G v and G Ω v. Among other things, we prove that if G is transitive, G v is finite, and the N-orbital {(v g, u g ) u Ω, g N(G)} is strongly connected (when viewed as a digraph on V ), then every simple homomorphic image of a subgroup of G v is also a homomorphic image of a subgroup of G Ω v. To demonstrate that the assumption on finiteness of G v cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G v is isomorphic to the modular group PSL(2, Z) = C 2 C 3. Keywords: permutation group, digraph, graph, stabiliser, modular group. 2

3 1 Introduction It is well known that every transitive permutation group G with a point-stabiliser H G is permutation isomorphic to the natural action of G on the cosets of H in G by multiplication. It is therefore not surprising that determining the size and the structure of the point-stabiliser in a transitive permutation group is one of the central problems in the theory of permutation groups. Special attention has been given to the relationship between the size of a pointstabiliser G v, and the size of an orbit Ω {v} of G v (called a G-suborbit). In particular, the question was raised by Sims, whether the size of G v in a finite primitive permutation group is bounded above in terms of the size of G-suborbits, and this was finally answered affirmatively in [3]. As we show in this paper, the structure of a vertex stabiliser G v seems to remain strongly related to the structure of the permutation group G Ω v (induced by the action of G v on a G v -orbit Ω) even if the group G is not primitive or finite. Let V be a (possibly infinite) set, let G Sym(V ), let v V, and let Ω be an orbit of the stabiliser G v. Further, let K = Ker(G v G Ω v ) denote the group of all permutations in G v fixing every element of Ω. Then G Ω v (viewed as an abstract group) is isomorphic to the quotient group G v /K. Since, in principle, the kernel K might be a rather complicated group, the vertex stabiliser G v could be considerably more complicated than G Ω v. However, it can be shown that under certain assumptions on G and Ω, the structure of G v (at least in the sense of the Jordan Hölder decomposition), is fully determined by the structure of the group G Ω v. For example, the following generalisation of an old result of Camille Jordan [6, Théorème 395, pp ] has been proved recently by Betten, Delandtsheer, Niemeyer, and Praeger. (Since Jordan s original result also appeared in [10] as Theorem 18.2, it is often mistakenly attributed to Wielandt. We thank the anonymous referee for this historical remark.) Theorem 1.1 ([1, Theorem 2.1]) Suppose that G is a transitive subgroup of Sym(V ) whose normaliser in Sym(V ) is finite and primitive. Let v V and let Ω V \ {v} be an orbit of G v. Then every composition factor (in the sense of the Jordan Hölder Theorem) of the group G v is isomorphic to a composition factor of some subgroup of G Ω v. It this paper we address the relationship between the groups G v and G Ω v in a slightly more general context of possibly infinite permutation groups with imprimitive normalisers. Even though the topic of the paper is essentially group theoretical, some basic graph theoretical language and techniques will be needed to formulate and prove the results. By a digraph on a vertex set V, we mean any non-empty subset Γ of the set V (2) = (V V )\{(v, v) v V }. For example, if G Sym(V ) is a permutation group, and is a G-orbital (that is, an orbit of G on V (2) ), then can be viewed as a digraph on V. The automorphism group of Γ, denoted by Aut(Γ), is the group of all permutations on V that (when viewed as permutations on V (2) ) preserve Γ. Clearly, a permutation group G Sym(V ) is a subgroup of Aut(Γ) if and only if Γ is a union of G-orbitals. For a set V and v V, consider the function Φ v mapping a digraph Γ V (2) to the neighbourhood Γ(v) = {u (v, u) Γ} of v in Γ. If G is a transitive permutation group on V, then Φ v induces a bijective correspondence between digraphs Γ on V for which G Aut(Γ), and unions of G v -orbits on V \ {v}. In this correspondence, G-orbitals (that is, digraphs on which G acts transitively) correspond 3

4 bijectively to G v -orbits. This shows that every statement concerning the action of G v on a G v -orbit can be expressed in terms of the action of G as a group of digraph automorphisms. A digraph Γ is locally finite if Γ(v) is finite for every v V, and is finite if its vertex set is finite. For a digraph Γ V (2), we let Γ = {(u, v) (v, u) Γ} and call Γ a graph if Γ = Γ. A directed path of length n in Γ from a vertex u V to a vertex v V is a finite sequence u = v 0, v 1,..., v n = v such that (v i 1, v i ) Γ for every i {1,..., n}. A path in Γ is a directed path in Γ Γ. A digraph Γ is (strongly) connected if for every (u, v) V (2) there exists a (directed) path between u and v. For a digraph Γ and for i N, let Γ (i) (v) denote the set of all vertices u for which the shortest directed path from v to u has length i. In particular, Γ (0) (v) = {v} and Γ (1) (v) = Γ(v). Let α be an ordinal number, let H be a subgroup of a group G, let G be a set of groups containing H and contained in G, and let β G β be a surjective function onto G from the set of all ordinals β less than or equal to α. Following [8], we say that G = G 0 G 1 G α = H is a complete descending series between G and H of type α whenever: 1. G β + G β for every ordinal β smaller than α; and 2. β<λ G β = G λ for every limit ordinal number λ smaller than or equal to α. The quotients G β /G β +, β [0, α), shall be referred to as the factors of the complete descending series. A complete descending series between a group G and the trivial group 1, will be called a complete descending series of G. Clearly, if α is a finite ordinal number, then a complete descending series of G is a subnormal series of G in the usual sense. We can now state the first main result of this paper. Theorem 1.2 Let V be an arbitrary set, let v V, let G be a (possibly infinite) transitive subgroup of Sym(V ), let N be the normaliser of G in Sym(V ), let be a G-orbital, and let Γ be the N-orbital containing. If Γ is strongly connected (when viewed as a digraph on V ), then there exists a complete descending series of the vertex-stabiliser G v with each factor isomorphic to a subgroup of G (v) v. Recall that an abstract group is a section of a group G if it is a homomorphic image of a subgroup of G. A section is called simple if it is a simple group. Note that a group is a simple section of a finite group G if and only if it is a composition factor of a subgroup of G. Observe that if, in addition to the assumptions of Theorem 1.2, we assume that G v is finite, we can conclude that every simple section of G v is also a section of G (v) v (see also Corollary 1.5). However, as the following proposition shows, this conclusion fails to be true in general. Proposition 1.3 There exists a transitive permutation group G with an orbital Γ isomorphic to the infinite cubic tree, such that G v is isomorphic to the free product C 2 C 3. 4

5 Since there are infinitely many finite simple groups that can be generated by an involution and an element of order 3 (like A n, n 5, for example), the vertex-stabiliser G v = C2 C 3 has simple sections which are not sections of the group G Γ(v) v = S 3. The details and the proof of Proposition 1.3 are supplied in Example 4.2. In spite of this example, not everything is lost. The following generalisation was suggested by the referee. It is well known that the symmetric group Sym(V ) carries a natural structure of a topological group in which the pointwise stabilisers of finite sets form a fundamental system of open neighbourhoods of the identity (see [2]). We call a permutation group on V closed if it is closed as a subset of Sym(V ). A section of a closed group G Sym(V ) is said to be closed if it is a quotient of a closed subgroup H of G by a closed normal subgroup. In particular, every finite subgroup in Sym(V ) (and every section thereof) is closed. We can now state the following generalisation of Theorem 1.1. Theorem 1.4 Let V be an arbitrary set, let v V, let G be a closed, transitive subgroup of Sym(V ), let N be the normaliser of G in Sym(V ), let be a locally finite G-orbital, and let Γ be the N-orbital containing. If Γ is strongly connected and locally finite, then every closed simple section of G v is a section of G (v) v. Since every finite subgroup of the topological group Sym(V ) is discrete, we have the following corollary of Theorem 1.4. Corollary 1.5 Let G be a transitive permutation group on a set V whose point-stabiliser G v is finite. Let N be the normaliser of G in Sym(V ), let be a G-orbital, and let Γ be the N-orbital containing. If Γ is strongly connected, then every simple section of G v is a section of G (v) v. Finally, observe that Theorem 1.4 indeed implies Theorem 1.1. A finite transitive permutation group G Sym(V ) is primitive if and only if every G-orbital is strongly connected (see, for example, [4, Theorem 3.2A and Lemma 3.2A]). If G, V, v and Ω are as in Theorem 1.1, and N is the normaliser of G in Sym(V ), then the N-orbital Γ = {(v g, u g ) u Ω, g N} is strongly connected (since N is primitive), Γ contains the G-orbital = {(v g, u g ) u Ω, g G}, and (v) = Ω. Hence, by Corollary 1.5, every simple section of G v is also a simple section of G Ω v. Since simple sections of a finite group are nothing but composition factors of its subgroups, the assertion of Theorem 1.1 follows. The methods used in this paper are similar to those outlined by Peter M. Neumann in [7]. In fact, as pointed out by the referee, an appropriate modification of these methods would lead to a simpler proof of Theorem 1.2. Instead of taking this route, we have chosen to take a detour and prove a more general, albeit a rather technical result (Lemma 4.1) which might prove useful when more information about the behaviour of vertex-stabilisers (in possibly intransitive permutation groups) is needed. If nothing else, the notion of a G-leash, upon which our proof of Theorem 1.4 is based, emerges naturally in the study of finite symmetric graphs. 5

6 2 Preliminaries Let us first review some basic notions and facts which will be needed later. Since we assume no finiteness conditions on the groups, sets and digraphs that occur, some use of transfinite set theory is unavoidable. We refer the reader to [5] for a detailed account of ordinal numbers and other set theoretical notions not defined here. The usual well-ordering of the set of ordinals will be denoted by <, and α β will mean that either α < β or α = β. We shall use the symbol α + to denote the successor of an ordinal α. The minimal ordinal number shall be denoted by 0 and its successor by 1. Accordingly, finite ordinals shall be identified with non-negative integers 0, 1, 2,.... For two ordinals α and β, we let [α, β) denote the set of ordinals η such that α η < β, and by [α, β] we mean [α, β + ). (Note that by this definition, β = [0, β) and β + = [0, β] for every ordinal β.) Recall that a non-zero ordinal is a limit ordinal if it is not the successor of any other ordinal. We shall assume that every set can be well ordered, and consequently, that for every set V there exists a well-ordering of V and an order-preserving bijection from V to [0, α). In this case, we shall say that V (together with the well-ordering of V ) gives rise to the ordinal α. As was mention in Section 1, the symmetric group Sym(V ) carries a natural structure of a topological group in which the pointwise stabilisers of finite sets form a fundamental system of open neighbourhoods of the identity. If the set V is finite or countable (that is, if V = {v 1, v 2, v 3,...}), then this topology is induced by a metric on Sym(V ) defined by d(g, h) = max{d (g, h), d (g 1, h 1 )}, where d (g, h) = { 0, if g = h 2 i, if v g j = vh j for all j < i but v i g v h i A nice feature of this metric is that it turns Sym(V ) into a complete metric space in which every translation λ a : g ag, for g, a G, is an isometry (see [2, Subsection 2.4]). Suppose now that a subgroup G Sym(V ) admits a connected, locally finite G-orbital Γ. Local finiteness of Γ implies that the cardinalities a i of the balls Σ i = j i (Γ Γ ) (j) (v) are finite for every v V and i N. Since Γ is connected, we may assume that V = {v i : i N} where Σ i = {v 1,..., v ai } for every i N. In particular, V is countable and so Sym(V ) admits a natural structure of a complete metric space, as described above. Observe that, relative to this metric, the diameter of each pointwise stabiliser G i = G (Σi ) is at most 2 a i. Moreover, since each G i has a finite index in G v = G 0, the vertex stabiliser G v is totally bounded. If, in addition, G is closed in Sym(V ), then G v is a complete, totally bounded metric space, and thus a compact topological group. Let us summarise this: Proposition 2.1 Let G be a transitive permutation group on a set V, let Γ be a strongly connected, locally finite G-orbital, and let G v denote the stabiliser of a point v V. If G is closed, relative to the natural topology on Sym(V ), then G v is compact. The following lemma is needed in the proof of Theorem 1.4. Recall that a section of a topological group is closed if it is a quotient of a closed subgroup by a closed normal subgroup. 6

7 Lemma 2.2 Let S be a closed simple section of a compact topological group G and let G = G 0 G 1 G 2 be a descending subnormal series of closed subgroups of G, such that i=0 G i = 1. Then S is a section of the quotient group G i /G i+1 for some integer i 0. Proof. By definition, there exists a subgroup H G and a group epimorphism π : H S whose kernel K is closed in H. For every integer i 0, let H i = G i H and S i = π(h i ). Since K and all H i are closed in a compact group G, they are also compact. Moreover, by taking a natural topology on the quotient group S, the projection π becomes a continuous epimorphism from a compact set, showing that the groups S i are compact (relative to the natural topology on S). Clearly, for every i 0, the group H i+1 is normal in H i and S i+1 is normal in S i. Moreover, since each H i is a subgroup of G i, we see that i=0 H i = 1. We shall now show that i=0 S i = 1. (Note that this last statement can fail to be true if the assumption on compactness is omitted.) Take an arbitrary element x i=0 S i, and for each i 0 choose y i H i such that π(y i ) = x. Since H is compact, the sequence (y i ) i 0 has a subsequence (y ij ) j 0 which converges to some y H. Since H i are closed, it follows that y i=0 H i, and so y = 1. Since π is continuous, we see that 1 = π(y) = π(lim j y ij ) = lim j π(y ij ) = x. Hence i=0 S i = 1, as claimed. Let i be the maximal integer for which S i = S. (Note that in view of the fact that i=0 S i = 1 such an integer indeed exists.) Since S is simple and S S i+1 S i = S, it follows that S i+1 = 1. This clearly implies that H i+1 K and H i /(H i K) = S. But then the Third Isomorphism Theorem implies that S = (H i /H i+1 )/((H i K)/H i+1 ), and therefore S is a homomorphic image of the group H i /H i+1. By the Second Isomorphism Theorem, we have that H i /H i+1 = (G i H)/(G i+1 H) = (G i H)G i+1 /G i+1, which is a subgroup of G i /G i+1. Hence S is a section of G i /G i+1. 3 A leash of a permutation group A base of a permutation group G Sym(V ) is a subset B V such that the pointwise stabiliser G (B) of B in G is trivial (see for example [4, Section 3.3]). Bases of permutation groups prove to be a useful tool in the analysis of finite primitive permutation groups. In Definition 3.1 below, we introduce a variation of this notion. For a digraph Γ V (2) and a set of vertices U in V, we let Γ(U) denote the union of U and all the neighbourhoods Γ(u), for u U. (Note that by this definition Γ(u) Γ({u}); in fact, Γ({u}) = {u} Γ(u).) By Γ[U] we denote the subdigraph of Γ spanned by U, that is, the digraph Γ U (2) on the vertex set U. Definition 3.1 Let Γ be a digraph on a vertex set V, let γ be an ordinal number, and let δ : [0, γ) V, η v η, be an arbitrary function. If for every η [1, γ) there exists η < η such that v η Γ(v η ), then we say that δ is a directed sequence in Γ of type γ. The vertex δ(0) is then called the beginning of δ. If Γ(Im(δ)) is a base of a group G Aut(Γ), then we say that δ is a G-leash in Γ. By N = {0, 1,...} we mean the set of natural numbers, well-ordered in the usual sense. (The ordinal arising from this well-ordering of N is denoted by ω.) 7

8 Lemma 3.2 Let Γ be a strongly connected digraph on a vertex set V, let G Aut(Γ), and let v V. Further, for every i N, let γ i be an ordinal arising from a well-ordering of Γ (i) (v). Then there exists a G-leash of type i N γ i with the beginning in v. In particular, if Γ is locally finite, then there exists a G-leash of type ω. Proof. For every i N, let < i be a well-ordering of Γ (i) (v) giving rise to the ordinal γ i. Since Γ is strongly connected, we have V = i N Γ(i) (v). Hence, there exists a well-ordering < on V giving rise to the ordinal number γ = i N γ i. (That is, for each u Γ (i) (V ) and w Γ (j) (V ), we let u < w whenever either i < j, or i = j and u < i w.) Moreover, there exists an order-preserving bijection δ : [0, γ) V. It is now clear that δ is a directed sequence in Γ. Since the image Im(δ) of δ is V, this directed sequence is also a G-leash for every G Aut(Γ). 4 A lemma and proofs of the results We will now state and prove a general lemma about vertex-stabilisers in groups of digraph automorphisms. Theorem 1.4 will then follow easily. Lemma 4.1 Let α be an ordinal number, let {Γ β β [0, α)} be a set of pairwise disjoint digraphs on a vertex set V, let G {Aut(Γ β ) β [0, α)}, and let Γ = {Γ β β [0, α)}. Suppose that there exists a G-leash δ in Γ of some type γ. For η [0, γ), let v η = δ(η), and for η [0, γ], let V η = {v ι ι [0, η)}. Further, let G 0 = G v0, and for every η [1, γ], let G η = G (Γ(Vη)). Then the following holds: (i) G v0 = G 0 G 1 G γ = 1 is a complete descending series for G v0 of type γ; (ii) G η /G η + = G Γ(vη) η G Γ(vη) v η for every η [0, γ). Further, for η [0, γ) and β [1, α], let G 0,η = G η and G β,η = {G η G (Γι(v η)) ι [0, β)}. Then the following holds: (iii) G η = G 0,η G 1,η G α,η = G η + is a complete descending series between G η and G η + of type α, with all the terms G β,η, β [0, α], normal in G η ; (iv) G β,η /G β +,η = G Γ β(v η) β,η G Γ β(v η) v η for every β [0, α). Consequently, there exists a complete descending series for G v0 of type αγ, with each of its factors isomorphic to a subgroup of some group G Γ β(u) u, β [0, α), u V. Proof. We shall first prove that for every η [0, γ) we have G η G vη. This is clearly true for η = 0. Now, if 0 < η, then, by definition of a directed sequence, there exists η < η such that v η Γ(v η ). On the other hand, G η is contained in every point-wise stabiliser G (Γ(vι)) for ι < η, and hence also in G (Γ(vη )). Therefore, G η fixes v η, as claimed. In particular, there exists a natural action of G η on Γ(v η ) (as well as on Γ β (v η ) for every β [0, α)). Since G η + = G η G (Γ(vη)) = Ker(G η G Γ(vη) η ), it follows that G η + G η and G η /G η + = G Γ(vη) η, completing the proof of part (ii). 8

9 Further, by the definition of a G-leash, it is obvious that G γ = {G (Γ(vι)) ι [0, γ)} = G (Γ(Vγ)) = G (Γ(Im(δ))) = 1. Now let λ [1, γ], and observe that η [0,λ) G η = η [0,λ) G (Γ(V η)) = G (Γ(Vλ )) = G λ. Part (i) now follows if we apply this equality for every limit ordinal number λ γ. Similarly, for any η [0, γ) we have G α,η = G η G ( {Γι(vη) ι [0,α)}) = G η G (Γ(vη)) = G η +. Observe also that for every λ [1, α], we have β [0,λ) G β,η = G λ,η. This fact (when applied for limit ordinals λ) is needed in the proof of part (iii). Now let β [0, α). Then G β +,η can be written either as G η G ( {Γι(vη) ι [0,β]}), or as G β,η G (Γβ (v η)). This shows that G β +,η = Ker(G η G {Γι(vη) ι [0,β]} η ) = Ker(G β,η G Γ β(v η) β,η ). In particular, G β +,η is normal in G β,η as well as in G η. Moreover, G β,η /G β +,η = G Γ β(v η) β,η, as claimed in part (iv). To complete the proof of part (iii), we need to show that G λ,η is normal in G η for every limit ordinal λ α. But, since G λ,η is the intersection of all G β +,η, β < λ, and since the latter are proven to be normal in G η, so is G λ,η. The last assertion of the lemma now follows easily from parts (i) - (iv). Proof of Theorem 1.2. Let V, v, G, N,, and Γ be as in the statement of Theorem 1.2. (That is, let let V be an arbitrary set, let v V, let G be a (possibly infinite) transitive subgroup of Sym(V ), let N be the normaliser of G in Sym(V ), let be a G-orbital, and let Γ be the N-orbital containing.) Suppose that Γ is strongly connected. Since G is normal in N(G), every element of the set D = { g g N(G)} is a G- orbital. Moreover, the sets in D are pairwise disjoint and their union is Γ. Let α be the ordinal number arising from a well-ordering of D. Then we can label the elements of D by the ordinals β [0, α), that is, D = {Γ β β [0, α)}. Since G is a transitive subgroup of N, the group N can be written as a product of G and the stabiliser N v. Consequently, D = { g g N v }. For u V and g N v consider the group G g (u) u = G u /G ( g ({u})). Since G is transitive, there exists an element h G such that u h = v. Observe that the conjugation by hg 1 maps G u to G v and G ( g ({u})) to G ( ({v})), and thus induces an isomorphism between the groups G g (u) u G Γ β u (u), for u V and β [0, α), are isomorphic to G (v) v. and G (v) v. Hence all the groups By Lemma 3.2, there exists a G-leash δ : [0, γ) V, such that δ(0) = v. Moreover, by Lemma 4.1, there exists a complete descending series for G v with each of its factors isomorphic to a subgroup of some group G Γ β(u) u proves Theorem 1.2., which is isomorphic to G (v) v, as shown above. This Proof of Theorem 1.4. By Theorem 1.2, there exists a complete descending series G v G 1 G 2 G α = 1 of the stabiliser G v, with each factor isomorphic to a subgroup of G (v) v. Since Γ is locally finite, this series can be chosen to have type ω (that is, α = ω). In view of Proposition 2.1, the stabiliser G v is compact (relative to the natural topology on Sym(V )), and hence, by Lemma 2.2, every closed simple section of G v is isomorphic to a section of a factor of the series ( ). Since such a factor is isomorphic to a subgroup of G (v) v closed simple section of G v is a section of the group G (v) v. 9 ( ), it follows that every

10 Example 4.2 Here we give an example of a permutation group the existence of which has been claimed in Proposition 1.3. Let Z denote the ring of integers, let Z[ 1 2 ] = { a : a, k Z}, 2 k let H = SL(2, Z), and let [ ] 0 1 s = GL(2, Z[ 1 [ d ]) and K = SL(2, Z)s = { 2 c ] [ ] a b : SL(2, Z)}. 2b a c d Observe that H K = Γ 0 (2), where Γ 0 (2) denotes the group of matrices in SL(2, Z) whose bottom-left entry is divisible by 2. It follows from [9, Corollary 2, pp. 80] that G 0 = H, K is the group SL(2, Z[ 1 2 ]), and what is more, G 0 is isomorphic to the free product of H and K amalgamated over Γ 0 (2). Let σ be the automorphism of G 0 induced by the conjugation by s, and let G = G 0, σ viewed as a subgroup of Hol(G 0 ). Since s 2 belongs to the centre of GL(2, Z[ 1 2 ]), we have that σ 2 = 1, and so [G : G 0 ] = 2. Let V be the set of the right cosets of H in G, and consider the transitive action of G on V by right multiplication. The kernel of this action is the core of H in G. Let us now show that core G (H) coincides with the centre Z = { I, I} of the group G 0 = SL(2, Z[ 1 2 ]). Observe first that Z H and Z G, implying that Z core G(H). To prove that core G (H) Z, take an arbitrary g core G (H) represented as a matrix in SL(2, Z) with entries a, b, c, d Z, and observe that for every k Z its conjugate [ 2 k k ] 1 [ a b c d ] [ 2 k k ] [ = a 2 2k ] b 2 2k c d is also an element of core G (H) SL(2, Z). But then b = c = 0, and so g { I, I}. Hence core G (H) = Z, as claimed. The transitive permutation group G Sym(V ) induced by the action of G on V is therefore isomorphic to G/Z = PSL(2, Z[ 1 2 ]) σ. Since the pointstabiliser of H V in G is H = SL(2, Z), the point-stabiliser of H in G is isomorphic to the group H/Z = PSL(2, Z), which is known to be isomorphic to the free product C 2 C 3. To conclude the example (and prove Proposition 1.3) it remains to show that G has an orbit on V (2) which is isomorphic to the infinite cubic tree. Let Γ V (2) be the G-orbit containing the element (H, Hσ). Since σ 2 = 1, we have that Γ = Γ, and so Γ can be viewed as a graph on V. Since the point-stabiliser of Hσ V in G is the group H σ = H s = K, it follows that the valency of Γ is [H : (H K)] = [SL(2, Z) : Γ 0 (2)] = 3. Finally, since G 0 = H (H K) K it follows that Γ has no cycles, and is therefore a tree, as claimed. 5 Some implications of Lemma 4.1 The statement of Lemma 4.1 can be applied in many situations. A few of them are considered below in a series of remarks. Throughout this section the notation and all the assumptions of Lemma 4.1 are maintained without further notice. In addition, we let v = v 0. We start with a few remarks regarding different finiteness conditions. Remark 1. Assume that the ordinal γ is finite (that is, assume that the G-leash in Γ is finite). Then the complete descending series in part (i) of Lemma 4.1 is finite, and hence a subnormal series for G v. If, in addition, Γ is locally finite, then the factors of this normal series are finite. Moreover: 10

11 Corollary 5.1 Let Γ be a disjoint union of finitely many locally finite digraphs Γ 1,..., Γ m on a vertex set V, let v V, and suppose G Aut(Γ i ) for every i {1,..., m}. If there exists a finite G-leash in Γ with its beginning in v, then the following statements hold: (i) G v is a finite group; (ii) if p is a prime divisor of G v, then p divides G Γ i(u) u for some u V and some i {1,..., m}; (iii) if G Γ i(u) u is solvable for every u V and every i {1,..., m}, then G v is solvable; (iv) if Γ i (u) 4 for every u V and every i {1,..., m}, then G v is solvable. Remark 2. Assume now that Γ is a graph. Let η [1, γ) and suppose η < η such that v η Γ(v η ). Since Γ = Γ, also v η Γ(v η ), and thus G Γ(vη) η G Γ(vη) v ηv η = (G Γ(vη) v η ) vη. That is, every factor of the complete descending series for G v in part (i) of Lemma 4.1, with the sole exception of the first one G 1 /G 0 (which is isomorphic to G Γ(v) v ), is a subgroup of a vertex stabiliser in a group Gu Γ(u). This has a further interesting consequence: Suppose that Γ is a graph and that G Γ(u) u is semiregular for every u V. (Recall that a permutation group is semiregular if the stabiliser of every point is trivial.) Then all the factors G β /G β +, β [1, γ) in the series G v = G 0 G 1 G γ = 1 are trivial. Suppose that G 1 is not trivial, and let µ = min{β [2, γ) G β G 1 }. If µ = β + for some β, then 1 = G β /G µ = G 1 /G µ, and thus G µ = G 1, a contradiction. On the other hand, if µ is a limit ordinal, then G µ = β<µ G β = β<µ G 1 = G 1, again contradicting the assumption on µ. Therefore, G 1 is trivial, and so G Γ(v) v following: = G 0 /G 1 = G0 = G v. We may therefore conclude the Corollary 5.2 Let Γ be a connected graph on a vertex set V, and let G Aut(Γ) be such that G Γ(u) u is semiregular for every u V. Then G v Γ(v) = G v for every v V. More generally: Corollary 5.3 Let Γ be a connected, locally finite graph on a vertex set V, let v V, let G be a closed subgroup of Aut(Γ), and let S be a closed simple section of G v. Then S is a section of Gv Γ(v) and w Γ(u)., or it is a section of the point-stabiliser (G Γ(u) u ) w in G Γ(u) u for some u V Proof. As in the proof of Theorem 1.4, it follows that S is a section of one of the factors G β /G β + of the complete descending series G v = G 0 G 1... G ω = 1. If this happens for β = 0, then S is a section of G 0 /G 1 Γ(v) = G v, as asserted. On the other hand, if β 1, then G β /G β + is a subgroup of (G u G w ) Γ(u) for some (u, w) Γ, and the result follows. Remark 3. As was shown in [7, Lemma 2], a finite vertex-transitive digraph is connected if and only if it is strongly connected. This shows that the word strongly connected in the 11

12 statements of Theorem 1.2, Theorem 1.4, and Lemma 3.2 can be substituted by connected, whenever Γ is a finite vertex transitive digraph. Acknowledgements. The authors would like to thank the editors who found a mistake in an earlier version of this paper, to Prof. Marston Conder for his valuable comments, and to the anonymous referee who suggested several generalisations and improvements of our original results. References [1] A. Betten, A. Delandtsheer, A. C. Niemeyer, and C. E. Praeger. On a theorem of Wielandt for finite primitive permutation groups, J. Group Theory 6 (2003), [2] P. J. Cameron, Oligomorphic Permutation Groups, London Math. Soc. Lecture Notes Ser. 152, Cambridge University Press (1990). [3] P. J. Cameron, C. E. Praeger, J. Saxl, and G. M. Seitz. On the Sims conjecture and distance transitive graphs, Bull. London Math. Soc. 15 (1983), [4] J. D. Dixon and B. Mortimer. Permutation Groups, Graduate Texts in Mathematics, vol. 163, Springer, New York (1996). [5] P. R. Halmos. Naive Set Theory, The University Series in Undergraduate Mathematics, D. Van Nostrand, New York (1960). [6] C. Jordan, Traité des substitutions et des équations algébriques, Gauthier Villars, Paris (1870). [7] P. M. Neumann, Finite Permutation Groups, edge-coloured graphs and matrices, in Topics in groups theory and computations (ed. M. P. J. Curran), Academic Press, London (1977). [8] D. J. S. Robinson. Finiteness Conditions and Generalized Soluble Groups, Part 1, Springer, New York (1972). [9] J.-P. Serre, Trees, Springer, New York (1980). [10] H. Wielandt, Finite Permutation Groups, Academic Press, New York (1964). 12