J Algebr Comb (2014) 40:179 185 DOI 101007/s10801-013-0482-5 On ntranstve graph-restrctve permutaton groups Pablo Spga Gabrel Verret Receved: 5 December 2012 / Accepted: 5 October 2013 / Publshed onlne: 26 October 2013 Sprnger Scence+Busness Meda New York 2013 Abstract Let Γ be a fnte connected G-vertex-transtve graph and let v be a vertex of Γ If the permutaton group nduced by the acton of the vertex-stablser G v on the neghbourhood Γ(v) s permutaton somorphc to L, then (Γ, G) s sad to be locally L A permutaton group L s graph-restrctve f there exsts a constant c(l) such that, for every locally L par (Γ, G) and a vertex v of Γ, the nequalty G v c(l) holds We show that an ntranstve group s graph-restrctve f and only f t s semregular Keywords Vertex-transtve Graph-restrctve Semregular 1 Introducton A graph Γ s sad to be G-vertex-transtve f G s a subgroup of Aut(Γ ) actng transtvely on the vertex set of Γ LetΓ be a fnte, connected, smple G-vertex-transtve graph and let v be a vertex of Γ If the permutaton group nduced by the acton of the vertex-stablser G v on the neghbourhood Γ(v)s permutaton somorphc to L, then (Γ, G) s sad to be locally L Note that, up to permutaton somorphsm, L does not depend on the choce of v, and, moreover, the degree of L s equal to the valency of Γ In[6, p 499], the second author ntroduced the followng defnton P Spga Departmento d Matematca Pura e Applcata, Unversty of Mlano-Bcocca, Va Cozz 53, 20126 Mlano, Italy e-mal: pablospga@unmbt G Verret (B) Faculty of Mathematcs, Natural Scences and Info Tech, Unversty of Prmorska, Glagoljaška 8, 6000 Koper, Slovena e-mal: gabrelverret@fmfun-ljs
180 J Algebr Comb (2014) 40:179 185 Defnton 11 A permutaton group L s graph-restrctve f there exsts a constant c(l) such that, for every locally L par (Γ, G) and for every vertex v of Γ, the nequalty G v c(l) holds To be precse, Defnton 11 s a generalsaton of the defnton from [6], where the group L s assumed to be transtve The problem of determnng whch transtve permutaton groups are graph-restrctve was also proposed n [6] A survey of the state of ths problem can be found n [3], where t was conjectured ([3, Conjecture 3]) that a transtve permutaton group s graph-restrctve f and only f t s semprmtve (A permutaton group s sad to be semregular f each of ts pont-stablsers s trval and semprmtve f each of ts normal subgroups s ether transtve or semregular) Havng removed the requrement of transtvty from the defnton of graphrestrctve, t s then natural to try to determne whch ntranstve permutaton groups are graph-restrctve The man result of ths note s a complete soluton to ths problem (whch we dd not expect, gven the abundance and relatve lack of structure of ntranstve groups) Theorem 12 An ntranstve and graph-restrctve permutaton group s semregular It s easly seen that a semregular permutaton group s graph-restrctve Indeed, f L s a semregular permutaton group of degree d and (Γ, G) s locally L, then for every arc vw of Γ the group G vw fxes the neghbourhood Γ(v) pontwse Snce Γ s connected, t follows that G vw = 1 and hence G v Γ(v) =d and L s graph-restrctve Thus Theorem 12 provdes a charactersaton of ntranstve graph-restrctve groups Corollary 13 An ntranstve permutaton group s graph-restrctve f and only f t s semregular Note that an ntranstve permutaton group s semregular f and only f t s semprmtve In partcular, Corollary 13 completely settles the ntranstve verson of [3, Conjecture 3], gvng remarkable new evdence towards ts veracty 2 Proof of Theorem 12 For the remander of ths paper, let L be a permutaton group on a fnte set Ω whch s nether transtve nor semregular We show that L s not graph-restrctve, from whch Theorem 12 follows 21 The constructon Let ω 1,,ω k Ω be a set of representatves of the orbts of L on Ω Snce L s not transtve, k 2 and, snce L s not semregular, we may assume wthout loss of
J Algebr Comb (2014) 40:179 185 181 Fg 1 Subgroup lattce for Sect 21 generalty that L ω1 1 Let n 2 be an nteger and let b 1 be the automorphsm of L ω1 L n ω 1 = L n+1 ω 1 defned by (x 0,x 1,,x n 1,x n ) b 1 = (x n,x n 1,,x 1,x 0 ), for each (x 0,,x n ) L n+1 ω 1 Smlarly, let b 2 be the automorphsm of L n ω 1 defned by (x 1,x 2,,x n 1,x n ) b 2 = (x n,x n 1,,x 2,x 1 ), for each (x 1,,x n ) L n ω 1 Clearly, b 1 and b 2 are nvolutons, that s, b 2 1 = 1 and b 2 2 = 1 Now, let b 3,, b k be cyclc groups of order 2 and consder the followng abstract groups: A := L L n ω 1, B 1 := ( L ω1 L n ω 1 ) b1, B 2 := L ω2 ( L n ω 1 b 2 ), B := L ω L n ω 1 b, C := L ω L n ω 1, for {3,,k}, for {1,,k}, where b 1,,b k / A For every {1,,k}, there s an obvous embeddng of C n both A and B Hence, n what follows, we regard C as a subgroup of both A and B Note that, for each {1,,k}, wehavea B = C, B : C =2 and A : C = L : L ω (see Fg 1) Lemma 21 The core of C 1 C k n A s 1 L n ω 1 Proof Let K be the core of C 1 C k n A Then K = (C 1 C k ) a = (Lω1 L ωk ) L a A a A( n ) a ω 1 Recall that L s a permutaton group on Ω and that ω 1,,ω k are representatves of the orbts of L on Ω We thus fnd that L ω1 L ωk s core-free n L and hence K = 1 L n ω 1
182 J Algebr Comb (2014) 40:179 185 Fg 2 The graph of groups Y Let T be the group gven by generators and relators T := A,B 1,,B k R, where R conssts only of the relatons n A,B 1,,B k together wth the dentfcaton of C n A and B, for every {1,,k} We wll obtan some basc propertes of T, whch can be deduced from any textbook on groups actng on graphs, such as [1, 2, 5] We have adopted the notaton and termnology of [2] and wll follow closely [2, I4] Usng ths termnology, the group T s exactly the fundamental group of the graph of groups Y shownnfg2 The vertces of Y are A, B 1,,B k and, for each {1,,k}, there s a (drected) edge C from A to B It follows from [2, I46] that the mages of A,B 1,,B k,c 1,,C k n T are somorphc to A,B 1,,B k,c 1,,C k, respectvely Ths allows us to dentfy A,B 1,,B k,c 1,,C k wth ther somorphc mages n T n what follows In partcular, for each {1,,k} we stll have the equaltes A B = C, B : C =2 and A : C = L : L ω n T LetT be the graph wth vertex set V T = T/A T/B 1 T/B k, (where denotes the dsjont unon) and edge-set ET = { {Ax, B x} x T, {1,k} } 22 Results about the group T and the graph T Clearly, the acton of T by rght multplcaton on V T nduces a group of automorphsms of T Under ths acton, the group T has exactly k + 1 orbts on V T, namely T/A, T/B 1,,T/B k, and k orbts on ET wth representatves {A,B 1 },,{A,B k } Ths nduces a (k + 1)-partton of the graph T Observe that the set of neghbours of A n T/B s {B a a A} As A : (A B ) = A: C = L : L w, we see that A has L : L w neghbours n T/B It follows that A has valency k =1 L : L ω = Ω A symmetrc argument, wth the roles of A and B reversed, shows that B has valency B : C =2 In partcular, T s a (2, Ω )-regular graph Lemma 22 The stablser of the vertex A n T s the subgroup A and the kernel of the acton on the neghbourhood of A s 1 L n ω 1 Proof The defnton of T mmedately shows that A s the stablser n T of the vertex A Moreover, the neghbourhood of A s T (A) ={B a {1,,k},a A}
J Algebr Comb (2014) 40:179 185 183 Fg 3 Illustraton for the proof of Lemma 23 Let K be the kernel of the acton of A on T (A) and let x K Clearly, B ax = B a f and only f axa 1 B, that s, axa 1 A B = C It follows by Lemma 21 that K = 1 L n ω 1 One of the most mportant and fundamental propertes of T s that t s a tree (see [2, I44]) We now deduce some consequences from ths pvotal result Lemma 23 For each {1,,k}, we have A A b = C Proof We argue by contradcton and assume that A A b C for some {1,,k} As B : C =2, we see that B normalses C and hence C <A A b In partcular, there exst a,a A \ C wth a = a b = b 1 ab It follows that A, B, Ab and B ab are dstnct vertces of T Now, the defnton of T shows that (A, B,Ab,B ab,ab 1 ab = A) s a cycle of length 4 n T (see Fg 3) Ths contradcts the fact that T s a tree and concludes the proof Lemma 24 The subgroup A s core-free n T In partcular, the group T acts fathfully on T/A Proof Let N be the core of A n T From Lemma 23, we obtan N C 1 C k, and t follows from Lemma 21 that N 1 L n ω 1 By constructon, the group b 1,b 2 nduces a transtve permutaton group on the n + 1 coordnates of L ω1 L n ω 1 As the frst coordnate of the elements of N s 1 and as N s nvarant under b 1,b 2,we see that every coordnate of N must be equal to 1, that s, N = 1 The lemma now follows As every vertex of T not n T/A has valency 2, we see that T s the subdvson graph of a tree T 0 wth vertex set T/A and valency Ω Clearly, T acts transtvely and, n vew of Lemma 24, fathfully on the vertces of T 0 The tree T 0 and the group T are our man ngredents for the proof of Theorem 12 (The auxlary graph T was ntroduced manly to make t more convenent to apply the results from [2]) Lemma 25 The stablser n T of the vertex A of T 0 s the subgroup A and the acton nduced by A on ts neghbourhood s permutaton somorphc to the acton of L on Ω Proof Let π : A L be the natural projecton onto the frst coordnate In other words, f a = (a 0,a 1,,a n ) A, wth a 0 L and wth a 1,,a n L ω1, then π(a) = a 0 Clearly, the kernel of π s 1 L n ω 1, whch by Lemma 22 s also the kernel of the acton of A on the neghbourhood of the vertex A Denote by T 0 (A) the neghbourhood of A n T 0 The defntons of T and T 0 yeld T 0 (A) ={Ab a
184 J Algebr Comb (2014) 40:179 185 {1,,k},a A} Letϕ : T 0 (A) Ω be the mappng ϕ : Ab a ω π(a) Weshow that ϕ s well-defned and njectve Indeed, Ab a = Ab a for some a,a A f and only f Ab a(a ) 1 b 1 = A, that s, a(a ) 1 A A b By Lemma 23, A A b = C Clearly, a(a ) 1 C f and only f π(a(a ) 1 ) L ω, that s, ω π(a) = ω π(a ) Ths shows that ϕ s well-defned and that t s njectve Clearly, ϕ s surjectve and hence t s a bjecton For every a,x A and for every {1,,k}, wehaveϕ((ab a)x) = (ϕ(ab a)) π(x) Asϕ s a bjecton, ths shows that the acton of A on T 0 (A) s permutaton somorphc to the acton of L on Ω Recall that a group X s sad to be resdually fnte f there exsts a famly {X m } m N of normal subgroups of fnte ndex n X wth m N X m = 1 Lemma 26 The group T s resdually fnte Proof As the groups A, B 1,,B k are fnte, t follows from [2, I47] that there exst a fnte group F and a group homomorphsm π : T F wth Ker π A = 1 and Ker π B = 1 for each {1,,k} Wrte K = Ker π Snce F s fnte, we have T : K < Snce K T, K A = 1 and K B = 1, t follows that the only element of K fxng a vertex of T s 1 and hence, by [2, I54], K s a free group In partcular, K s resdually fnte (see [4, 619] for example) It follows that there exsts a famly {K m } m N of normal subgroups of fnte ndex n K wth m N K m = 1 Let T m be the core of K m n T As T : K m = T : K K : K m <, wesee that T : T m < Moreover, snce T m K m,wehave m N T m = 1 and the lemma follows 23 Proof of Theorem 12 We now recall the defnton of a normal quotent of a graph Let Γ be a G-vertextranstve graph and let N be a normal subgroup of G Letv N denote the N-orbt contanng v VΓ Then the normal quotent Γ/N s the graph whose vertces are the N-orbts on VΓ, wth an edge between dstnct vertces v N and w N f and only f there s an edge {v,w } of Γ for some v v N and some w w N Observe that the group G/N acts transtvely on the graph Γ/N Lemma 27 There exsts a locally L par (Γ n,g n ) such that the stablser of a vertex of Γ n n G n has order L L ω1 n Proof By Lemma 26, T s resdually fnte and hence there exsts a famly {T m } m N of normal subgroups of fnte ndex n T wth m N T m = 1 Consder the set X = { a 1 b 1 a 2 b j a 3 a 1,a 2,a 3 A,,j {1,,k} } Observe that snce A s fnte, so s X In partcular, as m N T m = 1 and 1 X, there exsts m N wth X T m = 1 Let G n = T/T m and Γ n = T 0 /T m As T : T m <,
J Algebr Comb (2014) 40:179 185 185 the group G n and the graph Γ n are fnte Note that Γ n s connected and G n -vertextranstve We frst show that Γ n has valency Ω We argue by contradcton and suppose that Γ n has valency less than Ω Itfollows from the defnton of normal quotent that the vertex A of T 0 must have two dstnct neghbours n the same T m -orbt Recall that the neghbourhood of A n T 0 s {Ab a {1,,k},a A} In partcular, Ab a Ab j a and Ab an = Ab j a,for some, j {1,,k}, a,a A and n T m It follows that n a 1 b 1 Ab j a X and hence n X T m = 1, whch s a contradcton Let K be the kernel of the acton of G n on VΓ n Snce the valency of Γ n equals the valency of T 0, we see that Γ n s a regular cover of T 0 Snce Γ n s connected, t follows that K acts semregularly on VΓ n and hence K = T m By Lemma 25, (T 0,T) s locally L and hence so s (Γ n,g n ) Fnally, the stablser of the vertex AT m of Γ n s AT m /T m = A/(A Tm ) = A, whch has order A = L L ω1 n Proof of Theorem 12 By Lemma 27, for every natural nteger n 2, there exsts a locally L par (Γ n,g n ) wth (G n ) v = L L ω1 n,forv VΓ n As L ω1 > 1, ths shows that L s not graph-restrctve References 1 Dcks, W, Dunwoody, MJ: Groups Actngs on Graphs Cambrdge Studes n Advanced Mathematcs, vol 17 Cambrdge Unversty Press, Cambrdge (1989) 2 Goldschmdt, D: Graphs and groups In: Delgado, A, Goldschmdt, D, Stellmacher, B (eds) Groups and Graphs: New Results and Methods DMV Semnar, vol 6 Brkhäuser, Basel (1985) 3 Potočnk, P, Spga, P, Verret, G: On graph-restrctve permutaton groups J Comb Theory, Ser B 102, 820 831 (2012) 4 Robnson, DJS: A Course n the Theory of Groups Sprnger, New York (1980) 5 Serre, JP: Trees Sprnger, New York (1980) 6 Verret, G: On the order of arc-stablzers n arc-transtve graphs Bull Aust Math Soc 80, 498 505 (2009)