VERTEX-TRANSITIVE GRAPHS WITH ARBITRARILY

VERTEX-TRANSITIVE GRAPHS WITH ARBITRARILY LARGE VERTEX-STABILIZER Marston D. E. Conder and Cameron G. Walker

Abstract:

A construction is given for an innite family f;n g of nite vertex-transitive non-Cayley graphs of xed valency with the property that the order of the vertex-stabilizer in the smallest vertex-transitive group of automorphisms of ;n is a strictly increasing function of n. For each n the graph is 4-valent and arc-transitive, with automorphism group a symmetric group of large prime degree p > 22n +2 . The construction uses Sierpinski's gasket to produce generating permutations for the vertex-stabilizer (a large 2-group).

x1. Introduction In this paper we provide a positive answer to the following question:

Does there exist an innite family f;n g of nite vertex-transitive graphs of xed valency such that if Gn is the smallest group of automorphisms of ;n which acts transitively on vertices of ;n then the order of the stabilizer in Gn of a vertex of ;n increases as n !1?

This question was brought to our attention by Josef S iran , and attributed to Chris Godsil. We are grateful to Chris Godsil, Robert Jajcay, Cheryl Praeger, Josef S iran and Spyros Magliveras for communications on the topic. Without the condition on the smallest vertex-transitive group of automorphisms, the construction of vertex-transitive graphs with automorphism group having an arbitrarily large vertex-stabilizer is relatively easy. For example take the 4-valent graph with vertices 0 1 2 : : : 2n ; 1, and edges joining each of 2i and 2i + 1 to each of 2i + 2 and 2i + 3 modulo 2n (for 0  i 2m+2. Note that since there are innitely many primes congruent to 1 modulo 4, such a prime N can always be found.

De nition 3.1 : Let G be the symmetric group SN , in its natural action on the set

f1 2 : : : N g, and in this group dene three elements a b and c as follows: a = (1 3)(2 4)(5 7)(6 8) : : : (2m ; 7 2m ; 5)(2m ; 6 2m ; 4)(2m ; 3 2m ; 1)(2m ; 2 2m) (2m +1 N )(2m +2 2m +3)(2m +4 2m +5) : : : (N ; 5 N ; 4)(N ; 3 N ; 2), b = (1 2m +1 2 2m +2)(3 5)(4 6)(7 9)(8 10) : : : (2m ; 5 2m ; 3)(2m ; 4 2m ; 2) (2m +3 2m +4)(2m +5 2m +6) : : : (N ; 4 N ; 3)(N ; 2 N ; 1), and c = (1 2)(3 4)(5 6)(7 8) : : : (2m ; 7 2m ; 6)(2m ; 5 2m ; 4)(2m ; 3 2m ; 2)(2m ; 1 2m). 4

Note that a b and c are even permutations of orders 2 4 and 2 respectively, with b having a single 4-cycle, (N ;7)=2 transpositions, and three xed points (viz. 2m ; 1 2m and N ). This is perhaps best seen with the help of the diagram below, in which tranpositions of a are represented by thin lines, while cycles of b are represented by heavy polygons and dots, and the eect of c corresponds to reection in the vertical axis of symmetry:

Fig.2: Generating Permutations Within the alternating group AN the permutations a and b generate a subgroup which is transitive (since the diagram is connected) and indeed primitive (since N is prime). Also it may be seen from the diagram that c;1 ac = a while c;1 bc = b;1 .

Next, note that c is the product of transpositions tj = (2j ; 1 2j ) for 1  j  m. Letting this product correspond to the rst m entries of the mth row of the modied Sierpinski gasket (as illustrated in Fig.1), we work backwards to dene a sequence of additional permutations c1  c2 : : : cm as follows:

De nition 3.2 : For 1  i  m dene ci =

m Y j =i

tj dj;im;i , where tj is the transposition

(2j ; 1 2j ) in SN , and dj;im;i = m;i C(j;i)=2] (as given in Denition 2.1) for i  j  m. 5

For example, c1 = t1 t2 : : : tm = c while c2 = t2 t3t6 t7 : : : tm;6 tm;5tm;2 tm;1  and nally cm;1 = tm;1 tm while cm = tm = (2m ; 1 2m). Note that each of the ci other than cm is even, since drs = dr+1s and drr  0 mod 2 whenever r is even. Also note that the exponents dj;im;i may be taken modulo 2, and the only of these required for the denition of ci are the ones which appear in the rst half of the (m;i+1)st row of our modied Sierpinski gasket. The choice of these exponents is the key to our construction of the graph ;n , as will be seen later with the help of the two observations below.

Lemma 3.3 : For 1  j  m we have

8 b2 t > tj+1 : j ;1 tm

if j = 1 if j is odd if j 2 f2 4 : : : m ; 2g (a) a;1tj a = ttjj+1 if j is even if j 2 f3 5 : : : m ; 1g ;1 if j = m . Proof. This is a simple consequence of the denitions of the permutations a and b and the transpositions tj .



Lemma 3.4 : For 1  i  m we have 

odd (a) a;1ci a = cci c ifif ii is is even i;1 i

82 < b c1 if i = 1 (b) b;1ci b = : ci if i 2 f2 4 : : : mg ci;1 ci if i 2 f3 5 : : : m ; 1g .

Proof. First as c1 = c the cases where i = 1 for parts (a) and (b) are consequences of our earlier observation that c;1 ac = a and c;1 bc = b;1 . Next if i is odd then we have m m Y Y a;1 ci a = a;1 ( tj dj;im;i )a = (a;1tj a)dj;im;i

= = = =

by the denition of ci

j =i j =i m mY ;1 Y tj;1 dj;im;i by 3.3 and since the tj commute tj+1 dj;im;i even j =i+1 odd j =i mY ;1 m Y d j +1 ; im ; i tj+1 tj;1 dj;1;im;i as drs = dr+1s when r is even even j =i+1 odd j =i m mY ;1 Y d k ; im ; i tk dk;im;i after relabelling subscripts tk odd k=i even k=i+1 m Y tk dk;im;i = ci : k=i

A very similar argument shows b;1 ci b = ci when i is even, noting that since m ; i is even we have dm;im;i = m;i C(m;i)=2  0 mod 2 in this case. 6

On the other hand if i is even then also

ci;1 ci = = = = = =

m Y

m m Y Y d 0 m ; i +1 d d j ; i +1 m ; i +1 j ; im ; i tj tj = ti;1 tj dj;i+1m;i+1 +dj;im;i j =i;1 j =i j =i mY ;1 m Y +dj ;im;i d ; i +1 j ; im tj dj;i+1m;i+1 +dj;im;i ti;1 tj even j =i odd j =i+1 m mY ;1 Y d j ; i ; 1 m ; i ti;1 tj tj dj;i+1m;i by Lemma 2.2 even j =i+2 odd j =i+1 m mY ;1 Y tk;1dk;im;i after relabelling subscripts ti;1 tk+1 dk;im;i even k=i+2 odd k=i+1 m m ; 1 Y Y tk;1 dk;im;i tk+1 dk;im;i since d0m;i = 1 even k=i odd k=i+1 m Y ;1 dk;im;i (a tk a) = a;1 ci a : k=i

Finally the same procedure shows that if i is odd and i > 1 then ci;1 ci = b;1ci b, noting that dm;i+1m;i+1  0 mod 2 in this case and recalling that b;1tm b = tm. This completes the proof of Lemma 3.4. We are now in a position to dene the graph ;n .

De nition 3.5 : Let ;n be the graph ;(G H a), where G = SN and a 2 G are as dened

earlier, and H is the subgroup of SN generated by b and the m involutions c1  c2 : : : cm. Recall that the vertices of ;(G H a) may be taken as the right cosets of H in G, with two cosets Hx and Hy joined by an edge whenever xy;1 2 HaH , and the group G acts as a group of automorphisms of this graph by right multiplication on cosets. Before investigating its properties in the next Section, we make some observations about G, H and a.

Lemma 3.6 : The group G = SN is generated by the subgroup H and the element a. Proof. The subgroup generated by H and a contains b and a and is therefore transitive, of prime degree and therefore primitive, and contains the single 2-cycle cm = (2m;1 2m) and is therefore equal to SN (by Jordan's theorem 8 x13]).

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Lemma 3.7 : The subgroup generated by c1  c2 : : : cm is elementary abelian of order 2m. Proof. First the ci are products of commuting transpositions tj , and therefore generate an elementary abelian 2-group. Also by denition of the ci (and the properties of the Sierpinski gasket), for 1  k  m the subgroup generated by ck  : : : cm moves only the points 2k;1 2k : : : 2m;1 2m, and it follows that each such subgroup has order 2m;k+1.

Lemma 3.8 : The subgroup H is a 2-group of order 2m+2.

Proof. By part (b) of Lemma 3.4 we see that b2 centralizes hc1  c2 : : : cmi, and further, that b normalizes hb2 c1  c2 : : : cmi, therefore H = hb c1 c2 : : : cmi has order 2m+2.

Lemma 3.9 : H \ a;1Ha has index 4 in H .

Proof. By part (a) of Lemma 3.4 we see that a normalizes hc1  c2 : : : cmi, which is a subgroup of index 4 in H with transversal f1 b b2 b3g. Since each of a;1ba a;1b2a and a;1 b3a moves the point N , none of these three elements can lie in H and it follows that H \ a;1Ha = hc1 c2 : : : cmi.

x4. Properties of the graphs We begin with some of the basic properties which follow from the construction described in the previous Section:

Proposition 4.1 : The graph ;n dened in Section 3 has N !=2m+2 vertices, and is connected, 4-valent and arc-transitive, with SN as an arc-transitive group of automorphisms. The girth of ;n is 4.

Proof. Most of this follows from Lemmas 3.6, 3.8 and 3.9, and the fact that G = SN acts on ;n = ;(G H a) by right multiplication, with trivial kernel since AN is simple. Finally ab2 = (1 3 2 4)(2m+1 N 2m+2 2m+3) has order 4, therefore ;(G H a), has a circuit of length 4 with vertices H Ha Hab2a and Hab2ab2a (in order), and thus ;n has girth 4.

The stabilizer in G = SN of the vertex H is the subgroup H itself, of order 2m+2. This group acts transitively on the neighbour-set ;n (H ) = fHa Hab Hab2 Hab3g , and the stabilizer of the arc (H Ha) is the subgroup H \ a;1 Ha = hc1  c2 : : : cmi. As the latter subgroup is centralized by b2 , it is also the stabilizer of the arc (H Hab2), while the element c1 interchanges the other two arcs (H Hab) and (H Hab3). It follows that H acts as the dihedral group D4 of order 8 on the neighbour-set ;n (H ). 8

From Lemma 3.4 it is easy to see that every element g 2 G may be written in the form g = hw where h 2 hc1  c2 : : : cmi and w is a word in the elements a and b. In particular, every vertex of ;n is of the form Hw where w 2 ha bi, and it follows easily that the subgroup ha bi is transitive on vertices of ;n . (In fact we will see in Proposition 4.4 that ha bi = AN and this is the smallest vertex-transitive group of automorphisms of ;n .) Next for any positive integer s, an s-arc in a graph ; is an ordered (s + 1)-tuple of vertices (v0  v1 v2 : : : vs ) of ; such that any two consecutive vi are adjacent in ; and any three consecutive vi are distinct. By what we have seen above, the group SN does not act transitively on 2-arcs (ordered paths of length 2) in ;n , because of the existence of circuits of length 4. More of the nature of the action of SN on ;n is revealed below:

Lemma 4.2 : For 1  k  m, the subgroup of SN generated by ck  : : : cm is the stabilizer k;1

in SN of the k-arc (H (ab) k2 ]  H (ab) k2 ];1 : : : Hab H Ha Haba : : : H (ab) 2 ]a) of ;n . Moreover, this subgroup xes all vertices at distance up to  k2 ] from H but not all vertices at distance  k2 ] + 1 from H in ;n .

Proof. We use the following corollary of Lemma 3.4: if h 2 hci  ci+1 : : : cmi where i 3, and e is any integer, then abeh = h0 abe for some h0 2 hci;2  ci;1 : : :  cmi. Now consider any s-arc in ;n of the form (H Habe1  Habe2 abe1  : : : Habes :: abe2 abe1 ) where 1  s  m=2 and e1 2 f0 1 2 3g while ei 2 f1 2 3g for 2  i  s. The stabilizer in SN of the initial 1-arc (H Habe1 ) is either hc1  c2 : : : cmi or hb2 c1 c2 : : :  cmi, depending on whether e1 2 f0 2g or e1 2 f1 3g. By induction on s (and using Lemma 3.4) it follows that the stabilizer of the given s-arc contains hc2s;1 c2s : : : cmi or hc2s c2s+1 : : :  cmi, again depending on whether e1 2 f0 2g or e1 2 f1 3g. On the other hand, in the case where ei = 1 for all i 2, Lemma 3.4 shows that c2s;2 moves the nal vertex H (ab)s;1a of the s-arc (H Ha Haba : : : H (ab)s;2a H (ab)s;1a) to Hab3(ab)s;2a and similarly c2s;1 moves the nal vertex of the s-arc (H Hab H (ab)2 : : : H (ab)s;1 Habs) to Hab3(ab)s;1. The result follows by taking s =  k2 ].

Proposition 4.3 : The symmetric group SN is the full automorphism group of ;n . Proof. Assume the contrary, and let J be a subgroup of Aut ;n which properly contains G = SN  and let K be the stabilizer in J of the vertex labelled H in ;n : Then K contains H , and since ;n contains circuits of length 4 the subgroup K acts as D4 on ;n (H ) in the same way as H , and by induction on the length of a stabilizer sequence it follows that K is also a 2-group. Now let  2 K n H be an automorphism of ;n , chosen

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so that H has index 2 in hH i, in which case  normalizes H . By Lemma 4.2, and multiplying by a suitable element of H if necessary, we may suppose that  xes every vertex at distance up to m=2 from H in ;n , and further, that  xes the (m + 1)-arc (H (ab)m=2 H (ab)m=2;1 : : : Hab H Ha Haba : : : H (ab)m=2a). In particular, we may suppose that hi is the stabilizer in J of this (m + 1)-arc, which we will denote by M . Now M is xed by aa, and so aa 2 hi n H , which in turn implies ;1aa 2 H and therefore ;1a = a (since the stabilizer in H of M is trivial). It follows that  normalizes hH ai = G = SN , and then since Aut SN = SN (see 3 xII.5]) there must be a permutation  2 SN corresponding to  which normalizes H and centralizes a. But further, ;1 bb;1 lies in H and also xes every vertex Hx at distance up to m=2 from H in ;n (since  xes Hx and Hxb), so from Lemma 4.2 we deduce that ;1 bb;1 2 hcm i and therefore ;1b = ;1b 2 fb cmbg. The case ;1 b = cm b is impossible however, since b is even while cmb is odd, and therefore  centralizes b. Consequently  centralizes both a and b, from which it follows that  xes every vertex of ;n , a contradiction. Hence proof.

Proposition 4.4 : The smallest vertex-transitive group of automorphisms of ;n is the alternating group AN , and in this group the stabilizer of a vertex of ;n has order 2m+1.

Proof. First AN contains the vertex-transitive subgroup ha bi, since a and b are even, and hence AN itself is vertex-transitive on ;n . On the other hand, in the group SN the stabilizer of a vertex of ;n has order 2m+2 , and therefore any vertex-transitive subgroup has index at most 2m+2 in SN . But SN has no proper subgroup of index less than N other than AN (see 3 xII.5]), however, so by choice of N > 2m+2 the alternating group AN is the smallest vertex-transitive subgroup of Aut ;n : In particular, since each of the permutations ci other than cm is even, we nd ha bi = ha b c1 c2 : : : cm;1i = AN  and the stabilizer in AN of a vertex of ;n is hb c1 c2 : : : cm;1 i which has order 2m+1 .

x5. Final remarks Thus we have proved the following:

Theorem 5.1 : For every positive integer n there exists a nite arc-transitive 4-valent

graph ;n , with the property that in the smallest vertex-transitive group of automorphisms of ;n , the stabilizer of a vertex has order 22n +1.

In fact for each n there are innitely many such graphs, since in our construction there 10

are innitely many possibilities for the prime degree N . Moreover, the construction works also when N is not prime, as primitivity of the group generated by H and a may be proved using conjugates of the 2-cycle cm or the double transposition (ab2)2. Unfortunately since the action of the vertex-stabilizer is imprimitive on the neighbourset, our construction has little eect on progress towards settling Weiss's conjecture (see 6] and 7]). This conjecture remains open.

Acknowledgments The rst author gratefully acknowledges support from the Marsden Fund (grant number 95-UOA-MIS-0173), and the second author gratefully acknowledges support from a University of Auckland Postgraduate Scholarship.

References 1. Marston Conder, An innite family of 5-arc-transitive cubic graphs, Ars Combinatoria 25A (1988), 95{108. 2. Marston Conder & Peter Lorimer, Automorphism groups of symmetric graphs of valency 3, J. Combinatorial Theory Ser. B 47 (1989), 60{72. 3. B. Huppert, Endliche Gruppen I (Springer-Verlag, 1983). 4. R.C. Miller, The trivalent symmetric graphs of girth at most 6, J. Combinatorial Theory Ser. B 10 (1971), 163{182. 5. H.-O. Peitgen, H. J#urgens & D. Saupe, Chaos and Fractals: New Frontiers of Science (Springer-Verlag, 1992). 6. C.E. Praeger, Finite primitive permutation groups: a survey, Groups { Canberra 1989 (Springer Lecture Notes in Mathematics, vol. 1456, 1990), pp. 63{84.

7. Richard Weiss, s-transitive graphs, in: Algebraic Methods in Graph Theory (Coll. Math. Soc. Janos Bolyai 25, 1984), 827{847. 8. Helmut Wielandt, Finite Permutation Groups (Academic Press, New York, 1964). Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand

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