Primitive coherent configurations: On the order of uniprimitive permutation groups

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1 Primitive coheret cofiguratios: O the order of uiprimitive permutatio groups László Babai May 1, 2011 Abstract These otes describe the author s elemetary graph theoretic proof of the early tight exp4 l 2 boud o the order of primitive, ot doubly trasitive permutatio groups A. Math., The expositio icorporates a lemma by V. N. Zemlyacheko that simplifies the proof. The cetral cocept i the proof is primitive coheret cofiguratios, a combiatorial relaxatio of the actio of primitive permutatio groups. The expositio follows the authors 2003 REU lecture; simple observatios are listed as exercises. 1 Large primitive groups For large, the largest four primitive permutatio groups are S ad A, of order about!, ad S 2 k for = k 2 ad Sk S 2 for = k 2, of order about expc l. The classificatio of fiite simple groups allows oe to show that these are the largest ad eve to list the largest dow to size about expl 2 Camero [4], cf. Maróti [5]. We ca do reasoably well with elemetary meas. Theorem 1.1. Assume G S, A G, ad G is primitive. 1. Bochert, 1889 [3] G! +1/2! e 2 log. 2. Wieladt, 1934 [8], Praeger-Saxl, 1980 [7] G < 4 usig otrivial elemetary group theory 3. Babai, 1981 [1] If G is ot doubly trasitive the G < exp4 log 2 usig graph theory ad a simple probabilistic argumet. Based o a 2003 REU lecture; scribe: Tom Hayes. Revised: LB, May

2 4. Babai, 1982 [2] If G is doubly trasitive the G < exp exp c log usig elemetary group theory ad a simple probabilistic argumet 5. Pyber, 1993 [6]If G is doubly trasitive the G < exp c log 3 usig elemetary group theory ad the probabilistic argumet of [2] ad G < exp c log 2 usig additioally a elemetary group theoretic result of Wieladt [8] Exercise 1.2. Doubly trasitive implies primitive. The followig theorem is proved usig the classificatio of fiite simple groups; oe ca get close by elemetary meas. Theorem 1.3. If G S, G A is doubly trasitive the G < 1+log 2. Exercise 1.4. Verify that this boud is essetially tight for PSLd, q ad AGLd, q, actig o the correspodig projective ad affie spaces, resp., where q is fixed ad d. Remarks about symmetry ad regularity: symmerty coditios are give i terms of automorphisms; regularity coditios i terms of umerical parameters. Symmetry coditio imply regularity coditios e. g., vertex-trasitivity is a symmetry coditio, which implies that the graph is regular, a regularity coditio. The coverse is seldom true. We shall defie regularity coditios o a family of edge-colored digraphs which capture some combiatorial cosequeces of primitive group actio. Usig this traslatio, we shall prove a combiatorial result which implies a early optimal upper boud o the order of uiprimitive primitive but ot doubly trasitive permutatio groups. Picture of D 6. R 0 = = {x, x x Ω}, diagoal. Ω Ω = R 0 R 1 R r 1. r =# colors = # orbits of G o Ω Ω. D 6 has rak 4, r = 4. I this case, all orbitals are self-paired. Defiitio 1.5. A orbital Γ of a permutatio group G SymΩ is a orbit of G o the set of ordered pairs Γ Ω Ω. Γ is self-paired whe Γ = Γ 1 i. e., for x, y Γ there exists σ G such that x σ = y ad y σ = x. The rak r of a permutatio group is the umber of its orbitals. Exercise 1.6. If G is doubly trasitive, the rkg = 2. What do the two classes correspod to? Defiitio 1.7. COHERENT CONFIGURATION of rak r: X = Ω; R 0,..., R r 1, R i Ω Ω. Ω Ω = R 0... R r 1. X i = Ω, R i, i th color digraph, called a costituet digraph. The color of a pair x, y is defied as cx, y = i if x, y R i. To be coheret, the followig 3 axioms must be satisfied: A1: The diagoal is = R 0... R i0 1. Equivaletly, cx, x = cy, z y = z. 2

3 A2: i jr j = Ri 1. Termiology: R i is self-paired if R i = R 1, i. e., X i is udirected. A3: p i,j,k x, y R i #{z cx, z = j, cz, y = k} = p i,j,k Defiitio 1.8. For G SymΩ, XG := Ω; orbitals. We refer to these as the group case. Exercise 1.9. XG is a coheret cofiguratio. Exercise G AutXG, the group of color-preservig permutatios. π AutX if x, ycx, y = cx π, y π Remark There exist coheret cofiguratios without a group. I fact, there are expoetially may rak-3 coheret cofiguratios with o automorphisms. Well, we always lose i traslatio. The questio is how much. Exercise The umber of x y walks of a give color-compositio oly depeds o cx, y. E. g., how may walks from x to y of legth 4 are colored red, blue, purple, blue i order? Defiitio X is homogeeous if R 0 = i. e., x, ycx, x = cy, y. Exercise XG is homogeeous G is trasitive. Exercise If X is homogeeous, the every weak compoet of each X i is strogly coected. Exercise If X is homogeeous, the x ii-degree i x = out-degree i x = ρ i ρ i does ot deped o x. So X i is Euleria, ad ideed is regular. By the way, r 1 i=0 ρ i =, sice every vertex is coected to every other icludig itself i the graph X i, whose edge set cotais all 2 ordered pairs. Defiitio X is a primitive coheret cofiguratio if X is homogeeous ad ALL costituet digraphs X i, i 1 are coected. Exercise XG is primitive G is primitive. DO!!! Defiitio X is uiprimitive coheret cofiguratio if X is primitive ad rak 3. Exercise X is uiprimitive G is uiprimitive primitive but ot doubly trasitive. G SymΩ, Ψ Ω. Look at the poitwise stabilizer, G Ψ, = x Ψ G x. If G Ψ = {1}, the G Ψ, i fact G 1... Ψ + 1. Call such a Ψ a base of G. We shall prove, usig oly elemetary graph theoretic argumets, that Theorem If G is uiprimitive, the G < exp4 l 2. i 3

4 Lemma If G is uiprimitive, the Ψ Ω Ψ 4 l ad G Ψ = {1}. Examples: How large is the smallest base for various classes of permutatio groups? Defiitio z distiguishes x ad y if cx, z cy, z. Dx, y = {z cx, z cy, z} is the distiguishig set for x, y. Exercise If X = XG ad z Dx, y, the x, y are ot i the same orbit of G z. Obvious, because the group preserves the colors. Defiitio A distiguishig set of X is ay set Ψ Ω such that x yψ Dx, y. I other words, for every pair x, y, Ψ cotais a elemet which distiguishes them. Exercise For X = XG, if Ψ is a distiguishig set, the Ψ is a base for G. Theorem 1.21 will follow from the followig result. Theorem If X is a uiprimitive coheret cofiguratio, the there exists a distiguishig set Ψ such that Ψ < 4 l. This will be a immediate cosequece of the followig. From ow o, let us always assume X is a uiprimitive coheret cofiguratio. Theorem 1.28 Mai techical theorem. For every x, y, Dx, y /2. Proof: [Mai techical theorem Theorem 1.27]. Pick u 1,..., u m at radom, ad hope that we picked eough to hit each Dx, y. PrDx, y ot hit = 1 exp Dx, y m Dx, y m. Hece, by the Uio Boud, where D mi = mi x y Dx, y. Pr x, ydx, y ot hit < For this, it is sufficiet to show Dmi m exp 4 2 < exp exp + 2 l 1 D mim D mim + 2 l,

5 or equivaletly which follows from m D mi m + 2 l 0 2 l D mi 4 l =: m. The last iequality used the Mai techical theorem, which gives a lower boud o D mi. 2 Mi size of distiguishig sets We sped the rest of this class with provig the Mai techical theorem above. Exercise 2.1. Dx, y depeds oly o cx, y. Notatio 2.2. Let Di := Dx, y, where i = cx, y. X i = Ω; R i. Let X i = Ω; R i R 1 i be the correspodig udirected graph. Lemma 2.3. For i 1, if X i is ot the complete graph, the diamx i = 2. Proof: There exist x, y at distace 2 i X i, because there exist x, z ot adjacet i X i, but X i is coected by primitivity, ad so the third vertex of ay miimal x, z-path is at distace 2 from x. Now take ay u, v Ω, ot adjacet i X i. Need to show: dist X iu, v 2. Need to show: u, v have a commo eighbor i X i. cu, v {i, i 1 }. Implies # commo eighbors of u, v i X i is the same as for x, y. Exercise 2.4. If X is a regular graph of degree ρ ad diameter = 2, the ρ 1. Exercise ρ = 1 uder the above coditios implies ρ {2, 3, 7, 57}. Hit. Figure out a coectio to girth. This exercise is oly for studets who took the first half of this course. Lemma 2.6. i 1ρ i 1 1. Proof: If X i is the complete graph, the ρ i = 1/2 ad we are doe. Otherwise, use Lemma 2.3 ad Exercise 2.4. Notatio 2.7. We shal cosider the average distiguishig umber Also, let ρ max := max i ρ i. D = x y Dx, y. 1 5

6 Lemma 2.8. D ρ max Proof: Cout the umber of triples x, y, z such that z / Dx, y. This meas cx, z = cy, z = ρ i for some i. This is D = r 1 i=1 ρ iρ i 1 1 r 1 i=1 ρ ρ i 1 max < ρ max. 1 Lemma 2.9. Di dist X j idj. Proof: Let x 0, x 1,..., x d be a i X j path where cx 0, x d = i. Dx 0, x d d i=1 Dx i 1, x i. The size o the left side is Di; all stes o the right side have size Dj. Notatio diami := diamx i. Corollary Dj D/ diamj. Proof: Need: D diamjdj. Pick i such that Di D. The dist X j i diamx j = diamj. Corollary If diami = 2 the Di /2. Lemma 2.13 Zemlyacheko. If diami 3 the Di ρ i /3. Proof: Let x, y, z, w be a shortest path from z to w i X i. Let X i x = { eighbors of x i color i }. Claim X i x Dx, w ad Di Dx, w /3. The Lemma is immediate from the followig claim: The claim is easy: if some X i -eighbor u of x did ot distiguish x from w the cu, w = cu, x = i ±, so x u w would be a X i -path of legth 2, cotradictig the assumptio that dist i x, w = 3. Now Dx, w 3Di by Lemma 2.9. Exercise Suppose there exists a edge of color h betwee X i x ad X j x. The there exist at least maxρ i, ρ j such edges. Lemma h 0 xx distiguishes at least 1 pairs of color h. Proof: Let us costruct a graph H usig the set V = {0, 1,..., r 1} of colors as vertex set. Let wi, j be the umber of edges of color h or h 1 from X i x to X j x. Put a edge betwee i ad j if wi, j 0; assig weight wi, j to this edge. It follows from Exerciseco-ex that if there is a {i, j} edge the wi, j maxρ i, ρ j. 6

7 H is a coected graph. This follows from the primitivity of X why?. Let T be a spaig tree of H. Let us oriet T away from vertex color 0. x distiguishes τ edges of color h, where τ := total weight of edges of T. τ = i j Corollary Di 1/ρ i. Proof: wi, j r 1 ρ j = ρ j = 1. i j Cout the triples N = {x, y, z cx, y = i, z Dx, y} i two differet ways. Cout by x, y. The umber of pairs x, y such that cx, y = i is ρ i. For each such pair, there are Di choices for z. Thus, N = ρ i Di. Now cout by z. There are choices for z. Give z, there are at least 1 pairs x, y distiguished by z. Thus N = ρ i Di 1, ad so i=1 ρ i Di 1. Corollary If diami 3 the Di /3. Proof: Thus Multiplyig the expressios for Di from Lemma 2.13 ad Corollary 2.17, we get Di 2 ρ i 3 1 = 1. ρ i 3 1 Di 3 3. This result, combied with Corollary 2.12, completes the proof of the Mai Theorem. This proof is based o L. Babai: O the order of uiprimitive permutatio groups, Aals of Math , , as simplified by N. Zemlyacheko a year later. Cojecture For uiprimitive coheret cofiguratios, D mi = Ω ρ max. Note that this is true for the average rather tha the miimum size of distiguishig sets by Lemma 2.8. Aother ope questio: Cojecture For primitive coheret cofiguratios of rak r 4, D mi = Ω 1 1/r 1. Or at least D mi = Ω 1 fr, where fr 0. Note that the first statemet is true for r = 2. 7

8 Refereces [1] László Babai: O the order of uiprimitive permutatio groups. Aals of Mathematics , [2] László Babai: O the order of doubly trasitive permutatio groups. Ivetioes Math , [3] Alfred Bochert: Ueber die Zahl der verschiedee Werthe, die eie Fuctio gegebeer Buchstabe durch Vertauschug derselbe erlage ka. Mathematische Aale , [4] Peter J. Camero: Fiite permutatio groups ad fiite simple groups. Bull. Lodo Math. Soc , [5] Attila Maróti: O the orders of primitive groups. J. Algebra , [6] László Pyber: O the orders of doubly trasitive permutatio groups: elemetary estimates. J. Combiat. Theory. Ser. A , [7] Cheryl E. Praeger, Ja Saxl: O the orders of primitive permutatio groups. Bull. Lodo Math. Soc [8] Helmut Wieladt: Abschätzuge für de Grad eier Permutatiosgruppe vo vorgeschriebeem Trasitivitätsgrad. Dissertatio, Berli Schrifte des Math. Semiars ud des Istituts für agewadte Mathematik der Uiversität Berli ,