Direct products and wreath products

Chapter VI: Constructing new P -schemes from old ones

6.7 Direct products and wreath products

for allx∈ H₁\Gandy ∈H₂\G. Similarly, forH₁ ×H₂ ∈ P × P and(g, g )∈ G×G⁰, we have

c⁰⁰_H

1×H₂,(g,g⁰)◦φ_H1,H2(x, y) = φ_gH1g⁻¹_,g⁰_H2g⁰⁻¹(c_H1,g(x), c⁰_H

2,g⁰(y))

for allx ∈H₁\Gandy ∈H₂\G⁰. The various properties ofC × C⁰ (compatibility, regularity, invariance, antisymmetry, and strong antisymmetry) then follow from those ofC andC⁰in a straightforward manner.

Similarly, we define the direct product ofm-schemes:

Definition 6.10. Let Π = {P₁, . . . , P_m} and Π⁰ = {P₁⁰, . . . , P_m⁰ } be m-schemes on finite sets S and S⁰ respectively, where m ∈ N⁺. Define the m-collection Π×Π⁰ ={P₁⁰⁰, . . . , P_m⁰⁰}onS×S⁰in the following way: fork ∈[m], two elements z = ((x₁, y₁), . . . ,(x_k, y_k)), z⁰ = ((x⁰₁, y₁⁰), . . . ,(x⁰_k, y_k⁰)) ∈ (S ×S⁰)^(k) are in the same block ofP_k⁰⁰iff the following conditions are satisfied:

1. Fori, j ∈[k], it holds thatx_i =x_j iffx⁰_i =x⁰_j, andy_i =y_j iffy_i⁰ =y_j⁰.

2. Omit a minimal subset T of coordinates in[k] such that all x_i are distinct, and so are allx⁰_i. Letk⁰ =k− |T|. Suppose the remainingx-coordinates of z and z⁰ are x_i1, . . . , x_ik0 andx⁰_i1, . . . , x⁰_i

k0 respectively. Then (x_i1, . . . , x_ik0) and(x⁰_i1, . . . , x⁰_i

k0)are in the same block ofP_k⁰.9

3. The previous condition holds with x-coordinates replaced by y-coordinates andP_k⁰ replaced byP_k⁰0.

We have the following analogue of Lemma 6.20 whose proof is left to the reader.

Lemma 6.21. Them-collectionΠ×Π⁰ is an m-scheme onS×S⁰. Moreover, if ΠandΠ⁰are antisymmetric (resp. strongly antisymmetric), so isΠ×Π⁰. And ifΠ andΠ⁰have no matching, neither doesΠ×Π⁰.

Remark. The connection between Definition 6.9 and Definition 6.10 is as follows.

Given m ∈ N⁺, letP (resp. P⁰, P⁰⁰) be the system of stabilizers of depthmover G = Sym(S) (resp. G⁰ = Sym(S⁰), G⁰⁰ = Sym(S × S⁰)) with respect to the natural action of GonS (resp. G⁰ onS⁰, G⁰⁰ onS ×S⁰). Let P˜ be the system of

9The order of these coordinates does not matter by invariance ofΠ. Under the previous condition, the choice ofTdoes not matter either.

stabilizers of depth m with respect to theproduct action of G×G on S×S.10 ThenP ⊆ P × P˜ ⁰.11 So we obtain aP˜-scheme C˜fromC × C⁰. Using induction of P˜-schemes, we obtain aP⁰⁰-scheme C⁰⁰ (see Definition 6.4). Using the connection betweenm-schemes andP-schemes (see Theorem 2.1), we see that the construction ofC⁰⁰fromC andC⁰ corresponds to a construction of anm-scheme onS×S⁰ from those onSandS⁰. This is exactly Definition 6.10.

It is obvious that the direct product also preserves homogeneity and discreteness.

By taking iterated direct products, we can construct infinitely many antisymmetric homogeneous m-schemes with no matching if there exists a single one. As an application, we know that either the schemes conjecture (Conjecture 6.1) is true, or there exist infinitely many counterexamples.12

Corollary 6.6. For any m ∈ N⁺, there exist either infinitely many antisymmetric homogeneousm-schemes with no matching or none.

Wreath products. There exists another operation of P-schemes andm-schemes called thewreath product. While this operation is interesting on its own, we do not need it anywhere else in this thesis, except that it provides an alternative proof of Corollary 6.6. For this reason, we only give the definitions as well as the statements, and leave the proofs to the reader.

We first define the wreath product of groups.

Definition 6.11. Let G and G⁰ be groups and let Ω be a G⁰-set. Let G^Ω be the group consisting of all the functionsf : Ω →G. Its group operation is defined by (f f⁰)(x) =f(x)f⁰(x). Define thewreath productGoG⁰ as the group consisting of all the pairs(f, g)∈G^Ω×G⁰, with its group operation defined by

(f, g)(f⁰, g⁰) = (f·^gf⁰, gg⁰)

for(f, g),(f⁰, g⁰) ∈GoG⁰, where^gf⁰ : Ω → Gsendsx∈ Ωtof⁰(^g−1x). In other words, the groupGoG⁰is the semidirect productGo^ϕG⁰whereϕ:G⁰ →Aut(G^Ω)

10The product action is defined by^(g,g0)(x, x⁰) = (^gx,^g

0

x⁰)for(g, g⁰)∈G×G⁰ and(x, x⁰)∈ S×S⁰.

11To see this, note that for a subsetU ⊆S×S⁰ whose projections toSandS⁰ areU₁andU₂, respectively, we have(G×G⁰)_U =G_U1×G⁰_U

2.

12This claim also holds for the variant of the schemes conjecture (Conjecture 6.2) for the same reason.

sendsg ∈G to the automorphismf 7→ fofG^Ω. For convenience, we identifyG^Ω andG⁰ with subgroups ofGoG⁰and write(f, g)∈GoG⁰ asf g.

Use the following notations: let G and G⁰ be finite groups and let Ω be a finite G⁰-set. For a family H = {H_x : x ∈ Ω}of subgroups of Gindexed by Ωand a subgroupH⁰ ofG⁰ satisfying the following condition:

H_x =Gfor allx∈Ωnot fixed byG⁰, (6.1) writeH oH⁰ for the subset

{f g :f(x)∈H_xfor allx∈Ω, g ∈H⁰}

ofGoG, which is a subgroup ofGoGby (6.1). SupposeP and P⁰ are subgroup systems over finite groupsGandG⁰ respectively. DefineP o P⁰ to be the poset of subgroups ofGoG⁰consisting of the subgroupsHoH⁰for allH ={H_x ∈ P :x∈Ω}

andH⁰ ∈ P⁰ satisfying (6.1). ThenP o P⁰is a subgroup system overGoG⁰. ForH={H_x ∈ P :x∈Ω}andH⁰ ∈ P⁰ satisfying (6.1), we have a bijection

φ_H,H⁰ : Y

x∈Ω

H_x\G

!

×H⁰\G⁰ →(H oH⁰)\(GoG⁰) defined as follows: forf ∈Q

x∈ΩH_x\Gwhosex-factor isf_x ∈H_x\G, pickg_x ∈G such thatf_x =H_xg_x. Then definef⁰ : Ω→ Gsendingx∈ Ωtog_x. DefineφH,H⁰

such that it sends(f, Hg⁰)to(H oH⁰)f⁰g⁰ for g⁰ ∈G⁰. It can be shown thatφH,H⁰

is a well defined bijection. Finally, we define the wreath product of aP-collection and aP⁰-collection as follows.

Definition 6.12. For a P-collection C = {C_H : H ∈ P} and a P⁰-collection C⁰ ={C_H⁰ :H ∈ P⁰}, define the(P oP⁰)-collectionCoC⁰ ={C_HoH⁰⁰ 0 :HoH⁰ ∈ P oP⁰} by

C_HoH^{00 0} = (

φH,H⁰

Y

x∈Ω

Bx

!

×B⁰

!

:Bx ∈CHx forx∈Ω, B⁰ ∈C_H^{0 0} )

,

whereH={H_x :x∈Ω}. We callC o C⁰ thewreath productofC andC⁰. We have

Lemma 6.22. The wreath product C o C⁰ is a(P o P⁰)-scheme ifC is aP-scheme and C⁰ is a P⁰-scheme. Moreover, if C and C⁰ are antisymmetric (resp. strongly antisymmetric), then so isC o C⁰.

Similarly, we define the wreath product ofm-schemes:

Definition 6.13. Let Π = {P₁, . . . , P_m} and Π⁰ = {P₁⁰, . . . , P_m⁰ } be m-schemes on finite sets S and S⁰ respectively, where m ∈ N⁺. Define the m-collection ΠoΠ⁰ ={P₁⁰⁰, . . . , P_m⁰⁰}onS×S⁰ in the following way: fork ∈[m], two elements z = ((x₁, y₁), . . . ,(x_k, y_k)), z⁰ = ((x⁰₁, y₁⁰), . . . ,(x⁰_k, y_k⁰)) ∈ (S ×S⁰)^(k) are in the same block ofP_k⁰⁰iff the following conditions are satisfied:

1. Fori, j ∈[k], it holds thatyi =yj iffy⁰_i =y_j⁰.

2. For i ∈ [k], let T_i be set of indices j ∈ [k] satisfying y_i = y_j. Suppose T_i = {i₁, . . . , i_{`</sub>}, ordered in an arbitrary way. Then (x<sub>i1</sub>, . . . , x<sub>i`}) and (x⁰_i1, . . . , x⁰_{i`</sub>)are in the same block ofP<sub>`}.

3. Omit a minimal subsetT of coordinates in[k]such that ally_iare distinct. Let k⁰ =k−|T|. Suppose the remainingy-coordinates ofzandz⁰arey_i1, . . . , y_ik0

andy_i⁰₁, . . . , y_i⁰

k0 respectively. Then(y_i1, . . . , y_ik0)and(y_i⁰₁, . . . , y⁰_i

k0)are in the same block ofP_k⁰0.

We have the following analogue of Lemma 6.22.

Lemma 6.23. Them-collectionΠoΠ⁰ is anm-scheme onS×S⁰. Moreover, ifΠ andΠ⁰ are antisymmetric (resp. strongly antisymmetric), then so isΠoΠ⁰. And ifΠ andΠ⁰have no matching, then neither doesΠoΠ⁰.

Remark. The connection between Definition 6.12 and Definition 6.13 is as follows.

Given m ∈ N⁺, letP (resp. P⁰, P⁰⁰) be the system of stabilizers of depthmover G= Sym(S)(resp. G⁰ = Sym(S⁰),G⁰⁰ = Sym(S×S⁰)) with respect to the natural action ofGonS(resp. G⁰onS⁰,G⁰⁰onS×S⁰). LetP˜be the system of stabilizers of depthmwith respect to theimprimitive wreath product actionofGoG⁰onS×S⁰.13 ThenP ⊆ P o P˜ ⁰.14 So we obtain a P˜-scheme C˜fromC o C⁰. Using induction of P˜-schemes, we obtain aP⁰⁰-scheme C⁰⁰ (see Definition 6.4). Using the connection betweenm-schemes andP-schemes (see Theorem 2.1), we see that the construction

13The imprimitive wreath product action is defined by^(f,g)(x, x⁰) = (^f(gx0)x,^gx⁰)for(f, g)∈ GoG⁰and(x, x⁰)∈S×S⁰.

14To see this, consider a subsetU ⊆S×S⁰. Forx⁰ ∈S⁰, letU_x⁰ ={x∈ S : (x, x⁰)∈S⁰} andHx⁰ = GU_x0. Let H = {Hx⁰ : x⁰ ∈ S⁰} and let U⁰ be the projection ofU toS⁰. Then (GoG⁰)U =H oG⁰_U0. Moreover, ifx⁰ ∈S⁰is not fixed byG⁰_U0, thenx⁰ 6∈U⁰and henceUx⁰ =∅, which impliesH_x⁰ =G.

ofC fromC andC corresponds to a construction of anm-scheme onS×S from those onSandS⁰. This is exactly Definition 6.13.

C h a p t e r 7

SYMMETRIC GROUPS AND LINEAR GROUPS

Let G be a finite permutation group. Motivated by the P-scheme algorithms developed in Chapter 3 and Chapter 5, we are interested in the problem of bounding the integerd(G), introduced in Definition 2.8.

In this chapter, we study this problem for symmetric groups and linear groups with various special group actions.

Symmetric groups. For convenience, we introduce the following notation:

Definition 7.1. For n ∈ N⁺, define d_Sym(n) := d(G), where G is the symmetric groupSym(S)acting naturally on a finite setS of cardinalityn.1

Note thatd_Sym(n)is nondecreasing innby Corollary 6.4. The best known general upper bound fordSym(n)is

d_Sym(n)≤ 2

log 12

logn+O(1),

proven in [Gua09; Aro13] in different notations, based on the work of [Evd94;

IKS09]. In Section 7.1, we review this result and interpret it as a result about P-schemes.

In Section 7.3, we study the more general action ofSym(S)on the set ofk-subsets ofS, where1≤k ≤ |S|, and that on (an orbit of) the set of partitions ofS. These actions are called thestandard actionof symmetric groups, and play an important role in the study of minimal base sizes of primitive permutation groups (see, e.g., [LS99]). Our results for these group actions will be used in Chapter 8.

Linear groups. Let V be a vector space of dimension n ∈ N⁺ over a finite fieldF^q. We have the general linear group GL(V) consisting of all the invertible linear transformations of V over F^q. It is a subgroup of the general semilinear group ΓL(V), which consists of all the invertible semilinear transformations of V. Here we say a map φ : V → V is a semilinear transformation of V if

1ClearlydSym(n)only depends onnbut not onS.

φ(x+y) = φ(x) +φ(y)andφ(cx) =τ_φ(c)φ(x)hold for allx, y ∈ V andc∈ F^q, whereτ_φis an automorphism of the fieldF^q. We have thenatural actionofGL(V) and that ofΓL(V)onV − {0}, defined in the obvious way.

Denote byPV theprojective spaceassociated withV, i.e., PV is the set of equiv- alence classes of V − {0} where x, y ∈ V − {0} are equivalent iff x = cy for somec∈ F^×q. Define theprojective linear groupPGL(V) := GL(V)/F^×q and the projective semilinear group PΓL(V) := ΓL(V)/F^×q, whereF^×q is identified with the subgroup of thescalar linear transformationsofGL(V)(resp. ΓL(V)) so that c∈F^×q sendsx∈V tocx. The natural action ofGL(V)(resp. ΓL(V)) onV − {0}

induces an action ofPGL(V)(resp. PΓL(V)) onPV, called thenatural actionof PGL(V)(resp. PΓL(V)) onPV. Finally, whenV =Fⁿq, we also use the notations GL_n(q),ΓL_n(q),PGL_n(q), andPΓL_n(q).

We call the above groups GL(V), ΓL(V), PGL(V), and PΓL(V) linear groups. In Section 7.4, we investigated(G)for the natural action of a linear groupG. For convenience, we introduce the following notations:

Definition 7.2. Let V be a vector space of dimension n ∈ N⁺ over a finite field Fq. Define d_GL(n, q) := d(G), where G is the permutation group GL(V) acting naturally on V − {0}. Similarly define d_ΓL(n, q), d_PGL(n, q), and d_PΓL(n, q) by choosingGto be the permutation groupΓL(V),PGL(V),PΓL(V)acting naturally onV − {0},PV,PV, respectively.2

We show that the problems of bounding d_GL(n, q) d_ΓL(n, q), d_PGL(n, q), and dPΓL(n, q) are all equivalent: an upper bound f(n, q) for any one of them im- plies an upper boundf(n, q) +O(1)for the others. So it suffices to investigate just one of them.

Finally, we prove a bound d_GL(n, q)≤

logq logq+ (log 12)/4

n+O(1), slightly improving the trivial bounds.

Self-reduction. The results in Section 7.3 and Section 7.4 require a technique calledself-reduction of discreteness, which we introduce in Section 7.2. It reduces discreteness of a strongly antisymmetricP-scheme to discreteness of its restrictions

2Clearly these definitions only depend onnandqbut not onV.

to stabilizer subgroups. In many cases, such a reduction greatly simplifies the problem. Our results in Chapter 8 also rely heavily on this technique.

7.1 The natural action of a symmetric group

We introduce the following notations aboutm-schemes:

Definition 7.3. Forn∈N⁺, letm(n)(resp. m⁰(n)) be the smallest positive integer such that any non-discrete antisymmetric m(n)-scheme (resp. m⁰(n)-scheme) on [n]has a matching (resp. is not strongly antisymmetric).

It is easy to see that m(n) and m⁰(n) are nondecreasing in n. We also have dSym(n)≤m⁰(n)≤m(n)by Lemma 2.7 and Lemma 2.10.

It was proven [Gua09] and independently in [Aro13] thatm(n)≤

2 log 12

logn+ O(1). We review the proof of this bound, starting from the following lemma:

Lemma 7.1. LetΠ = {P₁, . . . , P_m}be an antisymmetricm-scheme on a finite set S where m ≥ 3. SupposeB ∈ P₁ satisfies|B| ≥ 3. Letxbe an element of B so thatΠ|_x ={P₁⁰, . . . , P_m−1⁰ }is an(m−1)-scheme onS− {x}(see Definition 6.3).

Then at least one of the two conditions is satisfied.

1. There existsB⁰ ∈P₁⁰ contained inBsatisfying|B⁰| ≤(|B| −1)/4.

2. There exist distinct elementsy, z ∈B− {x}such that for the(m−2)-scheme Π|x,y = {P₁⁰⁰, . . . , P_m−2⁰⁰ } on S − {x, y}, the block B⁰⁰ of P₁⁰⁰ containing z satisfies|B⁰⁰| ≤ (|B|+ 1)/12. Furthermore, (x, y), (y, z), and(z, x)are in the same block ofP₂.

Proof. By replacingΠ withΠk_B, we may assumeΠis homogeneous andS =B. By antisymmetry, we know |P₂| is even. If |P₂| ≥ 4, there exists B₁ ∈ P₂ of cardinality at most|B|(|B| −1)/4. LetB⁰ :={y ∈B : (x, y)∈B₁}. ThenB⁰ is a block ofP₁⁰by definition, and its cardinality is|B₁|/|B| ≤(|B| −1)/4by regularity ofΠ. And the first condition is met.

So assume|P2|= 2. ThenP2contains two blocksB1andB2of the same cardinality

|B|(|B| −1)/2. Choose y ∈ B − {x} such that (x, y) ∈ B1. Such an element y exists by regularity and homogeneity of Π. By Lemma 2.11 and Lemma 2.12, we have an antisymmetric association scheme P(Π) = P₂ ∪ {1_B} that has three blocks. By Lemma 2.20, the number of elements z ∈ B − {x, y} satisfying

(y, z),(z, x) ∈ B₁ is precisely (|B|+ 1)/4 > 0. The cardinality of the set T :=

{(a, b, c) : (a, b),(b, c),(c, a)∈B₁}is then|B₁|·(|B|+1)/4. Choosez ∈B−{x, y}

such that(x, y, z)∈T. LetB₁⁰,B₂⁰, andB₃⁰ be the blocks ofP₃containing(x, y, z), (y, z, x) and (z, x, y) respectively, which are all subsets of T. They have the same cardinality by invariance of Π, and are distinct by antisymmetry of Π. So

|B₁⁰| ≤ |T|/3 = |B₁| ·(|B|+ 1)/12. By regularity ofΠ, the cardinality of the set {u ∈ S − {x, y} : (x, y, u) ∈ B₁⁰} is |B₁⁰|/|B₁| ≤ (|B|+ 1)/12, and this set is exactly the blockB⁰⁰ ofP₁⁰⁰ containingz by definition. So the second condition is satisfied.

Lemma 7.1 implies the following recursive relation:

Lemma 7.2. Forn≥3, m(n)≤max

m

n−1 4

+ 1, m

n+ 1 12

+ 2

.

The inequality also holds form⁰(·)in replaced ofm(·).

Proof. Let Π = {P₁, . . . , P_m} be a non-discrete antisymmetric m-scheme on a finite setSof cardinalityn, wherem ≥3. Also assume

m ≥max

m

n−1 4

+ 1, m

n+ 1 12

+ 2

. We want to show thatΠhas a matching.

Choose B ∈ P₁ such that |B| > 1. Let x be an element of B and suppose Π|_x ={P₁⁰, . . . , P_m−1⁰ }. ThenΠ|_xis an antisymmetric(m−1)-scheme onS− {x}. Note thatΠkB is a homogeneous antisymmetricm-scheme onB by Lemma 6.14, which implies|B| ≥3. Then either of the two conditions in Lemma 7.1 is satisfied.

If the first condition is satisfied, there exists B⁰ ∈ P₁⁰ contained in B satisfying

|B⁰| ≤ (|B| −1)/4 ≤ (n−1)/4. If |B⁰| > 1, we see (Π|_x)k_B⁰ is a non-discrete antisymmetric(m−1)-scheme onB⁰. It has a matching since m−1 ≥ m((n− 1)/4) ≥ m(|B⁰|). SoΠ also has a matching by Lemma 6.3 and Lemma 6.14. On the other hand, if|B⁰|= 1, we letybe the unique element inB⁰ and letB₁ be the block ofP₂containing(x, y). Note that|B⁰|=|B₁|/|B|, which implies|B₁|=|B|. Asx, y ∈B, we haveπ²₁(B₁) =π₂²(B₁) =B. ThenB₁ is a matching ofΠ.

Next assume the second condition is satisfied. So there exist distinct elementsy, z ∈ B−{x}such that for the(m−2)-schemeΠ|x,y ={P₁⁰⁰, . . . , P_m−2⁰⁰ }onS−{x, y}, the

cardinality of the blockB ofP₁ containingzis at most(|B|+ 1)/12≤(n+ 1)/12. Furthermore,(x, y), (y, z), and(z, x)are in the same blockB₀ ofP₂. If|B⁰⁰| >1, we see(Π|_x,y)k_B⁰⁰ is a non-discrete antisymmetric(m−2)-scheme onB⁰⁰. It has a matching sincem−2≥ m((n+ 1)/12) ≥m(|B⁰⁰|). SoΠalso has a matching by Lemma 6.3 and Lemma 6.14. On the other hand, if |B⁰⁰| = 1, we let B₀⁰ be the block of P₃ containing (x, y, z). We have π₁³(B₀⁰) = π³₃(B⁰₀) = B₀ since (x, y),(y, z) ∈ B₀. Also note that|B⁰⁰| = |B₀⁰|/|B₀|, which implies |B₀| = |B₀⁰|. SoB₀⁰ is a matching ofΠ.

This proves the inequality for m(·). The proof form⁰(·)is similar, and we leave it to the reader.

Theorem 7.1([Gua09; Aro13]). For alln ∈N⁺, m(n)≤

2 log 12

logn+O(1).

More generally, an antisymmetric m-scheme Π = {P₁, . . . , P_m} on a finite setS always has a matching ifP₁ has a blockB of cardinalityk >1andm≥m(k). In particular it holds for sufficiently largem =

2 log 12

logk+O(1).

Proof. Notem(1) = 1andm(2) = 2. The first claim then follows from Lemma 7.2 and a simple induction. The second claim follows by consideringΠk_Band applying Lemma 6.14.

Theorem 7.1 implies a bound for d_Sym(n), and also a bound ford(G) by Corol- lary 6.3, whereGis an arbitrary permutation group on a set of cardinalityn: Corollary 7.1. LetGbe a permutation group on a set of cardinalityn ∈N⁺. Then d(G)≤d_Sym(n)≤

2 log 12

logn+O(1).

We conclude this section with the following technical lemma, which is used later in the proof of Theorem 7.5.

Lemma 7.3. Let G be a permutation group on a finite set S, and let P be the corresponding system of stabilizers of depthmwhere1≤m ≤ |S|. LetC ={C_H : H ∈ P}be a strongly antisymmetricP-scheme. SupposeC is non-discrete onG_x for some x ∈ S. Then there exists (x₁, . . . , x_m) ∈ S^(m) such thatC_Gx

1,...,xm has a block of cardinality at least2(^{log 124} )^m2−O(m).

Proof. Let P be the system of stabilizers of depth m with respect to the natural action of G⁰ := Sym(S) on S. Let C⁰ = {C_H⁰ : H ∈ P} be the induction of C to P⁰ (see Definition 6.4), which is strongly antisymmetric by Lemma 6.1 and is non-discrete onG⁰_xforx∈S in the lemma sinceC is non-discrete onG_x. Assume the lemma holds forSym(S), P⁰, and an m-tuple(y₁, . . . , y_m) ∈ S^(m), i.e., there existsB⁰ ∈C_G⁰ 0

y1,...,ym of cardinality at least2(^{log 124} )^m2−O(m). By Definition 6.4, we know B⁰ is of the form φ_G⁰_y1,...,ym,g(B), where g ∈ G⁰, φ_G⁰_y1,...,ym,g is an injection from(G∩gG⁰_y1,...,ymg⁻¹)\GtoG⁰_y1,...,ym\G⁰, andB is a block ofCG∩gG⁰_y

1,...,ymg⁻¹. Letxi =^gyifori∈[m]. ThenG∩gG⁰_y1,...,ymg⁻¹ =Gx1,...,xm. So(x1, . . . , xm)and B ∈CGx1,...,xm satisfy the condition in the lemma.

Thus we may assumeG = Sym(S)and it acts naturally onS. By Lemma 2.12, it suffices to show that for any non-discrete strongly antisymmetric m-scheme Π = {P₁, . . . , P_m}onS, the partitionP_mhas a block of cardinality at least2(^{log 124} )^m2−cm, where c = O(1). We prove this claim by induction on m. The case m = 1 is trivial. For m > 1, assume the claim for m⁰ < m. Let B₀ be a block of P₁ of cardinality k > 1. By Theorem 7.1, we have m ≤

2 log 12

logk +c⁰ for some c⁰ = O(1), or equivalently k ≥ 2^{log 122 (m−c0)}. Choose x ∈ B0 and consider the (m−1)-schemeΠ-schemeΠ⁰ := Π|x ={P₁⁰, . . . , P_m−1⁰ }onS− {x}. It is strongly antisymmetric by Lemma 6.3. Let B₁ be a block of P₁⁰ contained in B₀, which exists by compatibility of Π and the fact k > 1. If |B₁| = 1, we have seen in the proof of Lemma 7.2 that Π has matching, contradicting the assumption that Π is strongly antisymmetric. So |B₁| > 1. By Lemma 6.14, the homogeneous (m−1)-schemeΠ⁰k_B1 ={P₁⁰⁰, . . . , P_m−1⁰⁰ }onB₁is strongly antisymmetric. By the induction hypothesis, the partitionP_m−1⁰⁰ has a blockB⁰ ⊆B₁^(m−1) of cardinality at least2(^{log 124} )^{(m−1)2−c(m−1)}. AndB⁰ is also a block ofP_m−1⁰ ∈ Π⁰ by definition and compatibility ofΠ⁰. ThenPm ∈ Πhas a blockB containing(x, x1, . . . , xm−1)for all(x1, . . . , xm−1)∈B⁰. By regularity ofΠ, we have

|B|=|B0||B⁰| ≥2^{log 122 (m−c0)}·2(^{log 124} )^{(m−1)2−c(m−1)} ≥2(^{log 124} )^m2−cm for sufficiently largec=O(1).

7.2 Self-reduction of discreteness

In this section, we prove a “self-reduction” lemma, which states that discreteness of a strongly antisymmetricP-scheme is implied by discreteness of its restrictions to stabilizer subgroups.

We need the following technical lemma.

Lemma 7.4. Suppose G is a finite group, P is a subgroup system over G, and C ={C_H : H ∈ P} is aP-scheme. SupposeH₀, H₁, H₂ are subgroups inP such thatH₀ ⊆H₁∩H₂ andC|_H1, C|_H2 are both discrete onH₀. Fori= 0,1,2, letB_i be the block ofC_Hi containingH_ie∈H_i\G. Then(π_H0,H2)|_B0 ◦(π_H0,H1|_B0)⁻¹ is a well-defined bijection fromB₁toB₂sendingH₁e∈H₁\GtoH₂e∈H₂\G.

Proof. Note that π_H0,H1|_B0 is a surjective map from B₀ to B₁ sending H₀e to H₁e, and π_H0,H2|_B0 is a surjective map from B₀ to B₂ sending H₀e to H₂e. So it suffices to prove that these two maps are injective. The set B₀ ∩ (H₀\H₁) contains H₀e and is a block of C_H0|_H1 ∈ C|_H1 by the definition of restriction (Definition 6.2). By discreteness ofC|H1 onH0, this set is just the singleton{H0e}. On the other hand, the set H0\H1 ⊆ H0\G is precisely the preimage of H1e underπ_H0,H1 : H₀\G → H₁\G. SoB₀ ∩(H₀\H₁)is the preimage of H₁eunder π_H0,H1|_B0 :B₀ →B₁. By regularity ofC, the mapπ_H0,H1|_B0 is injective. Similarly π_H0,H2|_B0 is also injective.

The bijection in Lemma 7.4 can be used to separate elements in a strongly antisym- metricP-scheme:

Lemma 7.5. LetGbe a finite group acting on a finite setS, and letx ∈S. LetP be a subgroup system overG such thatG_x ∈ P, and let C = {C_H : H ∈ P}be aP-scheme. Suppose y = ^gxand z = ^g0x inS satisfy (1) G_y, G_z, G_y,z ∈ P and (2)C|_Gy andC|_Gz are both discrete onG_y,z. Then there exists a bijection between blocks of C_Gx sending G_xg⁻¹ to G_xg⁰⁻¹ that can be written as a composition of conjugations, projections and their inverses between blocks ofC_Gx, C_Gy,C_Gz and C_Gy,z. In particular, ifC is strongly antisymmetric, thenG_xg⁻¹ andG_xg⁰⁻¹ are in different blocks ofC_Gx.

Proof. LetB₀(resp. B₁,B₂) be the block ofC_Gy,z(resp. C_Gy,C_Gz) containingG_y,ze (resp. Gye,Gze). By Lemma 7.4, the mapπGy,z,Gz|B0◦(πGy,z,Gy|B0)⁻¹is a bijection fromB₁ toB₂ sendingG_yetoG_ze. LetB₁⁰ andB₂⁰ be the blocks ofC_Gx containing G_xg⁻¹ and G_xg⁰⁻¹ respectively. We have the conjugations c_Gx,g|_B⁰

1 : B₁⁰ → B₁ sendingG_xg⁻¹toG_yeandc_Gz,g⁰⁻¹|_B2 :B₂ →B₂⁰ sendingG_zetoG_xg⁰⁻¹. Then the map

c_Gz,g⁰⁻¹|_B2 ◦π_Gy,z,Gz|_B0 ◦(π_Gy,z,Gy|_B0)⁻¹ ◦c_Gx,g|_B⁰

1

is a bijection fromB₁⁰ toB⁰₂sendingG_xg⁻¹ toG_xg⁰⁻¹.

This provides a way of proving discreteness of a strongly antisymmetricP-scheme using discreteness of its restrictions to stabilizers. For example, if C is strongly antisymmetric and the conditions in Lemma 7.5 hold for all pairs(y, z)∈Gx×Gx, thenC is discrete onG_x. In fact, we only need to verify the conditions for a subset of pairs that form a connected graph:

Lemma 7.6(self-reduction lemma). LetG be a finite group acting on a finite set S, and letx ∈ S. Let P be a subgroup system overG such thatG_x ∈ P, and let C ={C_H :H ∈ P}be a strongly antisymmetricP-scheme. SupposeRis a subset ofS×S satisfying the following conditions:

1. For all(y, z) ∈R, it holds that (1)G_y, G_z, G_y,z ∈ P and (2)C|_Gy andC|_Gz are both discrete onG_y,z.

2. LetG_Rbe the undirected graph onSsuch that{y, z}is an edge iff(y, z)∈R or (z, y) ∈ R. ThenGx is contained in a connected component of G_R (in particular, this condition is satisfied ifG_Ris connected).

ThenC is discrete onG_x.

Proof. Fory ∈ S, denote byB_y the block ofC_Gy containingG_ye ∈ G_y\G. For (y, z)∈S×S, writey∼ zif there exists a bijectionτ :B_y →B_z sendingG_yeto G_zesuch thatτis a composition of maps of the formπ_H,H⁰|_Bor(π_H,H⁰|_B)⁻¹(where H, H ∈ P andB is block ofC_H). Then∼is an equivalence relation onS. By the first condition and Lemma 7.4, we havey∼zfor all(y, z)∈R. And by the second condition, we havey∼zfor all(y, z)∈Gx×Gx.

Consider any g, g⁰ ∈ G and let y = ^gx, z = ^g0x ∈ Gx. Let τ : By → Bz be a bijection sendingGyetoGzeas above. LetBandB⁰be the blocks ofCGxcontaining G_xg⁻¹ and G_xg⁰⁻¹ respectively. We have the conjugations c_Gx,g|_B : B → B_y sendingG_xg⁻¹ toG_yeandc_Gz,g⁰⁻¹|_Bz :B_z →B⁰ sendingG_zetoG_xg⁰⁻¹. Then the map

c_Gz,g⁰⁻¹|_Bz ◦τ ◦c_Gx,g|_B

is a bijection from B to B⁰ sending G_xg⁻¹ to G_xg⁰⁻¹. In particular, ifG_xg⁻¹ 6=

Gxg⁰⁻¹, thenB 6=B⁰ by strong antisymmetry ofC. Asg, g⁰ ∈Gare arbitrary, we knowC is discrete onG_x.