On κ -homogeneous, but not κ -transitive permutation groups
aa r X i v : . [ m a t h . L O ] M a r ON κ -HOMOGENEOUS, BUT NOT κ -TRANSITIVEPERMUTATION GROUPS SAHARON SHELAH AND LAJOS SOUKUP
Abstract.
A permutation group G on a set A is κ -homogeneous iff for all X, Y ∈ (cid:2) A (cid:3) κ with | A \ X | = | A \ Y | = | A | there is a g ∈ G with g [ X ] = Y . G is κ -transitive iff for any injective function f with dom( f ) ∪ ran( f ) ∈ (cid:2) A (cid:3) ≤ κ and | A \ dom( f ) | = | A \ ran( f ) | = | A | there is a g ∈ G with f ⊂ g .Giving a partial answer to a question of P. M. Neumann [4] weshow that there is an ω -homogeneous but not ω -transitive permu-tation group on a cardinal λ provided(i) λ < ω ω , or(ii) ω < λ , and µ ω = µ + and (cid:3) µ hold for each µ ≤ λ with ω = cf( µ ) < µ , or(iii) our model was obtained by adding ω many Cohen genericreals to some ground model.For κ > ω we give a method to construct large κ -homogeneous,but not κ -transitive permutation groups. Using this method weshow that there exists κ + -homogeneous, but not κ + -transitive per-mutation groups on κ + n for each infinite cardinal κ and naturalnumber n ≥ provided V = L . Introduction
Denote by S( A ) the group of all permutations of the set A . Thesubgroups of S( A ) are called permutation groups on A .We say that a permutation group G on A is κ -homogeneous iff forall X, Y ∈ (cid:2) A (cid:3) κ with | A \ X | = | A \ Y | = | A | there is a g ∈ G with g [ X ] = Y .We say that a permutation group G on A is κ -transitive iff for anyinjective function f with dom( f ) ∪ ran( f ) ∈ (cid:2) A (cid:3) ≤ κ and | A \ dom( f ) | = | A \ ran( f ) | there is a g ∈ G with f ⊂ g .In this paper we give a partial answer to the following question whichwas raised by P.N. Neumann in [4, Question 3]: Suppose that κ ≤ λ are infinite cardinals. Does there exist a per-mutation group on λ that are κ -homogeneous, but not κ -transitive? Date : Nov 24, 2019.2000
Mathematics Subject Classification.
Key words and phrases. permutation group, transitive, homogeneous.The first author was supported by European Research Council, grant no. 338821.Publication Number F1886.The second author was supported by NKFIH grants no. K113047 and K129211.
In section 2 we show that there exist ω -homogeneous, but not ω -transitive permutation groups on λ < ω ω in ZFC, and on any infinite λ if V = L (see Theorem 2.5).In section 3 we develop a general method to obtain large κ -homogeneous,but not κ -transitive permutation groups for arbitrary κ ≥ ω (see The-orem 3.4). Applying our method we show that if κ ω = κ , λ = κ + n for some n < ω , and (cid:3) ν holds for each κ ≤ ν < λ , then there is a κ -homogeneous, but not κ -transitive permutation group on λ (Corollary3.12).Finally in section 4, using some lemmas from section 3, we prove thatafter adding ω Cohen reals in the generic extension for each infinite λ there exist ω -homogeneous, but not ω -transitive permutation groupson λ (Theorem 4.1).Our notation is standard. Definition 1.1. If λ is fixed and f ∈ S ( A ) for some A ⊂ λ , we take f + = f ∪ (id ↾ ( λ \ A )) ∈ S ( λ ) . Given a family of functions, G , we say that a function y is G -large iff | y \ [ H| = | y | for each finite H ⊂ G .We say that a permutation group on A is κ -intransitive iff thereis a G -large injective function y with dom( y ) ∪ ran( y ) ∈ (cid:2) A (cid:3) κ and | A \ dom( y ) | = | A \ ran( y )) | = | A | .A κ -intransitive group is clearly not κ -transitive.2. ω -homogeneous but not ω -transitive Definition 2.1.
Given a set A we say that a family A ⊂ (cid:2) A (cid:3) ω is niceon A iff A has an enumeration { A α : α < µ } such that(N1) A is cofinal in (cid:10)(cid:2) A (cid:3) ω , ⊂ (cid:11) ,(N2) for each β < µ there is a countable set I β ∈ (cid:2) β (cid:3) ω such that forall α < β there is a finite set J α,β ∈ (cid:2) I β (cid:3) <ω such that A α ∩ A β ⊂ [ ζ ∈ J α,β A ζ . Theorem 2.2.
Assume that λ is an infinite cardinal, and A ⊂ (cid:2) λ (cid:3) ω isa nice family on λ . Then for each A ∈ A there is an ordering ≤ A on A such that(1) tp ( A, ≤ A ) = ω for each A ∈ A ,(2) if A, B ∈ A , then there is a partition { C i : i < n } of A ∩ B intofinitely many subsets such that ≤ A ↾ C i = ≤ B ↾ C i for all i < n . N κ -HOMOGENEOUS, BUT NOT κ -TRANSITIVE PERMUTATION GROUPS 3 Proof.
Fix an enumeration { A β : β < µ } of A witnessing that A isnice.We will define ≤ A β by induction on β < µ .Assume that ≤ A α is defined for α < β .By (N2) we can fix a countable set I β = { β i : i < ω } ∈ (cid:2) β (cid:3) ω suchthat for all α < β there is n α < ω such that A α ∩ A β ⊂ [ i Assume that λ is an infinite cardinal, A ⊂ (cid:2) λ (cid:3) ω is acofinal family and for each A ∈ A we have an ordering ≤ A on A suchthat(1) tp ( A, ≤ A ) = ω for each A ∈ A ,(2) if A, B ∈ A , then there is a partition { C i : i < n } of A ∩ B intofinitely many subsets such that ≤ A ↾ C i = ≤ B ↾ C i for all i < n .Then there is a permutation group on λ that is ω -homogeneous and ω -intransitive. S. SHELAH AND L. SOUKUP Proof. For A ∈ A let G A = { f + ∈ S( λ ) : f ∈ S( A ) ∧ there is a finite partition { C i : i < n } of A such that f ↾ C i is ≤ A -order preserving } . Let G be the permutation group on λ generated by [ {G A : A ∈ A} . Claim 2.3.1. G is ω -homogeneous. Indeed, let X, Y ∈ (cid:2) λ (cid:3) ω with | λ \ X | = | λ \ Y | = λ . Pick A ∈ A suchthat X ∪ Y ⊂ A and | A \ X | = | A \ Y | = ω .Let c be the unique ≤ A -monotone bijection between X and Y and d be the unique ≤ A -monotone bijection between A \ X and A \ Y . Thentaking g = c ∪ d we have g + ∈ G A ⊂ G and g + [ X ] = Y . Claim 2.3.2. G is ω -intransitive. Pick A ∈ A and choose B ∈ (cid:2) A (cid:3) ω such that | A \ B | = ω .Let b , b , . . . be the ≤ A -increasing enumeration of B . Define a bi-jection y : B → ω as follows: for i < ω and j < i let y ( b i + j ) = b i +1 − j . Observe that if c is ≤ A -monotone then |{ i < ω : |{ j < i : c ( b i + j ) = r ( b i + j ) }| ≥ }| ≤ . Indeed, if |{ j < i : c ( b i + j ) = y ( b i + j ) }| ≥ , then c should be ≤ A -decreasing, and if |{ i : { j < i : c ( b i + j ) = y ( b i + j ) } 6 = ∅}| ≥ , then y should be ≤ A -increasing.So y can not be covered by finitely many ≤ A -monotone functions.But for any h ∈ G , h ∩ ( A × A ) can be covered by finitely many ≤ A -monotone functions by (2) and by the construction of G .Thus y is G -large. (cid:3) To obtain nice families we recall some topological results. We saythat a topological space X is splendid (see [1]) iff it is countably com-pact, locally compact, locally countable such that | A | = ω for each A ∈ (cid:2) X (cid:3) ω .We need the following theorem: Theorem (Juhasz, Nagy, Weiss, [1]) . If(i) κ < ω ω , or(ii) ω < κ , cf( κ ) > ω and µ ω = µ + and (cid:3) µ hold for each µ < κ with ω = cf( µ ) < µ ,then there is a splendid space X of size κ .Remark . In [1, Theorem 11] the authors formulated a bit weaker result: if V = L and cf( κ ) > ω then there is a splendid space X of size κ .However, to obtain that results they combined “Lemmas 7, 9 and 16 N κ -HOMOGENEOUS, BUT NOT κ -TRANSITIVE PERMUTATION GROUPS 5 with the remark after Theorem 8” and their arguments used only theassumptions of the theorem above. Lemma 2.4. If X is a splendid space, U is the family of compact opensubsets of X , and Y ⊂ X , then U ⌈ Y = { U ∩ Y : U ∈ U } is nice on Y .Proof. Let A ∈ (cid:2) Y (cid:3) ω . Then A is countable, so it is compact. Since asplendid space is zero-dimensional, A can be covered by finitely manycompact open set, and so A can be covered by an element of U . Thus U ⌈ Y is cofinal in (cid:10)(cid:2) Y (cid:3) ω , ⊂ (cid:11) .To check (N2) observe that every U ∈ U is a countable compactspace, so it is homeomorphic to a countable successor ordinal. Thus U has only countably many compact open subsets. Hence U ↾ U iscountable which implies (N2) in the following stronger form:(N2 + ) for each β < µ there is a set I β ∈ (cid:2) β (cid:3) ω such that for all α < β there is ζ α ∈ I β such that A α ∩ A β = A ζ α ∩ A β . (cid:3) Remark . By [2, Corollary 2.2], if ( ω ω +1 , ω ω ) → ( ω , ω ) holds, then thecardinality of a splendid space is less than ω ω . So we need some newideas if we want to construct arbitrarily large nice families in ZFC. Theorem 2.5. If λ is an infinite cardinal, and(i) λ < ω ω , or(ii) ω < λ , and µ ω = µ + and (cid:3) µ hold for each µ ≤ λ with ω =cf( µ ) < µ .then there is an ω -homogeneous and ω -intransitive permutation groupon λ .Proof. Applying the Juhasz-Nagy-Weiss theorem for κ = λ if cf( λ ) >ω , and for κ = λ + if λ > cf( λ ) = ω , we obtain a splendid space on κ ≥ λ . So, by Lemma 2.4, we obtain a nice family on A on λ .Thus, putting together Theorems 2.2 and 2.3 we obtained the desiredpermutation group on λ . (cid:3) κ -homogeneous but not κ -transitive for κ > ω Write A⌈ X = { A ∩ X : A ∈ A} and A⌈ ∗ X = { T A ′ ∩ X : A ′ ∈ (cid:2) A (cid:3) <ω } . Definition 3.1. Let κ < λ be cardinals. We say that a cofinal family A ⊂ (cid:2) λ (cid:3) κ is locally small iff |A⌈ A | ≤ κ for all A ∈ A . Definition 3.2. If X, Y are subsets of ordinals with the same ordertypes, then let ρ X,Y be the unique order preserving bijection between X and Y . S. SHELAH AND L. SOUKUP Definition 3.3. If F is a set of functions, an F ∪ { x } -term t is asequence h h , . . . , h n − i , where h i = x or h i = x − or h i = f i or h i = f i − for some f i ∈ F . If g is function we use t [ g ] to denote the function h ′ ◦ h ′ ◦ · · · ◦ h ′ n − , where h ′ i = f i if h i = f i , f − i if h i = f − i , g if h i = x , g − if h i = x − .If H is a set of F ∪ { x } -terms, then write H [ g ] = { t [ g ] : t ∈ H } . We say that an F ∪ { x } -term t is an F -term iff neither x nor x − are in the t . If t is a F -term, then the function t [ g ] does not dependson g , so we will write t [ ] instead of t [ g ] in that situation.We say that a term t ′ is a subterm of a term t = h h , . . . , h n − i iff t ′ = h h i , h i , . . . , h i k i , where i < i < · · · < i k < n .The set of all F ∪ { x } -terms is denoted by T ERM ( F ∪ { x } ) .The set of all F -terms is denoted by T ERM ( F ) . Theorem 3.4. Assume that κ = κ + and there is a cofinal, locallysmall family A ⊂ (cid:2) λ (cid:3) κ . Then there is a permutation group G on λ which is κ -homogeneous, but not κ -transitive. Before proving this theorem we need some preparation. Lemma 3.5. Assume that(1) λ is a cardinal, H is a finite set of S ( λ ) ∪ { x } -terms, and H isclosed for subterms,(2) g is an injective function, dom( g ) ∪ ran( g ) ⊂ λ ,(3) α, α ∗ ∈ λ such that h α, α ∗ i / ∈ [ H [ g ] , (4) ζ ∈ λ \ dom( g ) and ζ ∈ λ \ ran( g ) ,(5) η ∈ λ \ ran( g ) and η ∈ λ \ dom( g ) such that η , η / ∈ { t [ g ]( α ) , t [ g ] − ( α ∗ ) : t ∈ H} . Let g = g ∪ {h ζ , η i} and g = g ∪ {h η , ζ i} . Then h α, α ∗ i / ∈ H [ g ] ∪ H [ g ] . Proof. We prove only h α, α ∗ i / ∈ H [ g ] . The proof of the other statementis similar.Assume on the contrary that h α, α ∗ i ∈ H [ g ] .Pick the shortest term t = h f , . . . , f n i from H such that t [ g ]( α ) = α ∗ .Write α n +1 = α and α i = h f i , . . . , f n i [ g ]( α ) for ≤ i ≤ n . Hence α = α ∗ . N κ -HOMOGENEOUS, BUT NOT κ -TRANSITIVE PERMUTATION GROUPS 7 Let i maximal such that α i is ζ or η . Since t [ g ]( α ) can not be α ∗ by (3), i is defined.Since α i = h f i , . . . , f n i [ g ]( α ) , it follows that α i = η by (5). So α i = ζ .Let j minimal such that α j is ζ or η . Since α j = ( h f , . . . , f j − i [ g ]) − ( α ∗ ) ,it follows that α j = η by (5). So α j = ζ by (5). Thus α i = α j = ζ ,and so α ∗ = h f , . . . , f j − , f i , . . . , f n i [ g ]( α ) . Since j < i , the term t ′ = h f , . . . , f j − , f i , . . . , f n i is shorter than t andstill α ∗ = t ′ [ g ]( α ) . So the length of t was not minimal. Contradiction. (cid:3) Lemma 3.6. Assume that(1) y ∈ S( κ ) ,(2) A ∈ (cid:2) λ (cid:3) κ , and B, C ∈ (cid:2) A (cid:3) κ such that | A \ B | = | A \ C | = κ ,(3) F ∈ (cid:2) S( λ ) (cid:3) κ such that | y \ [ H [ ] | = κ whenever H is a finite set of F -terms.Then there is g ∈ S( A ) such that(i) g [ B ] = C ,(ii) | y \ H [ g + ] | = κ whenever H is a finite set of F ∪ { x } -terms.Proof of Lemma 3.6. Write TASK = A × { dom , ran } and TASK = (cid:2) T ERM ( F ∪ { x } ) (cid:3) <ω × κ. Let { I , I } ∈ (cid:2)(cid:2) κ (cid:3) κ (cid:3) be a partition of κ , and fix enumerations { T i : i ∈ I } of TASK , and { T i : i ∈ I } of TASK .By transfinite induction, for i < κ we will construct a function g i and if i = j + 1 for some j ∈ K then we also pick an ordinal α j +1 ∈ κ for such that(a) g i is an injective function, dom( g i ) ∪ ran( g i ) ⊂ A ,(b) g i [ B ] ⊂ C and g i [ A \ B ] ⊂ A \ C ;(c) | g i | ≤ i ;(d) if i = j + 1 , j ∈ I and T j = h ζ , dom i , then ζ ∈ dom( g i ) ;(e) if i = j + 1 , j ∈ I and T j = h ζ , ran i , then ζ ∈ ran( g i ) ;(f) if i = j + 1 , j ∈ I and T j = hH j , χ j i , then(i) α j +1 ∈ κ \ { α j ′ +1 : j ′ ∈ I ∩ j } , and(ii) t [ g i ∪ id λ \ A ]( α j +1 ) is defined and t [ g i ∪ id λ \ A ]( α j +1 ) = y ( α j +1 ) for each t ∈ H j . S. SHELAH AND L. SOUKUP Let g = ∅ .If i is limit, then let g i = S j
Assume that κ = κ + and there is a cofinal, locally smallsubfamily C ⊂ (cid:2) λ (cid:3) κ . Then there is a family D ⊂ (cid:2) λ (cid:3) κ × (cid:2) λ (cid:3) κ such that(1) if h A, B i ∈ D , then B ∪ κ ⊂ A and | A \ B | = κ .Moreover, writing A = { A : h A, B i ∈ D} and B = { B : h A, B i ∈ D} (2) A is a cofinal, locally small subfamily of (cid:2) λ (cid:3) κ ,(3) B is cofinal in (cid:10)(cid:2) λ (cid:3) κ , ⊂ (cid:11) ,(4) { X ⊂ κ : | X | = | κ \ X | = κ } ⊂ B .Proof of Lemma 3.7. Fix a locally small, cofinal subfamily C ⊂ (cid:2) λ (cid:3) κ .We can assume that |{ C ∈ C : D ⊂ C }| = |C| for all D ∈ (cid:2) λ (cid:3) κ .Write µ = |C| . Then κ = κ + ≤ µ . So we can construct D byinduction such that A ⊂ C , κ ⊂ T A and B = C ∪ { X ⊂ κ : | X | = | κ \ X | = κ } . (cid:3) After that preparation we prove the main theorem of this section. Proof of Theorem 3.4. Fix D , A and B as in Lemma 3.7.For h A, B i ∈ D consider the structure M h A,B i = h A, <, B, { A ∩ X : A ∈ A}i .Fix D ′ ∈ (cid:2) D (cid:3) κ + such that writing A ′ = { A ′ : h A ′ , B ′ i ∈ D ′ } and B ′ = { B ′ : h A ′ , B ′ i ∈ D ′ } we have(a) ∀ h A, B i ∈ D ∃ h A ′ , B ′ i ∈ D ′ such that ρ A,A ′ is an isomorphismbetween M h A,B i and M h A ′ ,B ′ i .(b) { X ⊂ κ : | X | = | κ \ X | = κ } ⊂ B ′ .Pick K ∈ (cid:2) κ (cid:3) κ with | κ \ K | = κ . Choose y ∈ S ( κ ) such that y ( α ) = α for each α ∈ κ . Lemma 3.8 (Key lemma) . There are functions F = { f h A,B i : h A, B i ∈D ′ } such that(a) f h A,B i ∈ S( A ) ,(b) f h A,B i [ B ] = K ,moreover, taking S = (cid:8) ρ C ,C : h A , B i , h A , B i ∈ D ′ , C ∈ A⌈ ∗ A , C ∈ A⌈ ∗ A ,ρ C ,C [ A⌈ C ] = A⌈ C } , if H is a finite collection of F ∪ S -terms, then | y \ [ H [ ] | = κ. Before proving the Key lemma, we show how the Key Lemma com-pletes the proof of Theorem 3.4.So assume that the Key lemma holds.For each h A, B i ∈ D pick h A ′ , B ′ i ∈ D ′ such that ρ A,A ′ is an isomor-phism between M h A,B i and M h A ′ ,B ′ i . We assume that h A ′ , B ′ i = h A, B i for h A, B i ∈ D ′ .Let g h A,B i = ρ A ′ ,A ◦ f h A ′ ,B ′ i ◦ ρ A,A ′ ∈ S ( A ) . Let G be the permutation group on λ generated by G = { g h A,B i + : h A, B i ∈ D} . Lemma 3.9. G is κ -homogeneous.Proof of Lemma 3.9. It is enough to show that for each X ∈ (cid:2) λ (cid:3) κ thereis g ∈ G with g [ X ] = K .So fix X ∈ (cid:2) λ (cid:3) κ . Pick h A, B i ∈ D such that X ⊂ B .Then Z = g h A,B i [ X ] ⊂ g h A,B i [ B ] =( ρ A ′ ,A ◦ f h A ′ ,B ′ i ◦ ρ A,A ′ )[ B ]=( ρ A ′ ,A ◦ f h A ′ ,B ′ i )[ B ′ ] = ρ A ′ ,A [ K ] = K. Since | Z | = | κ \ Z | = κ , there is C such that h C, Z i ∈ D ′ . Then f h C,Z i [ Z ] = K . Thus g h C,Z i + [ Z ] = K because h C ′ , Z ′ i = h C, Z i and so f h C,Z i = g h C,Z i .Thus K = ( g h C,Z i + ◦ g h A,B i + )[ X ] . (cid:3) Lemma 3.10. G is not κ -transitive.Proof of Lemma 3.10. We prove that y h for any h ∈ G .Assume that h = ( g +0 ) ℓ ◦ ( g +1 ) ℓ ◦ · · · ◦ ( g + n − ) ℓ n − , where g i = g h A i ,B i i = ρ A ′ i ,A i ◦ f A ′ i ,B ′ i ◦ ρ A i ,A ′ i and ℓ i ∈ {− , } for i < n .Since g + i \ g i is the identity function on λ \ A i , we have h ⊂ [ { ( g i ) ℓ i ◦ ( g i ) ℓ i ◦ · · · ◦ ( g i k − ) ℓ ik − : k < n, i < i < · · · < i k − < n } . Fix k ≤ n and i < i < · · · < i k − < n .Observe that if ℓ i = − then ( g i ) ℓ i = ( ρ A ′ i ,A i ◦ f A ′ i ,B ′ i ◦ ρ A i ,A ′ i ) − = ρ A ′ i ,A i ◦ ( f A ′ i ,B ′ i ) − ◦ ρ A i ,A ′ i . So ( g i ) ℓ i ◦ ( g i ) ℓ i ◦ · · · ◦ ( g i k − ) ℓ ik − = ρ A ′ i ,A i ◦ ( f A ′ i ,B ′ i ) ℓ i ◦ ρ A i ,A ′ i ◦ ρ A ′ i ,A i ◦ ( f A ′ i ,B ′ i ) ℓ i ◦ ρ A i ,A ′ i ◦ For j < k let ρ ∗ j = ρ A ij ,A ′ ij ◦ ρ A ′ ij +1 ,A ij +1 . N κ -HOMOGENEOUS, BUT NOT κ -TRANSITIVE PERMUTATION GROUPS11 Observe that ρ ∗ j = ρ ρ Aij +1 ,A ′ ij +1 [ A ij ∩ A ij +1 ] ,ρ Aij ,A ′ ij [ A ij ∩ A ij +1 ] ∈ S . (See Figure 1.) ρ A i j , A ′ i j ρ A ′ ij +1 ,A ij +1 ρ ∗ j A i j A i j +1 A ′ i j +1 A ′ i j A i j ∩ A i j +1 Figure 1. The function ρ ∗ j Thus ( g i ) ℓ i ◦ ( g i ) ℓ i ◦ · · · ◦ ( g i k − ) ℓ ik − = ρ A i ,A ′ i ◦ ( f A ′ i ,B ′ i ) ℓ ◦ ρ ∗ ◦ ( f A ′ i ,B ′ i ) ℓ ◦ ρ ∗ ◦ . . . ◦ ( f A ′ ik − ,B ′ ik − ) ℓ ik − ◦ ρ A ′ ik − ,A ik − . Since ρ A ℓ ,A ′ ℓ ↾ κ = id ↾ κ , we have (cid:0) ( g i ) ℓ i ◦ ( g i ) ℓ i ◦ · · · ◦ ( g i k − ) ℓ ik − (cid:1) ∩ κ × κ ⊂ ( f A ′ i ,B ′ i ) ℓ ◦ ρ ∗ ◦ ( f A ′ i ,B ′ i ) ℓ ◦ ρ ∗ ◦ . . . ◦ ( f A ′ ik − ,B ′ ik − ) ℓ ik − But ( f A ′ i ,B ′ i ) ℓ ◦ ρ ∗ ◦ ( f A ′ i ,B ′ i ) ℓ ◦ ρ ∗ ◦ · · · ◦ ( f A ′ ik − ,B ′ ik − ) ℓ ik − = t [] for the F ∪ S -term t = D ( f A ′ i ,B ′ i ) ℓ , ρ ∗ , ( f A ′ i ,B ′ i ) ℓ , ρ ∗ , . . . , ( f A ′ ik − ,B ′ ik − ) ℓ ik − E .Since there are only finitely many sequences i < . . . i k − < n , weobtain that h ∩ κ × κ is covered by the union of finitely many F ∪ S -terms.But y is not covered by the union of finitely many F ∪ S -terms. So y witnesses that G is not κ -transitive. (cid:3) Proof of the Key Lemma 3.8. Write D ′ = {h A α , B α i : α < κ + } .By transfinite induction, we define functions { f α : α < κ + } such thattaking F <β = { f γ : γ < β } and S <β = { ρ C ,C : δ, γ < β, C ∈ A⌈ ∗ A δ , C ∈ A⌈ ∗ A γ , ρ C ,C [ A ↾ C ] = A ↾ C } , we have(i) f α ∈ S( A α ) ,(ii) f α [ B α ] = K ,(iii) if H is a finite collection of F <α +1 ∪ S <α +1 -terms, then | y \ H [ ] | = κ. Assume that we have constructed f β for β < α . Then we have: if H is a finite collection of F <α ∪ S <α -terms, then | y \ H [ ] | = κ. ( ∗ )To continue the construction we need a bit more. Claim 3.10.1. If H is a finite collection of F <α ∪ S <α +1 -terms, then | y \ H [ ] | = κ. Proof. First observe that if ρ i = ρ A i ,A ∗ i for i < , then ρ ◦ ρ = ρ ρ − [ A ∗ ∩ A ] ,ρ [ A ∗ ∩ A ] . ( ‡ )Let t = h t , t , . . . , t n i be an element of H . Since ρ C ,C ↾ κ = id ↾ κ , t [ ] ∩ κ × κ = h t , . . . t n i [ ] ∩ κ × κ if t ∈ S <α +1 . So we can assume that t ∈ F <α . Similar argumentgive that we can assume that t n ∈ F <α .Now assume that h t i , . . . , t j i = (cid:10) f α i , ρ C i +1 ,D i +1 , ρ C i +2 ,D i +2 , . . . , ρ C j − ,D j − , f α j (cid:11) Then, by ( ‡ ) ρ C i +1 ,D i +1 ◦ ρ C i +2 ,D i +2 ◦ · · · ◦ ρ C j − ,D j − = ρ E i ,E j . for some E i ∈ A⌈ C i +1 and E j ∈ A⌈ D j − .Thus we can assume that j = i + 2 and h t i , t i +1 , t i +2 i = h f α , ρ E ,E , f α i . Now f α ◦ ρ E ,E ◦ f α = f α ◦ ρ A α ∩ E ,A α ∩ E ◦ f α and ρ A α ∩ E ,A α ∩ E ∈ S <α .Thus there is a F <α ∪ S <α -terms s t such that t [ ] ∩ ( κ × κ ) = s t [ ] ∩ ( κ × κ ) . Since | y \ S { s t [ ] : t ∈ H}| = κ by ( ∗ ), the Claim holds. (cid:3) Since the claim holds, we can apply Lemma 3.6 for the family F = F <α ∪ S <α +1 to obtain f α as g .So we proved the Key Lemma 3.8. (cid:3) So we proved theorem 3.4 (cid:3) The following theorem is hidden in [3]: N κ -HOMOGENEOUS, BUT NOT κ -TRANSITIVE PERMUTATION GROUPS13 Theorem 3.11. If κ ω = κ , λ = κ + n for some n < ω , and (cid:3) ν holds foreach κ ≤ ν < λ , then there is a cofinal, locally small family in (cid:2) λ (cid:3) κ . Indeed, in subsection 2.4 of [3] the author defines the weakly rounded subsets of λ = κ + n , in Lemma 2.4.1 he shows that the family of weaklyrounded sets is cofinal, finally on page 52 he proves a Claim whichclearly implies that the family of weakly rounded sets is locally small.Putting together Theorems 3.4 and 3.11 we obtain the followingcorollary. Corollary 3.12. If κ ω = κ , λ = κ + n for some n < ω , and (cid:3) ν holds foreach κ ≤ ν < λ , then there is a κ -homogeneous, but not κ -transitivepermutation group on λ . ω -homogeneous but not ω -transitive permutationgroups in the Cohen model For f ∈ S( κ ) let supp( f ) = { α : f ( α ) = α } . Write S ω ( λ ) = { f ∈ S( λ ) : | supp( f ) | ≤ ω } . Theorem 4.1. If P = Fin(2 ω , then V P | = “for each λ ≥ ω there is an ω -homogeneousand ω -intransitive permutation group on λ .” The proof of this theorem is based on the following Lemma.Let us recall that if g ∈ S( ω ) then g + = g ∪ (id ↾ ( λ \ ω )) . Lemma 4.2. Assume that V ⊂ V are ZFC models and λ ≥ ω is acardinal in V . If(1) ∀ X ∈ (cid:0)(cid:2) λ (cid:3) ω (cid:1) V ∃ Y ∈ (cid:0)(cid:2) λ (cid:3) ω (cid:1) V X ⊂ Y ,(2) V | = G is an ω -homogeneous permutation group on ω , G ⊃ S ω ( ω ) V , and r ∈ S( ω ) is G -large,then in V the permutation group G ∗ on λ generated by { g + : g ∈ G } ∪ S ω ( λ ) V is ω -homogeneous, and r is G ∗ -large.Proof. We will work in V .First we show that G ∗ is ω -homogeneous.If X, Y ∈ (cid:2) λ (cid:3) ω first pick X , Y ∈ (cid:2) λ (cid:3) ω ∩ V with X ⊂ X and Y ⊂ Y such that | X \ X | = | Y \ Y | = ω . Fix f, h ∈ S ω ( λ ) V with f [ X ] = ω and h [ Y ] = ω . Since G is ω -homogeneous, there is g ∈ G with g (cid:2) f [ X ] (cid:3) = h [ Y ] . Then ( h − ◦ g + ◦ f )[ X ] = Y and h − ◦ g + ◦ f ∈ G ∗ .Before proving that r is G ∗ -large we need some preparation. Write G + = { g + : g ∈ G } . Claim 4.2.1. If h , . . . h k ∈ S ω ( λ ) V and A ∈ (cid:2) ω (cid:3) ω then there is h ∈ S ω ( ω ) V such that ( h ◦ · · · ◦ h k ) ∩ ( A × A ) ⊂ h. Proof of the Claim 4.2.1. By (1) we can assume that A ∈ V , and so h ′ = ( h ◦ · · · ◦ h k ) ∩ ( A × A ) ∈ V . Since h ′ is a countable injective func-tion with dom( h ′ ) ∪ ran( h ′ ) ⊂ ω it can be extended to a permutation h ∈ S ω ( ω ) V . (cid:3) If F is a set of functions, let hF i gen = { f ◦ · · · ◦ f n − : n ∈ ω, f i ∈ F for i < n } . Claim 4.2.2. For each t ∈ (cid:10) G + ∪ S ω ( λ ) V (cid:11) gen t there is a finite set H ⊂ (cid:10) G ∪ S ω ( λ ) V (cid:11) gen such that t ⊂ [ H . Proof of the Claim 4.2.2. If t = f ◦ · · · ◦ f n − , let H = { id λ }∪{ f ′ i ◦· · ·◦ f ′ i j ◦· · ·◦ f ′ i k : k ≤ n, i < · · · < i j < · · · < i k < n } , where f ′ i = f i if f i ∈ S ω ( λ ) V , and f i = g if f i = g + for some g ∈ G ,and id λ denotes the identity function on λ .Pick α ∈ λ such that t ( α ) = α .Write α n = α and α i = f i ( α i +1 ) for i = n − , . . . , . Let ≤ i
For each s ∈ (cid:10) G ∪ S ω ( λ ) V (cid:11) gen and countable set A ∈ (cid:2) ω (cid:3) ω there is u ∈ (cid:10) G ∪ S ω ( ω ) V (cid:11) gen such that s ∩ ( A × A ) ⊂ u. Proof of the Claim 4.2.3. Since both G and S ω ( λ ) V are groups we canassume that s = g ◦ h ◦ · · · ◦ g n ◦ h n , where g i ∈ G and h i ∈ S ω ( λ ) V .Write A n = A , and let B i = h i [ A i +1 ] ∩ ω and A i = g i [ B i ] for i = n − , . . . , .By Claim 4.2.1 for each i there is h ′ i ∈ S ω ( ω ) V such that h i ∩ ( A i +1 × B i ) ⊂ h ′ i .Let u = g ◦ h ′ ◦ · · · ◦ g n ◦ h ′ n .We show that s ∩ ( A × A ) ⊂ u .Fix α ∈ A . Let α n = α and for i = n − , . . . , let β i = h i ( α i +1 ) and α i = g i ( β i ) . If s ( α ) is defined and s ( α ) ∈ A , then for each i < n we have β i ∈ B i and α i ∈ A i , and so u ( α ) is also defined and u ( α ) = s ( α ) . (cid:3) Putting together Claims 4.2.2 and 4.2.3 we obtain that N κ -HOMOGENEOUS, BUT NOT κ -TRANSITIVE PERMUTATION GROUPS15 Claim 4.2.4. For each g ∈ G ∗ there is a finite subset H g of G suchthat g ∩ ( ω × ω ) ⊂ [ { h ↾ ω : h ∈ H g } . Claim 4.2.4 yields that r is G ∗ -large.So we proved the G ∗ is ω -intransitive which completes the proof ofthe lemma. (cid:3) By Lemma 4.2 the following theorem yields theorem 4.1. Theorem 4.3. If P = Fin(2 ω , then V P | = “there is an ω -homogeneousand ω -intransitive permutation group G on ω with G ⊃ S ω ( ω ) V ”. Proof. Given sets X and Y let us denote by Bij p ( X, Y ) the set of allfinite bijections between subsets of X and Y .We will define an iterated forcing system with finite support h P ν : 0 ≤ ν ≤ ω , Q ν : − ≤ ν < ω i and an increasing sequence of permutation groups h G ν : ν < ω i , G ν ⊳ S( ω ) V Pν , simultaneously.Take G = S ω ( ω ) V and P = Q − = Bij p ( ω, ω ) . Denote by r thegeneric permutation of ω given by the V -generic filter over P . Bystandard density arguments it is easy to see that r is G -large. Nowwe carry out the inductive construction as follows: • for each ν < ω we pick X ν , Y ν , Z ν ∈ ( (cid:2) ω (cid:3) ω ) V Pν with X ν ∪ Y ν ⊂ Z ν and | Z ν \ X ν | = | Z \ Y ν | = ω , • put Q ν = { p ∪ p : p o ∈ Bij p ( X ν , Y ν ) , p ∈ Bij p ( Z ν \ X ν , Z ν \ Y ν ) } , Q ν = h Q ν , ⊃i and g ν = S G ν , where G ν is the Q ν -generic filterover V P ν , • take G ν +1 as the subgroup of S( ω ) V Pν +1 generated by G ν ∪{ g ν + } . • for limit ν let G ν = S ζ<ν G ζ .We use a bookkeeping function to ensure that every pair X, Y ∈ ( (cid:2) ω (cid:3) ω ) V P ω will be chosen as X ν , Y ν in some step. Then G = S ν< ω G ν will be ω -homogeneous.So the question is whether we guarantee that r is G ν -large duringthe induction.If ν is a limit ordinal, then G ν = S ζ<ν G ζ , so if r is G ζ -large for ζ < ν , then r is G ν -large as well.Assume now that r is G ν -large and prove that r is G ν +1 -large as well.The following lemma clearly implies this statement. In this lemmawe use some notations introduced in Definition 3.3 in the previoussection. Lemma 4.4. If H is a finite set of G ν ∪ { x } -terms, p ∈ Q ν , M isa natural number, then there is a condition q ≤ p in Q ν and there is α ∈ ω \ M such that t [ q ]( α ) is defined for each t ∈ H and t [ q ]( α ) = r ( α ) .Proof of the lemma. We can assume that H is closed for subterms.We know that | r \ S H [ ] | = ω because r is G ν -large.Since H is closed for subterms, y ∩ [ H [ ] = y ∩ [ H [id ω \ Z ν ] . Since | p | < ω , we have | y \ [ H [ p ∪ id ( λ \ Z ν ) ] | = ω. So we can pick α ∈ ω \ M such that ( ∗ ) for each t ∈ H either t [ p ∪ id λ \ Z ν ]( α ) is undefined or t [ p ∪ id λ \ Z ν ]( α ) = r ( α ) .Now in finitely many steps, using Lemma 3.5, we can extend thefunction p ∈ Q ν to a function q ∈ Q ν such that ( ∗ ) t [ q ∪ id λ \ Z ν ]( α ) is defined and t [ q ∪ id λ \ Z ν ]( α ) = r ( α ) for each t ∈ H .Indeed, if t [ q ′ ∪ id λ \ Z ν ]( α ) is not defined, where t = h t , . . . , t n i thenthere is i < n such that either ζ i = h t i +1 , . . . , t n i [ q ′ ∪ id λ \ Z ν ]( α ) is defined, t i = x and ζ i / ∈ dom( q ′ ) or ζ = h t i +1 , . . . , t n i [ g ′ ∪ id λ \ Z ν ]( α ) is defined, t i = x − and ζ i / ∈ ran( q ′ ) . In both cases, using Lemma 3.5, we can extend q ′ to q ′′ such that h t i , . . . , t n i [ q ′′ ∪ id λ \ Z ν ]( α ) is defined and h α, r ( α ) i / ∈ H [ q ′′ ∪ id λ \ Z ν ] . Sowe proved Lemma 4.4. (cid:3) So r is G ν +1 -large.Thus, by transfinite induction, we proved that r is G -large whichcompletes the proof of the theorem. (cid:3) References [1] I. Juhász, Zs. Nagy , W. Weiss, On countably compact, locally countable spaces ,Periodica Math. Hung, 10.2–3 (1979) 193–206.[2] I. Juhász, S. Shelah, L. Soukup, More on countably compact, locally countablespaces , Israel J Math 62 (1988) 302-310[3] R. W. Knight, A topological application of flat morasses , Fund. Math. 194(2007), 45-66[4] P. M. Neumann, Homogeneity of infinite permutation groups , Bull. LondonMath. Soc. (1988), 305-312 N κ -HOMOGENEOUS, BUT NOT κ -TRANSITIVE PERMUTATION GROUPS17 Institute of Mathematics, Hebrew University, JerusalemAlfréd Rényi Institute of Mathematics, Budapest, Hungary E-mail address : [email protected] URL : ˜˜