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On Orbit Equivalent Permutation Groups ON ORBIT EQUIVALENT PERMUTATION GROUPS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Keyan Yang ***** The Ohio State University 2008 Dissertation Committee: Approved by Prof. Akos´ Seress, Advisor Prof. Ronald Solomon Advisor Prof. Michael Davis Graduate Program in Mathematics ABSTRACT Two permutation groups G, H ≤ Sym(Ω) are called orbit equivalent if they have the same orbits on the power set of Ω. Primitive orbit equivalent permutation groups were determined by Seress. In this thesis we prove results toward the classification of two-step imprimitive, orbit equivalent permutation groups, which is the next natural step in the program of classifying all transitive, orbit equivalent pairs. Along the way, we also prove that with a short explicit list of exceptions, all primitive groups have at least four regular orbits on the power set of the underlying set. ii Dedicated to Binyi iii ACKNOWLEDGMENTS I would like to thank my advisor Akos´ Seress for his patience, guidance and con- stant encouragement which made this work possible. I am grateful for the countless enlightening discussions and suggestions as well as the intellectual support he has provided. I am grateful to Professor Cai-heng Li for his helpful discussions. I have appre- ciated the many conversations we have had. I would like to thank Professor Ron Solomon, who has always been ready to share his knowledge with me. I have appreci- ated his enthusiasm and generosity in helping me. I would like to thank the member of my dissertation committee, Professor Michael Davis for his invaluable time. I am thankful for the Department of Mathematics for their support. I especially thank Cindy Bernlohr for her patience. Last but not the least I wish to thank my family for their constant support and encouragement. iv VITA 1978 . Born in Hubei, China 2000 . B.Sc. in Mathematics, China University of Geosciences, China 2003 . MS. in Mathematics, Fudan University, China 2003-Present . Graduate Teaching Associate, The Ohio State University PUBLICATIONS 1. A.´ Seress and K. Yang, On orbit equivalent, two-step imprimitive permutation groups. To appear in ”Computational Group Theory and the Theory of Groups” Contemporary Mathematics vol. 470, American Mathematical Society, 2008. 2. K. Yang and F. Li, Classification of torsion-free sheaves on a rational curve with a triple point, Journal of Fudan University. 2004 Vol.43 No.3 P.366-370. v FIELDS OF STUDY Major Field: Mathematics Specialization: Combinatorics vi TABLE OF CONTENTS Abstract . ii Dedication . ii Acknowledgments . iv Vita......................................... v List of Tables . ix CHAPTER PAGE 1 Introduction . 1 1.1 Introduction . 1 1.2 Structure of the thesis and main results . 2 2 Preliminaries . 7 2.1 Permutation groups . 7 2.2 Primitive groups . 7 2.3 Wreath products . 8 2.4 The O’Nan-Scott Theorem . 9 2.5 Previous results . 11 3 Regular orbits of primitive groups . 13 3.1 Product action . 13 3.2 An estimate . 15 3.3 Proof of Theorem 1.1 . 16 4 General results on orbit equivalent groups . 22 4.1 Basic results . 22 4.2 Groups that are orbit equivalent to a regular group . 23 vii 4.3 Maximal blocks . 26 4.4 Orbit equivalent pairs of wreath products . 31 5 A ∈ (γ) and B ∈ (γ) ........................... 33 5.1 Clean subgroups isomorphic to An . 33 5.2 Proof of Theorem 1.2 . 37 5.3 Sm o Sn ............................ 41 5.4 Sn × Sm ........................... 43 5.4.1 Regular orbits of Sn × Sm .................... 44 5.4.2 Groups that are orbit equivalent to Sn × Sm . 47 6 Miscellaneous orbit equivalent 2-step imprimitive permutation groups . 55 6.1 A ∈ (α) and B ∈ (β) .................... 55 6.2 A ∈ (α) and B ∈ (γ)..................... 67 6.3 A ∈ (β) and B ∈ (α) .................... 70 6.4 A ∈ (β) and B ∈ (γ) .................... 78 7 Orbit equivalent permutation groups up to degree 15 . 82 8 GAPcode.................................. 84 Bibliography . 89 viii LIST OF TABLES TABLE PAGE 6.1 Groups with no regular orbit . 71 ix CHAPTER 1 INTRODUCTION 1.1 Introduction The action of a permutation group G on Ω naturally induces an action on the power set of Ω. Two permutation groups G, H ≤ Sym(Ω) are called orbit equivalent, in notation G ≡ H, if they have the same orbits on the power set. In general, two orbit equivalent permutation groups are not necessarily equal. An open problem posed about 30 years ago was to determine all transitive orbit equivalent pairs. A trivial observation is that if G ≡ H, then the group generated by G and H is also orbit equivalent to G and H. Hence, it is enough to consider the pairs (G, H) such that H < G. Clearly, if G ≤ Sym(Ω) has a regular orbit, i.e an orbit of size |G| on P(Ω), then no proper subgroup H < G is orbit equivalent to G. In [18], Siemons proved that two orbit equivalent permutation groups must share the block systems of imprimitivity. Much research work [4] [17] [20] was done on regular orbits of primitive groups and primitive orbit equivalent pairs. Cameron, Neumann, and Saxl [4] proved that with the exception of finitely many primitive groups, a primitive group with no regular orbits on the power set of the permutation domain must contain An, but they did not give estimates on the number of exceptional 1 groups. Seress [17] determined all primitive groups with no regular orbits and all primitive orbit equivalent pairs of groups . Although the primitive case was completely solved, little was known about transi- tive orbit equivalent pairs. Regarding the orbit structure of a transitive group, in [19], Siemons proved the orbits of G acting on the power set were completely determined by the orbits of G on the n∗−subsets of Ω, where (1/2)(n − 1) ≤ n∗ ≤ (1/2)(n + 1). In this dissertation, we start to investigate the group structure of transitive orbit equivalent pairs. We take the natural step and focus on two-step imprimitive groups. This is the first step towards transitive groups from primitive groups. 1.2 Structure of the thesis and main results In Chapter 2, we provide basic theory and notation for permutation groups. We also list the previous results concerning orbit equivalent permutation groups. In Chapter 3, we prove the following result concerning regular orbits of primitive groups. Theorem 1.1. Suppose that a primitive group G ≤ Sym(Ω) has a regular orbit, but no more than three regular orbits on P(Ω). Let n = |Ω| and let k be the number of reg- ular orbits on P(Ω). Then (G, n, k) must be one of the following: (C2, 2, 1), (C3, 3, 2), (D14, 7, 2), (AGL(1, 8), 8, 3), (PSL(2, 11), 12, 2), (PSL(2, 13), 14, 2), (A7, 15, 2), or 4 (C2 .S5, 16, 2). We divide primitive permutation groups into the following three classes. 2 (α) Primitive groups with at least one regular orbit on the power set. By Theo- rem 1.1, with eight exceptions, any primitive group with a regular orbit has at least four regular orbits. (β) Primitive groups G ≤ Sn with no regular orbits but (G, n) 6= (An, n) ,(Sn, n). (γ)(An, n) and (Sn, n). In chapter 4, we prove some basic properties of orbit equivalent permutation groups. An imprimitive group G with block system Σ and ∆ as a block is called two-step Σ ∆ imprimitive, if both the induced group G and the restriction G∆ on the block ∆ are primitive. Starting in Chapter 5, we investigate two-step imprimitive orbit equivalent groups Σ ∆ based on the structure of their primitive constituents G and G∆. In Chapter 5, we give the pairs H < G with G ≡ H when the primitive con- stituents of the two-step imprimitive group G are of type (γ) according to our clas- sification. The following results were proved. Σ ∆ Theorem 1.2. Let H < G and H ≡ G. Suppose that (G , n) and (G∆, |∆|) are of type (γ). Suppose b := |∆| ≥ 2 and n := |Σ| ≥ 9 and let K and L be the kernels of the action on Σ of G and H , respectively. Then one of the following holds. n−1 n 1. b = 2 and C2 ≤ L ≤ K ≤ C2 or 1 ≤ L ≤ K ≤ C2. n−1 n−1 2. b = 3 and C3 ≤ L ≤ K ≤ C3 .C2. 3 n n−1 n n−1 n n 3. b = 4 and K4 .C3 ≤ L ≤ K ≤ K4 .C3 .C2, K4 .C3 ≤ L ≤ K ≤ K4 .C3.C2, n−1 n−1 or K4 .C3 ≤ L ≤ K ≤ K4 .C3.C2. Here K4 is the Klein 4−group. ∼ 4. Ab × An ≤ H < G ≤ Sb × Sn and K∆ = K. 5. Ab o An ≤ G ≤ Sb o Sn and if b∈ / {5, 6, 9} then Ab o An ≤ H. As two interesting special cases, we determine the groups that are orbit equivalent to Sm × Sn and Sm o Sn with the standard imprimitive action on [m] × [n]. ∆ In the first half of Chapter 6, , we study the case when the restriction of G∆ is of type (α). If (GΣ, n) is not of type (γ), we have the following. Theorem 1.3. Let G ≤ Sym(∆ × [n]) be a two-step imprimitive group and suppose ∆ Σ that (A = G∆, |∆|) ∈ (α) but (A, |∆|) 6= (C2, 2) and (B, n) = (G , n) 6∈ (γ). If H < G and H ≡ G then (G, H) must be one of the following: (i) G = A o B with (A, |∆|) = (D14, 7) and (B, n) = (D10, 5), (AΓL(1, 9), 9), (AGL(2, 3), 9), or (PΓL(2, 9), 10).
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