ON SYLOW 2-SUBGROUPS OF FINITE SIMPLE GROUPS OF ORDER UP TO 210 DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Sergey Malyushitsky, M.S. ***** The Ohio State University 2004 Dissertation Committee: Approved by Professor Koichiro Harada, Adviser Professor Akos Seress Adviser Professor Ronald Solomon Department of Mathematics ABSTRACT A 2-group ( a group of order a power of 2 ) is called realizable if it occurs as a Sylow 2-subgroup of a finite simple group. The purpose of this thesis is to study all realizable groups of order up to 210. From the classification of all simple groups of finite order we know all realizable groups of order up to 210 as Sylow 2-subgroups of known finite simple groups. However without the use of classification determining all realizable 2-groups is very difficult. In the first part of the thesis we present an argument that produces all realizable groups of order up to 32, by eliminating one by one all 2-groups that can not occur as a Sylow 2-subgroup of a finite simple group. When the number of 2-groups of given order becomes too large to handle it by hand we attempt to use a computer for repetitive checks on a large number of 2-groups. The second part of the thesis is devoted to describing all realizable 2-groups of order up to 210 using the classification of all finite simple groups. We determine the identification number of each group in the Small Groups Library in GAP4 and compute the power-commutator presentation of each realizable group S of type G. We also determine the conjugacy classes of involutions of S and their fusion in G, maximal subgroups of S, maximal abelian and maximal elementary abelian subgroups of S, Z(S), S/Z(S), maximal quotient groups of S and maximal normal extra-special subgroups of S. ii This is dedicated to my family. iii ACKNOWLEDGMENTS I would like to express my deep gratitude to my adviser, Professor Koichiro Harada, for his guidance and support. I would like to thank Professor Akos Seress for his advice on various problems related to GAP4. I would also like to thank Professor Ronald Solomon and Professor Sia Wong for their comments and suggestions. And finally, I would like to thank my family, and especially my wife Svetlana, for their love and support. iv VITA 1972 . .Born - Minsk, Belarus 1997 . .M.S. Mathematics 2000 . .M.S. Computer Science 1993-2001 . Graduate Teaching Associate, The Ohio State University. 2001-2002 . Software Engineer, IMAKE Software & Services, Inc. 2002-present . .Graduate Teaching Associate, The Ohio State University. FIELDS OF STUDY Major Field: Mathematics v TABLE OF CONTENTS Page Abstract . ii Dedication . iii Acknowledgments . iv Vita......................................... v List of Tables . ix List of Figures . x Chapters: 1. Introduction . 1 2. Quoted results and consequences . 11 2.1 Notation . 11 2.2 Quoted results . 12 2.3 Main consequences . 14 3. Determining realizable 2-groups . 18 3.1 2-groups of order ≤ 8......................... 18 3.2 2-groups of order 16 . 19 3.3 2-groups of order 32 . 22 3.4 2-groups of order 64 . 30 vi 4. Properties of Realizable groups of order 26 ................. 37 4.1 Realizable group of type L2(64) . 37 6 7 4.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ) ......... 38 4 5 4 4.3 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ),U3(q), q ≡ 2 + 1(mod 25)................................ 40 4.4 Realizable group of type Sz(8) . 42 4.5 Realizable group of type U3(4) . 43 2 4.6 Realizable group of type A8,A9,L4(2),U4(2),S4(q), q ≡ 2 ± 1(mod 23).................................... 44 4.7 Realizable group of type L3(4) . 46 2 3 4.8 Realizable group of type M12, D4(q), q ≡ 2 ± 1(mod 2 ), G2(q), q ≡ 22 ± 1(mod 23)............................. 47 5. Realizable Groups of Order 27 ........................ 50 5.1 Realizable group of type L2(128) . 50 7 8 5.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ) ......... 51 5 6 5.3 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ) and U3(q), q ≡ 25 + 1(mod 26)............................. 53 3 4 3 5.4 Realizable group of type L3(q), q ≡ 2 + 1(mod 2 ),U3(q), q ≡ 2 − 1(mod 24)................................ 54 5.5 Realizable group of type A10 ...................... 56 3 5.6 Realizable group of type M22, M23, McL, L4(q), q ≡ 5(mod 2 ), 3 U4(q), q ≡ 3(mod 2 ).......................... 58 5.7 Realizable group of type J2, J3 .................... 60 6. Realizable Groups of Order 28 ........................ 63 6.1 Realizable group of type L2(256) . 63 6.2 Realizable group of type S4(4) . 65 8 9 6.3 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ) ......... 67 6 7 6 6.4 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ),U3(q), q ≡ 2 + 1(mod 27)................................ 69 3 4 3 6.5 Realizable group of type G2(q), q ≡ 2 ± 1(mod 2 ),D4(q), q ≡ 2 ± 1(mod 24)................................ 71 3 4 6.6 Realizable group of type S4(q), q ≡ 2 ± 1(mod 2 ) ......... 72 6.7 Realizable group of type Ly ...................... 75 7. Realizable Groups of Order 29 ........................ 77 7.1 Realizable group of type L2(512) . 77 vii 9 10 7.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 )......... 79 7 8 7 7.3 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ),U3(q), q ≡ 2 + 1(mod 28)................................ 80 4 5 4 7.4 Realizable group of type L3(q), q ≡ 2 + 1(mod 2 ),U3(q), q ≡ 2 − 1(mod 25)................................ 81 4 4 7.5 Realizable group of type L4(q), q ≡ 7(mod 2 ),U4(q), q ≡ 9(mod 2 ) 83 3 3 7.6 Realizable group of type L5(q), q ≡ 3(mod 2 ),U5(q), q ≡ 5(mod 2 ) 85 2 3 7.7 Realizable group of type S6(q), q ≡ 2 ± 1(mod 2 ) ......... 87 2 3 7.8 Realizable group of type S6(2),A12,A13,O7(q), q ≡ 2 ± 1(mod 2 ) . 90 7.9 Realizable group of type L3(8) . 93 7.10 Realizable group of type U3(8) . 95 7.11 Realizable group of type HS ..................... 96 7.12 Realizable group of type O0Nan .................... 98 8. Realizable Groups of Order 210 ....................... 100 8.1 Realizable group of type L2(1024) . 100 10 11 8.2 Realizable group of type L2(q), q ≡ 2 ± 1(mod 2 ) . 102 8 9 8 8.3 Realizable group of type L3(q), q ≡ 2 − 1(mod 2 ),U3(q), q ≡ 2 + 1(mod 29)................................ 103 4 5 4 8.4 Realizable group of type D4(q), q ≡ 2 ± 1(mod 2 ),G2(q), q ≡ 2 ± 1(mod 25)................................ 105 4 5 8.5 Realizable group of type S4(q), q ≡ 2 ± 1(mod 2 ) . 106 8.6 Realizable group of type He, M24,L5(2) . 108 8.7 Realizable group of type Co3 ..................... 111 4 4 8.8 Realizable group of type L4(q), q ≡ 9(mod 2 ),U4(q), q ≡ 7(mod 2 ) 113 8.9 Realizable group of type U5(2) . 116 8.10 Realizable group of type Sz(32) . 118 − 2 3 8.11 Realizable group of type A14,A15,O8 (q), q ≡ 2 ± 3(mod 2 ) . 120 Bibliography . 124 viii LIST OF TABLES Table Page 1.1 Realizable 2-groups of order 22 and their types. 1 1.2 Realizable 2-groups of order 23 and their types. 2 1.3 Realizable 2-groups of order 24 and their types. 2 1.4 Realizable 2-groups of order 25 and their types. 2 1.5 Realizable 2-groups of order 26 and their types. 3 1.6 Realizable 2-groups of order 27 and their types. 3 1.7 Realizable 2-groups of order 28 and their types. 4 1.8 Realizable 2-groups of order 29 and their types. (For groups of type L3(8) and U3(8) the ID number identifying them in the Small Groups Library in GAP4 is not computed as the number of 2-groups of the same order, rank and p-class is too high to complete the comprehensive search for these groups.) . 4 1.9 Realizable 2-groups of order 210 and their types. (The group ID number is not given because the 2-groups of order 210 are not enumerated in the Small Groups Library in GAP4.) . 5 1.10 The table shows the number of realizable 2-groups of given order along with the total number of 2-groups of the same order and also the num- ber of 2-groups of class 2. 5 ix LIST OF FIGURES Figure Page 1.1 Realizable groups inclusion tree. 10 x CHAPTER 1 INTRODUCTION The main purpose of the dissertation is to investigate the internal structure of realizable 2-groups of order up to 210 and describe the most interesting internal prop- erties of such groups. A group S is called realizable if it occurs as a Sylow-2 subgroup of a finite simple group G. We then call S of type G. Using the full classification of simple groups, see [7], one can list all realizable 2-groups of order ≤ 210. (see Tables 1.1 - 1.9) Each table shows all realizable 2-groups of given order.