A CHARACTERIZATION OF THE 2-FUSION SYSTEM OF L4(q) DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Justin Lynd, B.S. Graduate Program in Mathematics The Ohio State University 2012 Dissertation Committee: Dr. Ronald Solomon, Advisor Dr. Matthew Kahle Dr. Jean-Francois Lafont Dr. Richard Lyons c Copyright by Justin Lynd 2012 ABSTRACT We study saturated fusion systems F on a finite 2-group S with involution cen- tralizer having a unique component on a dihedral group and containing the Baumann 2 subgroup of S. Assuming F = O (F), O2(F) = 1, and the centralizer of the com- ponent is a cyclic 2-group, it is shown that F is uniquely determined as the 2-fusion system of L4(q) for some q ≡ 3 (mod 4). This should be viewed as a contribution to a program recently outlined by Aschbacher for the classification of simple fusion systems at the prime 2. The analogous problem in the classification of finite simple groups of component type (the L2(q), A7 standard component problem) was one of the last to be completed, and was ultimately only resolved in an inductive context with heavy machinery. Thanks primarily to the hypothesis concerning the Baumann subgroup and the absence of cores, our arguments by contrast require only 2-fusion analysis and transfer. We prove a generalization of the Thompson transfer lemma in the context of fusion systems, which is applied often. ii To David Reid Dillon iii ACKNOWLEDGMENTS To: Ronald Solomon, for knowing precisely when to push and when to let be, providing continual support, being a careful mentor and constant advocate, sharing his over- whelming breadth of knowledge and insight, and dedicating large amounts time and effort throughout this project (and others prior); I always knew I made the right choice. Richard Lyons, for carefully reading this thesis and offering many probing questions and helpful suggestions; here's to the next three years. Matthew Kahle and Jean Lafont, for serving on my dissertation committee. Mathematical mentors, colleagues, and friends: Andy Chermak, Dan File, Mike Geline, George Glauberman, David Green, Ellen Henke, Ian Leary, Bob Oliver, Silvia Onofrei, Sejong Park, Julianne Rainbolt, Radu Stancu, Matt Welz, erica Whitaker, James Wilson, among others for mathematical discussions, guidance and support, and allowing me the pleasure of your company. The OSU Mathematics Department for providing a comfortable, yet stimulating en- vironment in which to work. The OSU Graduate School for a Presidential Fellowship which allowed for the com- pletion of this dissertation. iv Kristen, for patience, unending support, and many sleepless nights and days. Mackenzie and Liam, for continually teaching me about myself. Mom, for love, wisdom, and guidance. David Dillon, the primary reason I stand where I stand. Moy, for friendship, conversations, and good times (minus the retinal tear). Thank you. v VITA 1981 . Born in Huntington, West Virginia 2004 . B.S. Mathematics, Marshall University 2004-2010, 2012 . Graduate Research/Teaching Associate, Math- ematics, The Ohio State University 2011 . Presidential Fellow, The Ohio State Uni- versity PUBLICATIONS Adam Allan, Michael Dunne, Harold Ellingsen, and John Jack. Classification of the group of units in the Gaussian integers modulo n, The Pi Mu Epsilon Journal, Issue 12:9, (2008), 513-519. Justin Lynd and Sejong Park. Analogues of Goldschmidt's thesis for fusion systems, J. Algebra 324 (2010), 3487{3493. Justin Lynd. 2-subnormal quadratic offenders and Oliver's p-group conjecture, Pro- ceedings of the Conference on Algebraic Topology, Group Theory, and Representation Theory, Proc. Edin. Math. Soc., accepted. vi FIELDS OF STUDY Major Field: Mathematics Specialization: Finite Group Theory vii TABLE OF CONTENTS Abstract . ii Dedication . ii Acknowledgments . iv Vita . vi CHAPTER PAGE 1 Introduction . 1 1.1 The L2(q), A7 problem in finite groups . 5 1.2 The proof of the main theorem . 7 1.3 Outline . 9 1.4 Notation . 11 2 Fusion and linking systems . 12 2.1 Automorphism groups of p-groups . 12 2.2 Ln(q) and Un(q) for n 6 4....................... 16 2.3 Fusion systems . 18 2.4 Subsystems and normality . 28 2.5 The hyperfocal and residual subsystems . 31 2.6 Centralizers . 36 2.7 Simple fusion systems . 38 2.8 Linking systems . 43 3 Thompson Transfer Lemma for fusion systems . 48 4 The involution centralizer . 56 4.1 Reduced and tame fusion systems . 56 4.2 Structure of the involution centralizer . 59 5 Proof of Theorem A . 63 5.1 The 2-central case . 63 viii 5.2 The 2-rank 3 case . 70 5.3 The 2-rank 4 case: jQj =2 ...................... 74 5.4 The 2-rank 4 case: jQj > 2 ...................... 82 5.5 Proof of Theorem A . 96 Bibliography . 97 ix CHAPTER 1 INTRODUCTION Let p be a prime number. If G is a finite group with Sylow p-subgroup S, two subsets X and Y of S are fused in G if they are conjugate in G, i.e. if there exists g 2 G with gXg−1 = Y . A subgroup H of G controls fusion in S if whenever two subsets of S are fused in G, then they are fused in H. The study of p-fusion in finite group theory began with the work of Frobenius and Burnside on control of fusion in the last decade of the 19th century. In the latter half of the 20th, the analysis of 2-fusion was a central technique in the classification of the finite simple groups. Now, motivated by categorical models of p-fusion originated by Lluis Puig (see [Pui06]), fusion theory has evolved to become both the foundation for the investigation of p-completed classifying spaces of finite groups and related spaces, as well as a unifying tool which motivates long-standing local-to-global conjectures in modular representation theory. For good introductions to this subject and its applications, see [AKO11] and [Cra11b]. A fusion system on a finite p-group S is a category F whose objects are the subgroups of S, and whose morphisms satisfy HomS(P; Q) ⊆ HomF (P; Q) ⊆ Inj(P; Q) for P , Q 6 S in such a way that every morphism of F is the composition of an isomorphism followed by an inclusion. Here, for any group X which contains S, the set HomX (P; Q) is the set of group homomorphisms from P to Q which are 1 induced by conjugation by an element of X, and Inj(P; Q) is the set of injective group homomorphisms from P to Q. If G is a finite group and S 2 Sylp(G), then there is a fusion system F = FS(G) with HomF (P; Q) = HomG(P; Q) and we say that F is realized by G. Sometimes we write Fp(G) for FS(G) when the Sylow p-subgroup S is unimportant. A fusion system as described above is too general an object to say many meaningful things about. For this reason one imposes axioms which force S to act like a Sylow p-subgroup of F, and we speak of a saturated fusion system. These axioms are easily checked via Sylow's Theorem for the fusion system of a finite group. See Chapter 2 for more details. The emerging \structure theory" of saturated fusion systems, continuing to un- fold as it has over the last decade, parallels that of finite groups. Axioms and im- portant foundational ideas have originated and have been subsequently simplified in work of Puig [Pui06], Broto, Levi, and Oliver [BLO04], Stancu [Sta06], and Roberts and Shpectorov [RS09]. Many standard objects and key concepts in group theory have been written in the language of fusion systems. For example, there are no- tions of the hyperfocal subsystem, Op(F), and the residual subsystem, Op0 (F), of a fusion system [Pui06, BCG+07], constrained fusion systems [BCG+05], normal sub- systems [Lin06, Asc08, Cra11b], and the layer E(F) and generalized Fitting subsys- ∗ tem F (F) = Op(F)E(F) of a fusion system [Asc08, Asc11a]. Once saturation is determined for appropriate subsystem analogues of important subgroups as above, classical theorems of group theory often have analogues in fusion systems which ad- mit easier proofs and which require less machinery than their counterparts in group theory. Some examples of this phenomenon lie in the work of Kessar and Linckel- man on Glauberman's ZJ-theorem [KL08], D´ıaz,Glesser, Mazza, Park, and Stancu on Glauberman-Thompson Theorems on control of fusion [DGMP09] and Yoshida's transfer theorem [DGPS11], Aschbacher's E-balance theorem [Asc11a], Park and the 2 author's work on Goldschmidt's thesis [LP10], and Onofrei-Stancu on Stellmacher's W -functors [OS09], among others to appear or in progress. One particularly salient result from finite group theory whose analogue does not hold the context of saturated fusion systems is the following theorem, proved by Solomon [Sol74]: There is no finite simple group with an involution centralizer isomor- phic to Spin7(q) for any odd q. Indeed, motivated by predictions of Benson [Ben94] that Solomon's non-existent sporadic groups should nevertheless have 2-completed classifying spaces, Levi and Oliver construct in [LO02] a simple saturated fusion system FSol(q) on the Sylow 2-subgroup Sq of Spin7(q) for each odd q which has ∼ the property that CF (z) = FSq (Spin7(q)) for an involution z 2 Z(Sq). Moreover, these fusion systems are not realizable as the fusion system of any finite group by Solomon's Theorem. Such fusion systems are called exotic. While exotic fusion systems at odd primes seem to be prevalent and have been constructed by several au- thors [BLO04,BLO06,RV04,Rui07,CP10], the Solomon systems are the only known simple exotic systems at the prime 2.