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FUSION SYSTEMS on P-GROUPS with an EXTRASPECIAL SUBGROUP of INDEX P

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FUSION SYSTEMS on P-GROUPS with an EXTRASPECIAL SUBGROUP of INDEX P FUSION SYSTEMS ON p-GROUPS WITH AN EXTRASPECIAL SUBGROUP OF INDEX p by RAUL´ MORAGUES MONCHO A thesis submitted to the University of Birmingham for the degreee of DOCTOR OF PHILOSOPHY School of Mathematics College of Engineering and Physical Sciences University of Birmingham August 2018 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Abstract In this thesis we classify saturated fusion systems on p-groups S containing an extraspecial subgroup of index p for an arbitrary odd prime p. We prove that if 6 F is a saturated fusion system on S with Op(F) = 1 then either jSj ≤ p or S is isomorphic to a unique group of order pp−1. We either classify the fusion systems or cite references to show that F is known in all cases except when S is a Sylow p−1 p-subgroup of SL4(p), which remains as future work. When jSj = p with p ≥ 11 we describe new infinite exotic families related to those constructed by Parker and Stroth. A mis padres Consuelo y Juan Jos´e. ACKNOWLEDGEMENTS First and foremost, I would like to thank my supervisor Professor Chris Parker, for his invaluable guidance and patience. Thanks for the suggestion of this project and incisive comments throughout its development, which have been vital to the completion of this thesis. I am very grateful for the support of my family and friends, for all their support throughout these years. To my parents Consuelo and Juan Jos´e,to whom I dedicate this thesis, as well as Pablo, Yassin and Hao among many others. My thanks to the colleagues of the Algebra Group with whom I have shared these last four years, for many fruitful conversations. A special mention to Valentina and Ryan, who were always helpful and could brighten my day. Thanks also to the people of the School of Mathematics, from the everyday work and the great times playing football, badminton, or whichever activity someone cooked up. Thanks to David, Alessandro, Rob, Gemma, Chloe, Alberto, and to all the people I have met over these four years, for having made this a great experience. Finally, I thankfully acknowledge financial support from the University of Birmingham. CONTENTS Introduction 1 1 Group Theory Background 16 1.1 Conjugation and commutators . 16 1.2 Regular p-groups . 19 1.3 Extraspecial p-groups . 21 1.4 Isomorphisms between small classical groups . 25 1.5 p-groups with an extraspecial subgroup of index p .......... 27 1.6 Isomorphism classes of p-groups with an extraspecial subgroup of index p .................................. 32 1.7 Actions of groups on p-groups . 39 1.8 Transvections . 41 1.9 Quadratic action and Thompson Replacement Theorem . 42 1.10 Strongly p-embedded subgroups . 49 1.11 Subgroups of GLr(p) with a strongly p-embedded subgroup . 57 2 An introduction to fusion systems 63 2.1 Alperin's fusion theorem . 69 2.2 Local theory of fusion systems . 73 3 Small F-essential candidates 83 3.1 Abelian p-groups of rank 2 . 85 3.2 p-groups of order p4 ........................... 89 4 Fusion systems on p-groups with an extraspecial subgroup of index p: reduction 94 4.1 F-essential subgroups contained in Q ................. 98 4.2 Hypothesis C: some F-essential subgroup E 2 M has ZE ≤ Q ... 101 4.2.1 Φ(E) = FE leads to a Sylow p-subgroup of G2(p)...... 109 4.2.2 Φ(E) = ZE leads to a Sylow p-subgroup of SL4(p)...... 110 4.3 Hypothesis D: every F-essential subgroup E 2 M has ZE Q ... 114 4.3.1 E \ Q maximal abelian in Q leads to a Sylow p-subgroup of SU4(p).............................. 115 4.3.2 E \ Q not maximal abelian in Q ............... 117 4.4 Summary of the reduction . 119 5 Fusion systems on a Sylow p-subgroup of SU4(p) 123 5.1 Structure of S for both SL4(p) and SU4(p).............. 124 5.2 F-essential subgroups . 127 − 5.3 Natural Ω4 (p)-module calculations . 130 5.4 Classification of the fusion systems on a Sylow p-subgroup of SU4(p) 137 6 Fusion systems on S of order pp−1 147 7 Simple fusion systems when jSj = p4 163 8 Conclusion 176 A Group extensions i A.1 Diagrams and the short five lemma . ii A.2 Semidirect products . iv A.3 Group extensions . vi A.4 Low dimensional cohomology . xi A.5 Extensions with nonabelian normal subgroup . xiv B Finite simple groups whose Sylow p-subgroups contain an ex- traspecial subgroup of index p xviii B.1 Groups of Lie type in defining characteristic . xxv B.2 Alternating groups . xxx B.3 Groups of Lie type in cross characteristic when p 6= 2 . xxxi B.3.1 Classical groups . xxxi B.3.2 Exceptional groups . xxxviii B.4 Sporadic groups . xlii B.5 Fusion systems of the finite simple groups . xlvii B.6 Almost simple groups . liv C Magma code lvi C.1 Orbits calculation . lvi C.2 Reduction to SL4(3) . lvii C.3 Magma code to find subgroups that can be FS(G)-essential . lviii List of References lix LIST OF TABLES ∼ 1+4 ∼ 1.1 Split extensions of Q = p+ by K = Cp................ 34 7.1 Simple fusion systems on p-groups of order p4 with A not F-essential.165 7.2 Reduced fusion systems on p-groups of order p4 with A F-essential. 165 7.3 [COS17, Table 2.1] . 170 7.4 [COS17, Table 4.1, rows 1 and 5] . 170 7.5 [COS17, Table 2.2] with rk(A) = m = 3, e = 1. 172 7.6 [COS17, Table 4.3, row 2] for p =5................... 173 B.1 Finite simple groups whose Sylow p-subgroups have an extraspecial subgroup of index p when p is odd. xx B.2 Cyclotomic polynomials expressing the r0-part of the orders of uni- versal versions of Lie type groups and their centres. xxxviii B.3 Sporadic finite simple groups whose Sylow p-subgroups have an extraspecial subgroup of index p.................... xliii LIST OF NOTATION Throughout this thesis p denotes a prime, all groups are finite, and we write maps on the right hand side. Our notation is conventional except where stated otherwise. • CG(H) = fg 2 G j gh = hg for all h 2 Hg. g • NG(H) = fg 2 G j H = Hg. • [x; y] = x−1y−1xy is the commutator of x and y. • [H; K] = h[h; k] j h 2 H; k 2 Ki. 0 • γ2(G) = G = [G; G], the derived subgroup of G, and γi+1(G) = [γi(G);G]. • Z1(G) = Z(G) and Zi(G) is the i-th term of the upper central series of G. • Φ(G) is the Frattini subgroup of G. • Sylp(G) is the set of Sylow p-subgroups of G. T Op(G) = S is the largest normal p-subgroup of G. • S2Sylp(G) • Op0 (G) is the largest normal subgroup of G of order coprime to p. • Op(G) is the smallest normal subgroup of G of order p-power index. p0 • O (G) = hS j S 2 Sylp(G)i, the normal closure in G of S is the smallest normal subgroup of G of index coprime to p. pi • Ωi(G) = hg 2 G j g = 1i. i i • fi(G) = Gp = hgp j g 2 Gi. mp(G) • mp(G) is the p-rank of G, that is, p is order of the largest elementary abelian subgroup of G. Not to confuse with the rank of a p-group jP=Φ(P )jp. • F ∗(G) is the generalised Fitting subgroup of G. • Hom(G; H) is the set of group homomorphisms φ : G ! H. • Aut(G) is the group of automorphisms φ : G ! G. • Inn(G) is the group of inner automorphisms of G. g −1 • For g 2 G the map cg 2 Inn(G) is defined by h 7! hcg = h = g hg. • Out(G) = Aut(G)= Inn(G) is the outer automorphism group of G. • A × B is the direct product of A and B. • An = A × · · · × A is the direct product of n copies of A. • A ◦C B is a central product of A and B with C = A \ B ≤ Z(A ◦C B). ∼ • A:B denotes a group extension of A by B where A E A:B and A:B=A = B. • A oφ B or A : B denote a semidirect product or split extension of A by B with nontrivial action (if specified) φ : B ! Aut(A), except in Appendix A, where we write B nφ A. • A:B denotes an extension of A by B which does not split. The following will denote specific groups: • Cn is a cyclic group of order n. 1+2n 1+2n • If p is odd, p+ is the extraspecial group of order p and exponent p. 1+2n 1+2n 2 • If p is odd, p− is the extraspecial group of order p and exponent p . • Dn is the dihedral group of order n.
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