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The Centre of a Block THE CENTRE OF A BLOCK A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences 2016 Inga Schwabrow School of Mathematics 2 Contents Abstract 5 Declaration 7 Copyright Statement 9 Acknowledgements 11 Introduction 13 1 Modular Representation Theory 17 1.1 Basic notation and setup . 17 1.2 Conjugacy class sums, characters and Burnside's formula . 18 1.3 Blocks . 22 1.4 Defect and defect groups . 25 1.5 Jacobson radical and Loewy length . 28 1.6 The Reynolds ideal . 35 2 Equivalences of blocks 37 2.1 Brauer correspondence . 37 2.2 Morita equivalence . 38 2.3 Derived equivalence . 38 2.4 Stable equivalence of Morita type . 40 2.5 Properties of blocks with TI defect groups . 41 2.6 On using the centre to show no derived equivalence exists . 43 3 Blocks with trivial intersection defect groups 44 3 3.1 The Mathieu group M11 with p =3.................... 45 3.2 The McLaughlin group McL, and Aut(McL), with p = 5 . 49 3.3 The Janko group J4 with p =11...................... 52 3.4 The projective special unitary groups . 54 3.5 A question of Rickard . 60 3.6 On the existence of perfect isometries in blocks with TI defect groups . 62 4 On the Loewy length of the Suzuki groups and the small Ree groups in defining characteristic 68 4.1 Relating structure constants . 68 4.2 Suzuki Groups . 70 4.3 The Ree groups . 76 5 More sporadic simple groups 108 5.1 Mathieu groups . 109 5.2 Janko groups . 111 5.3 The Higman-Sims group . 112 5.4 The McLaughlin group . 113 5.5 The Held group . 113 5.6 The Rudvalis group . 114 5.7 The O'Nan group . 115 5.8 A conjecture . 115 6 Blocks with normal, abelian defect groups 116 6.1 A reduction for the Loewy length calculation . 116 6.2 Calculating some examples . 121 n 6.3 General formulae for the Loewy length of (Cp) o E ........... 126 Bibliography 132 A GAP Code 138 B Some structure constants in Ree groups 147 Word count 21714 4 The University of Manchester Inga Schwabrow Doctor of Philosophy The Centre of a Block June 6, 2016 Let G be a finite group and F a field. The group algebra FG decomposes as a direct sum of two-sided ideals, called the blocks of FG. In this thesis the structure of the centre of a block for various groups is investigated. Studying these subalgebras yields information about the relationship between two block algebras and, in certain cases, forms a vital tool in establishing the non-existence of an important equivalence in the context of modular representation theory. In particular, the focus lies on determining the Loewy structure for the centre of a block, which so far has not been studied in detail but is fundamental in gaining a better understanding of the block itself. For finite groups G with non-abelian, trivial intersection Sylow p-subgroups, the analysis of the Loewy structure of the centre of a block allows us to deduce that a stable equivalence of Morita type does not induce an algebra isomorphism between the centre of the principal block of G and its Sylow normaliser. This was already known for the Suzuki groups; the techniques will be generalised to extend the result to cover 2 the Ree groups of type G2(q). In addition, the three sporadic simple groups with the trivial intersection property, M11, McL and J4, together with some small projective special unitary groups are studied with respect to showing the non-existence of an isomorphism between the centre of the principal block and the centre of its Brauer correspondent. Finally, the Loewy structure of centres of various principal blocks are calculated. In particular, some small sporadic simple groups and groups with normal, elementary abelian Sylow p-subgroups are considered. For the latter, some specific formulae for the Loewy length are derived, which generalises recent results on groups with cyclic Sylow p-subgroups. 5 6 Declaration No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 7 8 Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis) owns certain copyright or related rights in it (the \Copyright") and she has given The University of Manchester certain rights to use such Copyright, including for administrative purposes. ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic copy, may be made only in accordance with the Copyright, Designs and Patents Act 1988 (as amended) and regulations issued under it or, where appropriate, in accordance with licensing agreements which the University has from time to time. This page must form part of any such copies made. iii. The ownership of certain Copyright, patents, designs, trade marks and other intel- lectual property (the \Intellectual Property") and any reproductions of copyright works in the thesis, for example graphs and tables (\Reproductions"), which may be described in this thesis, may not be owned by the author and may be owned by third parties. Such Intellectual Property and Reproductions cannot and must not be made available for use without the prior written permission of the owner(s) of the relevant Intellectual Property and/or Reproductions. iv. Further information on the conditions under which disclosure, publication and com- mercialisation of this thesis, the Copyright and any Intellectual Property and/or Reproductions described in it may take place is available in the University IP Policy (see http://documents.manchester.ac.uk/DocuInfo.aspx?DocID=487), in any rele- vant Thesis restriction declarations deposited in the University Library, The Univer- sity Library's regulations (see http://www.manchester.ac.uk/library/aboutus/regul- ations) and in The University's Policy on Presentation of Theses. 9 10 Acknowledgements First of all, I would like to thank my supervisor, Dr. Charles Eaton, for his support, guidance and assistance given to me throughout my PhD. Without the many meetings and helpful discussions, none of this research would have been possible. I would also like to thank my Master supervisor at the University of Southampton, Dr. Tim Burness, for introducing me to the representation theory of finite groups, an extremely rich and beautiful area of pure mathematics. Additional thanks go to Professor Radha Kessar for making me aware of the paper discussed in Section 3.4.11 and Dr. Benjamin Sambale for some useful discussions on Loewy lengths including the use of GAP in the calculations. I greatly appreciate the Engineering and Physical Sciences Research Council for funding my PhD research. Many thanks go to all staff and fellow students at the School of Mathematics, University of Manchester, who have helped me in any way over the last three and a half years. Finally, I thank my family for their continuous support throughout my studies and especially Julian Brough, for his patience and support, which helped me overcome many tough situations and finish this thesis. 11 12 Introduction The representation theory of finite groups has many fascinating and deep open prob- lems. Many of them originate in the work of Richard Brauer, who began the systematic study of modular representations of a finite group, in which the characteristic of the field divides the order of the group. His conjectures formulated in 1963 [Bra63] have led to the development of many new concepts and methods to investigate properties of finite groups. This work often concerns local representation theory, which asserts that fundamental information about the representation theory of G is encoded in some local subgroup of G, that is, the normaliser of some non-trivial p-subgroup. Let p be a fixed prime number. Let G be a finite group whose order is divisible by p. Let O be a complete valuation ring whose residue field k has characteristic p. The group algebra OG decomposes into blocks; hence there is an inherent desire to study these blocks up to some notion of equivalence. Different levels of equivalence between two block algebras A and B have been defined and studied; we are most interested in the notion of a derived equivalence, which has the following consequences: Derived equivalence ) Perfect Isometry ) Z(A) ∼= Z(B): This motivates us to study the centre of a block. So far, this algebra is not well understood and so we explicitly calculate its structure for various groups. A group is said to have trivial intersection (TI) Sylow p-subgroups groups, if any two distinct Sylow p-subgroups intersect trivially. If a block B of G has TI defect groups P , then there exists a stable equivalence of Morita type between B and its Brauer correspondent b in NG(P ). This equivalence induces an algebra homomorphism between the respective stable centres, where Zst(A) := Z(A)=Zpr(A). Although Z(B) and Z(b) have the same dimension [BM90, Theorem 9.2], and the stable centres are isomorphic, i.e. Zst(B) ∼= Zst(b) [Bro94, Proposition 5.4], surprisingly, there is in 13 14 general no isomorphism between the two centres of B and b. The abelian defect group conjecture proposed by Brou´ein 1990 [Bro90, Question 6.2] provides a structural explanation for the close relationship between certain blocks. The conjecture claims that there is a derived equivalence between a block B with abelian defect groups and its Brauer correspondent b in NG(P ) where P 2 Sylp(G).
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