ON <i>p</i>- SOLUBLE GROUPS AND THE NUMBER OF SIMPLE MODULES ASSOCIATED WITH A GIVEN BRAUER PAIR

Barker, Laurence

doi:10.1093/qmath/48.2.133

Cited by 14 publications

(2 citation statements)

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“…(We emphasize this fact because a comment in [1] obscured it.) This idea is developed in Robinson [3,Section 4], where a stronger conjecture is presented, and it is shown that when the stabilizer N G QY b of a maximal element QY b of CB controls strong fusion in the self-centralizing Brauer subpairs of QY b, the stronger conjecture implies that the number of irreducible ordinary characters in B of a given defect d equals the number of irreducible ordinary characters of defect d lying in the block of N G QY b in Brauer correspondence with B.…”

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confidence: 98%

Alperin's fusion theorems and G-posets

Barker¹

1998

Journal of Group Theory

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Some G-posets comprising Brauer pairs or local pointed groups belong to a class of G-posets which satisfy a version of Alperin's fusion theorem, and as a consequence, have simply connected orbit spaces.One of the two purposes of this paper is to unify several versions of Alperin's fusion theorem. The other is to appreciate the apparently technical conclusion topologically. Let G be a ®nite group. A G-poset, recall, is a partially ordered set upon which G acts as automorphisms. One form of Alperin's fusion theorem derives from Sylow's theorem together with the nilpotency of ®nite p-groups. These two properties of psubgroups are expressed axiomatically in de®ning a Sylow G-poset to be a ®nite set X equipped with G-stable relations t and such that (i) is a partial ordering, and is the transitive closure of t,(ii) G acts transitively on the maximal elements of X , and for any x e X , the stabilizer N G x acts transitively on the elements y e X which are maximal subject to x t y.Given an upwardly closed G-subposet Y of a Sylow G-poset X (for all y e Y and x e X with y x, we have x e Y ), then Y is a Sylow G-poset. Although our results are expressed in an abstract setting, and the proofs require no specialist knowledge, the following account of a motivation for the notion of a Sylow G-poset assumes some familiarity with p-local representation theory, particularly as discussed in Kno È rr±Robinson [2] and The Âvenaz [7]. Let p be a prime divisor of jGj, and B a positive-defect p-block of G. Let BB be the G-poset of non-trivial Brauer pairs associated with B, and LB the G-poset of non-trivial local pointed groups associated with B. (The trivial Brauer pair associated with B is the unique minimal one.) By results of Alperin, Broue Â, Puig in The Âvenaz [7, 40.10, 40.15, 48.1], BB and LB are Sylow. The upwardly closed G-subposet CB of BB consisting of * The author wishes to thank the Alexander von Humboldt Foundation for funding a visit to the Friedrich-Schiller-Universita Èt, during which some of this work was done.Brought to you by |

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confidence: 98%

Alperin's fusion theorems and G-posets

Barker¹

1998

Journal of Group Theory

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“…contradiction shows that D , b is, after all, a B-subpair. 2 The conclusion of this part of the discussion is that the Fong correspon- Ž Returning to our proof of the main theorem, we obtain for our fixed idempotent of RG , and we let ␤ denote the block 1 . RG .…”

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confidence: 99%