Alperin's fusion theorems and G-posets

Abstract

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 |

The Coset Poset and Probabilistic Zeta Function of a Finite Group

“…For (a), we trivially have A P ⊆ C P , so A P ≤ N * P . To prove the opposite inclusion, we use Alperin's fusion theorem [1]; see also Barker [4] for a short proof of Alperin's theorem. Suppose x ∈ C P .…”

Homological Algebra on Graded Posets

“…Webb conjectured that the orbit space S p (G)/G (as topological space) is contractible. This conjecture was first proven by Symonds in [40], generalized for blocks by Barker [4,5] and extended to arbitrary (saturated) fusion system by Linckelmann [28].…”

“…Denotemos por S p (G) el complejo de Brown para el primo p, que fue introducido en [11]. Webb conjeturó que el espacio de órbitas S p (G)/G es contráctil (como espacio topológico), lo cual fue probado por Symonds en [40], extendido a bloques por Barker [4,5] y extendido a sistemas de fusión (saturados) arbitrarios por Linckelmann [28].…”

“…The simplicial complex whose simplices consist of all of the totally ordered subsets of P is exactly the barycentric subdivision [38, 3.3] of K, sdK. By [15,3,4] there are homeomorphisms |K| ∼ = |sdK| ∼ = |NP|. Thus, as the realization of a simplicial complex K we can consider either of |K|, |sdK| or |NP|.…”