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In this paper we determine the group of endotrivial modules for certain symmetric and alternating groups in characteristic p. If p = 2, then the group is generated by the class of Ω n (k) except in a few low degrees. If p > 2, then the group is only determined for degrees less than p 2. In these cases we show that there are several Young modules which are… (More)
In this paper, we determine a presentation by explicit generators and relations for the Dade group of all (almost) extraspecial p-groups. The proof of the main result uses the cohomolog-ical properties of the Tits building corresponding to the natural geometric structure of the lattice of subgroups of such p-groups.
The Dade group D(P) of a finite p-group P , formed by equivalence classes of endo-permutation modules, is a finitely generated abelian group. Its torsion-free rank equals the number of conjugacy classes of non-cyclic subgroups of P and it is conjectured that every non-trivial element of its torsion subgroup D t (P) has order 2, (or also 4, in case p = 2).… (More)
We prove analogues of results of Glauberman and Thompson for fusion systems. Namely, given a (saturated) fusion system F on a finite p-group S, and in the cases where p is odd or F is S 4-free, we show that Z(N F (J(S))) = Z(F) (Glauberman), and that if C F (Z(S)) = N F (J(S)) = F S (S), then F = F S (S) (Thompson). As a corollary, we obtain a stronger form… (More)
The group of endotrivial modules has recently been determined for a finite group having a normal Sylow p-subgroup. In this paper, we give and compare three different presentations of a torsion-free subgroup of maximal rank of the group of endotrivial modules. Finally, we illustrate the constructions in an example.
We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for n p 2 , the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The… (More)
The source of a simple kG-module, for a finite p-solvable group G and an algebraically closed field k of prime characteristic p, is an endo-permutation module (see [Pu1] or [Th]). L. Puig has proved, more precisely, that this source must be isomorphic to the cap of an endo-permutation module of the form Q/R∈S Ten P Q Inf Q Q/R (M Q/R), where M Q/R is an… (More)
We show that K∞ and K ∞ control transfer in every fusion system on a finite p-group when p ≥ 5, and that they control weak closure of elements in every fusion system on a finite p-group when p ≥ 3. This generalizes results of G. Glauberman concerning finite groups.