Let G ¼ ðP; LÞ be a parapolar space which is locally A nÀ1; 3 ðKÞ for some integer n > 6 and K a field. There exists a class D of 2-convex subspaces, each isomorphic to D 5; 5 ðKÞ, such that every symplecton of G is contained in a unique element of D. Let G ¼ ðP; LÞ be a parapolar space which is locally A nÀ1; 4 ðKÞ for n ¼ 7 or an integer n d 9 and some field K.Assume that G satisfies the extra condition called the Weak Hexagon Axiom. Then there exists a class D of 2-convex subspaces, each isomorphic to D 6; 6 ðKÞ, such that every symplecton of G is contained in a unique element of D. In both of the above cases G is the homomorphic image of a truncated building.2000 Mathematics Subject Classification. 51B25, 51E24 P We prove the following:Brought to you by |

No abstract

We define new collections of p-subgroups for a finite group G and p a prime dividing its order. We study the homotopy relations among them and with the standard collections of p-subgroups and determine their ampleness and sharpness properties.

As a counterpart for the prime 2 to Glauberman's $ZJ$-theorem, Stellmacher proves that any nontrivial 2-group $S$ has a nontrivial characteristic subgroup $W(S)$ with the following property. For any finite $\Sigma_4$-free group $G$, with $S$ a Sylow 2-subgroup of $G$ and with $O_2(G)$ self-centralizing, the subgroup $W(S)$ is normal in $G$. We generalize Stellmacher's result to fusion systems. A similar construction of $W(S)$ can be done for odd primes and gives rise to a Glauberman functor.Comment: LaTeX file, 19 page

Let G be group; a finite p-subgroup S of G is a Sylow p-subgroup if every finite p-subgroup of G is conjugate to a subgroup of S. In this paper, we examine the relations between the fusion system over S which is given by conjugation in G and a certain chamber system C, on which G acts chamber transitively with chamber stabilizer N G (S).Next, we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system. We determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system. We give an application to fusion systems with parabolic families of classical type.