49 research outputs found
Centralizers of Subsystems of Fusion Systems
When is a -local finite group and
(T,\mathcal{E},\mathcal{\L}_0) is weakly normal in
we show that a definition of
given by Aschbacher has a simple interpretation from which one can deduce
existence and strong closure very easily. We also appeal to a result of Gross
to give a new proof that there is a unique fusion system
on .Comment: 9 page
Centralizers of normal subgroups and the -Theorem
Glauberman's -theorem and analogous statements for odd primes show that,
for any prime and any finite group with Sylow -subgroup , the
centre of is determined by the fusion system
. Building on these results we show a statement that seems a
priori more general: For any normal subgroup of with
, the centralizer is expressed in terms of the
fusion system and its normal subsystem induced by .Comment: 3 pages; to appear in the Journal of Algebr
Tournaments, 4-uniform hypergraphs, and an exact extremal result
We consider -uniform hypergraphs with the maximum number of hyperedges
subject to the condition that every set of vertices spans either or
exactly hyperedges and give a construction, using quadratic residues, for
an infinite family of such hypergraphs with the maximum number of hyperedges.
Baber has previously given an asymptotically best-possible result using random
tournaments. We give a connection between Baber's result and our construction
via Paley tournaments and investigate a `switching' operation on tournaments
that preserves hypergraphs arising from this construction.Comment: 23 pages, 6 figure
Trees of Fusion Systems
We define a `tree of fusion systems' and give a sufficient condition for its
completion to be saturated. We apply this result to enlarge an arbitrary fusion
system by extending the automorphism groups of certain of its subgroups
Bounding the Number of Hyperedges in Friendship -Hypergraphs
For , an -uniform hypergraph is called a friendship
-hypergraph if every set of vertices has a unique 'friend' - that
is, there exists a unique vertex with the property that for each
subset of size , the set is a hyperedge.
We show that for , the number of hyperedges in a friendship
-hypergraph is at least , and we
characterise those hypergraphs which achieve this bound. This generalises a
result given by Li and van Rees in the case when .
We also obtain a new upper bound on the number of hyperedges in a friendship
-hypergraph, which improves on a known bound given by Li, van Rees, Seo and
Singhi when .Comment: 14 page
Weights for -local compact groups
In this short note, we initiate the study of -weights for an
-local compact group over a discrete -toral group
with discrete torus . Motivated by Alperin's Weight Conjecture for simple
groups of Lie-type, we conjecture that when is (algebraically)
connected, that is every element of is -fused into , the
number of weights of is equal to the number of ordinary
irreducible characters of its Weyl group. By combining the structure theory of
with the theory of blocks with cyclic defect group, we are able
to give a proof of this conjecture in the case when is simple and
. We also propose and give evidence for an analogue of the
Alperin-McKay conjecture in this setting.Comment: 11 page
Algorithms for fusion systems with applications to p-groups of small order
For a prime , we describe a protocol for handling a specific type of
fusion system on a -group by computer. These fusion systems contain all
saturated fusion systems. This framework allows us to computationally determine
whether or not two subgroups are conjugate in the fusion system for example. We
describe a generation procedure for automizers of every subgroup of the
-group. This allows a computational check of saturation. These procedures
have been implemented using MAGMA. We describe a program to search for
saturated fusion systems on -groups with
and . Employing these computational methods we
determine all such fusion system on groups of order where . This gives the
first complete picture of which groups can support saturated fusion systems on
small -groups of odd order