26 research outputs found
The classification of torsion endo-trivial modules
This paper is a major step in the classification of endotrivial modules over
p-groups. Let G be a finite p-group and k be a field of characteristic p. A
kG-module M is an endo-trivial module if {\End_k(M)\cong k\oplus F} as
kG-modules, where F is a free module. The classification of endo-trivial
modules is the crucial step for understanding the more general class of
endo-permutation modules. The endo-permutation modules play an important role
in module theory, in particular as source modules, and in block theory where
they appear in the description of source algebras. Endo-trivial modules are
also important in the study of both derived equivalences and stable
equivalences of group algebras and block algebras.
The collection of isomorphism classes of endo-trivial modules modulo
projectives is an abelian group under tensor product. The main result of this
paper is that this group is torsion free except in the case that G is cyclic,
quaternion or semi-dihedral. Hence for any p-group which is not cyclic,
quaternion or semi-dihedral and any finitely generated kG-module M, if M
\otimes_k M \otimes_k ... \otimes_k M \cong k \oplus P for some projective
module P and some finite number of tensor products, then M \cong k \oplus Q for
some projective module Q. The proof uses a reduction to the cases in which G is
an extraspecial or almost extraspecial p-group, proved in a previous paper of
the authors, and makes extensive use of the theory of support varieties for
modules.Comment: 61 pages, published versio
Genuinely nonabelian partial difference sets
Strongly regular graphs (SRGs) provide a fertile area of exploration in
algebraic combinatorics, integrating techniques in graph theory, linear
algebra, group theory, finite fields, finite geometry, and number theory. Of
particular interest are those SRGs with a large automorphism group. If an
automorphism group acts regularly (sharply transitively) on the vertices of the
graph, then we may identify the graph with a subset of the group, a partial
difference set (PDS), which allows us to apply techniques from group theory to
examine the graph. Much of the work over the past four decades has concentrated
on abelian PDSs using the powerful techniques of character theory. However,
little work has been done on nonabelian PDSs. In this paper we point out the
existence of \textit{genuinely nonabelian} PDSs, i.e., PDSs for parameter sets
where a nonabelian group is the only possible regular automorphism group. We
include methods for demonstrating that abelian PDSs are not possible for a
particular set of parameters or for a particular SRG. Four infinite families of
genuinely nonabelian PDSs are described, two of which -- one arising from
triangular graphs and one arising from Krein covers of complete graphs
constructed by Godsil \cite{Godsil_1992} -- are new. We also include a new
nonabelian PDS found by computer search and present some possible future
directions of research.Comment: 24 page
Fusion systems on -groups with an extraspecial subgroup of index
In this thesis we classify saturated fusion systems on -groups containing an extraspecial subgroup of index for an arbitrary odd prime . We prove that if is a saturated fusion system on with = 1 then either or is isomorphic to a unique group of order \(p^{p-1}\
Elementary abelian subgroups in p-groups of class 2
All the results in this work concern (finite) p-groups. Chapter 1 is concerned with classifications of some classes of p-groups of class 2 and there are no particularly new results in this chapter, which serves more as an introductory chapter. The "geometric" method we use for these classifications differs however from the standard approach, especially for p-groups of class 2 with cyclic center, and can be of some interest in this situation. This "geometry" will for instance, prove to be particularly useful for the description of the automorphism groups performed in Chapter 3. Our main results can be found in chapters 2 and Chapter 3. The results of Chapter 2 have a geometric flavour and concern the study of upper intervals in the poset Ap(P) for p-groups P. We already know from work of Bouc and Thévenaz [8], that Ap(P)≥2 is always homotopy equivalent to a wedge of spheres. The first main result in Section 2.4, is a sharp upper bound, depending only on the order of the group, to the dimension of the spheres occurring in Ap(P)≥2. More precisely, we show that if P has order pn, then H~k(Ap(P)≥2) = 0 if k ≥ ⎣n-1/2⎦. The second main result in this section is a characterization of the p-groups for which this bound is reached. The main results in Section 2.3 are numerical values for the number of the spheres occurring in Ap(P)≥2 and their dimension, when P is a p-group with a cyclic derived subgroup. Using these calculations, we determine precisely in Section 2.5, for which p-groups with a cyclic center, the poset Ap(P) is homotopically Cohen-Macaulay. Section 2.7 is an attempt to generalize the work of Bouc and Thévenaz [8]. The main result of this section is a spectral sequence E1rs converging to H~r+s(Ap(P)>Z), for any Z ∈ Ap(P). We show also that this spectral sequence can be used to recover Bouc and Thévenaz's results [8]. In Section 2.8, we give counterexamples to results of Fumagalli [12]. As an important consequence, Fumagalli's claim that Ap(G) is homotopy equivalent to a wedge of spheres, for solvable groups G, seems to remain an open question. The results of Chapter 3 are more algebraic and concern automorphism groups of p-groups. The main result is a description of Aut(P), when P is any group in one of the following two classes: p-groups with a cyclic Frattini subgroup. odd order p-groups of class 2 such that the quotient by the center is homocyclic
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Diameter, Girth And Other Properties Of Highly Symmetric Graphs
We consider a number of problems in graph theory, with the unifying theme being the properties of graphs which have a high degree of symmetry.
In the degree-diameter problem, we consider the question of finding asymptotically large graphs of given degree and diameter. We improve a number of the current best published results in the case of Cayley graphs of cyclic, dihedral and general groups.
In the degree-diameter problem for mixed graphs, we give a new corrected formula for the Moore bound and show non-existence of mixed Cayley graphs of diameter 2 attaining the Moore bound for a range of open cases.
In the degree-girth problem, we investigate the graphs of Lazebnik, Ustimenko and Woldar which are the best asymptotic family identified to date. We give new information on the automorphism groups of these graphs, and show that they are more highly symmetrical than has been known to date.
We study a related problem in group theory concerning product-free sets in groups, and in particular those groups whose maximal product-free subsets are complete. We take a large step towards a classification of such groups, and find an application to the degree-diameter problem which allows us to improve an asymptotic bound for diameter 2 Cayley graphs of elementary abelian groups.
Finally, we study the problem of graphs embedded on surfaces where the induced map is regular and has an automorphism group in a particular family. We give a complete enumeration of all such maps and study their properties
Fusion systems on -groups of sectional rank 3
In this thesis we study saturated fusion systems on -groups having sectional rank 3, for odd. We obtain a complete classification of simple fusion systems on p-groups having sectional rank 3 for ≥ 5, exhibiting a new simple exotic fusion system on a 7-group of order 75. We introduce the notion of pearls, defined as essential subgroups isomorphic to the groups C X and (for odd), and we illustrate some properties of fusion systems involving pearls. As for = 3, we determine the isomorphism type of a certain section of the 3-groups considered
Model theory of finite and pseudofinite groups
This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory