Direct sum | EPFL Graph Search

The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more elementary kind of structure, the abelian group. The direct sum of two abelian groups A and B is another abelian group A\oplus B consisting of the ordered pairs (a,b) where a \in A and b \in B. To add ordered pairs, we define the sum (a, b) + (c, d) to be (a + c, b + d); in other words addition is defined coordinate-wise. For example, the direct sum \Reals \oplus \Reals , where \Reals is real coordinate space, is the Cartesian plane, \R ^2 . A similar process can be used to form the direct sum of two vector spaces or two modules. We can also form direct sums with any finite number of summands, fo

Finite metacyclic groups as active sums of cyclic subgroups

Nadia Romero Romero

The notion of active sum provides an analogue for groups of what the direct sum is for abelian groups. One natural question then is which groups are the active sum of a family of cyclic subgroups. Many groups have been found to give a positive answer to this question, while the case of finite metacyclic groups remained unknown. In this note we show that every finite metacyclic group can be recovered as the active sum of a discrete family of cyclic subgroups.

Elsevier2014

Modules d'endo-p-permutation

Jean-Marie Urfer

This dissertation is concerned with the study of a new family of representations of finite groups, the endo-p-permutation modules. Given a prime number p, a finite group G of order divisible by p and an algebraically closed field k of characteristic p, we say that a kG-module M is an endo-p-permutation module if its endomorphism algebra Endk(M) is a p-permutation kG-module, that is a direct summand of a permutation kG-module. This generalizes the notion, first introduced by E. Dade in 1978, of endo-permutation modules for p-groups . For P a p-group, E. Dade defined an abelian group structure on the set of isomorphism classes of indecomposable endo-permutation kP-modules with vertex P and he proved that the complete description of the structure of this group is equivalent to the classification of endo-permutation kP-modules. This group of isomorphism classes is now called the Dade group of the p-group P. The problem of describing the Dade group for an arbitrary p-group was recently solved by S. Bouc. This opens the question of studying endo-p-permutation modules, which are the natural generalization to arbitrary finite groups of endo-permutation modules. In the following text, we present the basic properties of endo-p-permutation modules and give a characterization of indecomposable endo-p-permutation modules with vertex P via properties of their sources modules. In particular, when the normalizer of P controls p-fusion, we are able to give a complete classification of sources of indecomposable endo-p-permutation modules with vertex P, using Bouc's description of the Dade group. When p is odd, we also give an alternative proof of a theorem of Dade concerned with extensions of endo-permutation modules, using our previous results. We present a consequence of this theorem of Dade on the structure of the multiplicity module associated to an indecomposable endo-p-permutation module. Finally we study some concrete examples of endo-p-permutation modules such as relative syzygies and relative Heller translates. We prove also that the Green correspondent of an indecomposable kNG(P)- endo-p-permutation module with vertex P is not in general an endo-p-permutation kG-module. The study of such representations is motivated by the important role they play in certain areas of representations theory. For instance, endo-permutation modules, and more generally endo-p-permutation modules (as is proved here), appear in the study of simple modules for p-solvable groups.

EPFL2006

Varieties of modules for Z/2Z x Z/2Z

Let k be an algebraically closed field of characteristic 2. We prove that the restricted nilpotent commuting variety C, that is the set of pairs of (n x n)-matrices (A, B) such that A(2) = B-2 = [A, B] = 0, is equidimensional. C can be identified with the 'variety of n-dimensional modules' for Z/2Z x Z/2Z, or equivalently, for k[X, Y]/(X-2, Y-2). On the other hand, we provide an example showing that the restricted nilpotent commuting variety is not equidimensional for fields of characteristic > 2. We also prove that if e(2) = 0 then the set of elements of the centralizer of e whose square is zero is equidimensional. Finally, we express each irreducible component of C as a direct sum of indecomposable components of varieties of Z/2Z x Z/2Z-modules. (c) 2007 Elsevier Inc. All rights reserved.

Elsevier2007

Modules d'endo-p-permutation

Jean-Marie Urfer

EPFL2006