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Concept# Solvable group

Summary

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.
Motivation
Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0). This means associated to a polynomial f \in F[x] there is a tower of field extensionsF = F_0 \subseteq F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m=Ksuch that
# F_i = F_{i-1}[\alpha_i] where \alpha_i^{m_i} \in F_{i-1}, so \alpha_i is a solution to the equation x^{m_i} - a where a \in F_{i-1}

# F_m con

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We exhibit a counterexample to a fiber theorem stated by F. Fumagalli [J. Algebra 283 (2) (2005), 639-654] and show how it affects the rest of Fumagalli's paper. As a consequence, whether the poset A_p(G) is homotopy equivalent to a wedge of spheres for any finite solvable group G seems to remain an open question.

2009All 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.

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.