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Albert Ruiz

Publications and source records attributed to Albert Ruiz.

9 recordsLinked to original sources
arXiv2021

Realizability and tameness of fusion systems

A saturated fusion system over a finite $p$-group $S$ is a category whose objects are the subgroups of $S$ and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over $S$ is realized by a finite group $G$ if $S$ is a Sylow $p$-subgroup of $G$ and morphisms in the category are those induced by conjugation in $G$. One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the classification of finite simple groups, but we also give more precise formulations whose proof is independent of the classification.

math.GR↗
arXiv2020

Simplicity of fusion systems of finite simple groups

We determine for which known finite simple groups $G$ and which primes $p$ the $p$-fusion system of $G$ is simple. This means first collecting together the results that were already known (and correcting two errors made in an earlier study of this question), and then handling the remaining cases. At the same time, we develop some new tools to use when determining $O^{p'}(\mathcal{F})$ for arbitrary saturated fusion systems.

math.GR↗
arXiv2017

Reduced fusion systems over $p$-groups with abelian subgroup of index $p$: III

We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian $p$-groups with an abelian subgroup of index $p$. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete $p$-toral groups with an abelian subgroup of index $p$. In all of these cases (finite or infinite), we reduce the problem to one of listing all $\mathbb{F}_pG$-modules (for $G$ finite) satisfying certain conditions: a problem which was solved in the earlier paper by Craven, Oliver, and Semeraro using the classification of finite simple groups.

math.GR↗
arXiv2017

Some new examples of simple $p$-local compact groups

In this paper we present new examples of simple $p$-local compact groups for all odd primes. We also develop the necessary tools to show saturation, simpleness and the non-realizability as $p$-compact groups or compact Lie groups, which can be applied in a more general framework.

math.AT↗
arXiv2011

Cohomological uniqueness, Massey products and the modular isomorphism problem for 2-groups of maximal nilpotency class

Let $G$ be a finite 2-group of maximal nilpotency class, and let $BG$ be its classifying space. We prove that iterated Massey products in $H^*(BG;\F_2)$ do characterize the homotopy type of $BG$ among 2-complete spaces with the same cohomological structure. As a consequence we get an alternative proof of the modular isomorphism problem for 2-groups of maximal nilpotency class.

math.AT↗
arXiv2010

Cohomological uniqueness of some $p$-groups

In this paper we consider classifying spaces of a family of $p$-groups and we prove that mod $p$ cohomology enriched with Bockstein spectral sequences determines their homotopy type among $p$-completed CW-complexes. We end with some applications to group theory.

math.AT↗
arXiv2004

All p-local finite groups of rank two for odd prime p

In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analisis of the possible saturated fusion systems in terms of the outer automorphism group ant the proper F-radical subgroups. Also, for each case in the classification, either we give a finite group with the corresponding fusion system or we check that it corresponds to an exotic p-local finite group, getting some new examples of these for p = 3.

math.AT↗