arXiv subjects

Bob Oliver

Publications and source records attributed to Bob Oliver.

29 records · Page 2Linked to original sources

arXiv2016

Automorphisms of fusion systems of sporadic simple groups

We prove here that with a very small number of exceptions, when $G$ is a sporadic simple group and $p$ is a prime such that the Sylow $p$-subgroups of $G$ are nonabelian, then $Out(G)$ is isomorphic to the outer automorphism groups of the fusion and linking systems of $G$. In particular, the $p$-fusion system of $G$ is tame in the sense of [AOV1], and is tamely realized by $G$ itself except when $G\cong M_{11}$ and $p=2$. From the point of view of homotopy theory, these results also imply that $Out(G)\cong Out(BG^\wedge_p)$ in many (but not all) cases.

math.GR↗

arXiv2016

A remark on the construction of centric linking systems

We give examples to show that it is not in general possible to prove the existence and uniqueness of centric linking systems associated to a given fusion system inductively by adding one conjugacy class at a time to the categories. This helps to explain why it was so difficult to prove that these categories always exist, and also helps to motivate the procedure used by Chermak when he did prove it.

math.GR↗

arXiv2016

Reductions to simple fusion systems

We prove that if $\mathcal{E}\trianglelefteq\mathcal{F}$ are saturated fusion systems over $p$-groups $T\trianglelefteq S$, such that $C_S(\mathcal{E})\le T$, and either $Aut_{\mathcal{F}}(T)/Aut_{\mathcal{E}}(T)$ or $Out(\mathcal{E})$ is $p$-solvable, then $\mathcal{F}$ can be "reduced" to $\mathcal{E}$ by alternately taking normal subsystems of $p$-power index or of index prime to $p$. In particular, this is the case whenever $\mathcal{E}$ is simple and "tamely realized" by a known simple group. This answers a question posed by Michael Aschbacher, and is useful when analyzing involution centralizers in simple fusion systems, in connection with his program for reproving parts of the classification of finite simple groups by classifying certain 2-fusion systems.

math.GR↗

arXiv2016

Automorphisms of fusion systems of finite simple groups of Lie type

For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic, with a very short list of exceptions. When p is different from the defining characteristic, the situation is much more complex, but can always be reduced to a case where the natural map from Out(G) to outer automorphisms of the fusion or linking system is split surjective. This work is motivated in part by questions involving extending the local structure of a group by a group of automorphisms, and in part by wanting to describe self homotopy equivalences of the p-completion of the classifying space BG in terms of Out(G).

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arXiv2013

An algebraic model for finite loop spaces

The theory of p-local compact groups, developed in an earlier paper by the same authors, is designed to give a unified framework in which to study the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, as well as some other families of a similar nature. It also includes, and in many aspects generalizes, the earlier theory of p-local finite groups. In this paper we show that the theory extends to include classifying spaces of finite loop spaces. Our main theorem is in fact more general and states that in a fibration whose base spaces if the classifying space of a finite group, and whose fibre is the classifying space of a p-local compact group, the total space is, up to p-completion the classifying space of a p-local compact group.

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arXiv2012

Existence and uniqueness of linking systems: Chermak's proof via obstruction theory

We present a version of a proof by Andy Chermak of the existence and uniqueness of centric linking systems associated to arbitrary saturated fusion systems. This proof differs from the one by Chermak in that it is based on the computation of higher derived functors of certain inverse limits. This leads to a much shorter proof, but one which is aimed mostly at researchers familiar with homological algebra.

math.GR↗

arXiv2011

Equivalences between fusion systems of finite groups of Lie type

We prove, for certain pairs G,G of finite groups of Lie type, that the p-fusion systems for G and G' are equivalent. In other words, there is an isomorphism between a Sylow p-subgroup of G and one of G' which preserves p-fusion. This occurs, for example, when G=H(q) and G'=H(q') for a simple Lie type H, and q and q' are prime powers, both prime to p, which generate the same closed subgroup of the p-adic units. Our proof uses homotopy theoretic properties of the p-completed classifying spaces of G and G', and we know of no purely algebraic proof of this result.

math.GR↗

arXiv2010

Reduced, tame and exotic fusion systems

We define here two new classes of saturated fusion systems, reduced fusion systems and tame fusion systems. These are motivated by our attempts to better understand and search for exotic fusion systems: fusion systems which are not the fusion systems of any finite group. Our main theorems say that every saturated fusion system reduces to a reduced fusion system which is tame only if the original one is realizable, and that every reduced fusion system which is not tame is the reduction of some exotic (nonrealizable) fusion system.

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arXiv2005

Subgroup families controlling p-local finite groups

A p-local finite group consists of a finite p-group S, together with a pair of categories which encode ``conjugacy'' relations among subgroups of S, and which are modelled on the fusion in a Sylow p-subgroup of a finite group. It contains enough information to define a classifying space which has many of the same properties as p-completed classifying spaces of finite groups. In this paper, we examine which subgroups control this structure. More precisely, we prove that the question of whether an abstract fusion system F over a finite p-group S is saturated can be determined by just looking at smaller classes of subgroups of S. We also prove that the homotopy type of the classifying space of a given p-local finite group is independent of the family of subgroups used to define it, in the sense that it remains unchanged when that family ranges from the set of F-centric F-radical subgroups (at a minimum) to the set of F-quasicentric subgroups (at a maximum). Finally, we look at constrained fusion systems, analogous to p-constrained finite groups, and prove that they in fact all arise from groups.

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arXiv2005

Extension of p-local finite groups

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arXiv2005

Construction of 2-local finite groups of a type studied by Solomon and Benson

A p-local finite group is an algebraic structure with a classifying space which has many of the properties of p-completed classifying spaces of finite groups. In this paper, we construct a family of 2-local finite groups, which are exotic in the following sense: they are based on certain fusion systems over the Sylow 2-subgroup of Spin_7(q) (q an odd prime power) shown by Solomon not to occur as the 2-fusion in any actual finite group. Thus, the resulting classifying spaces are not homotopy equivalent to the 2-completed classifying space of any finite group. As predicted by Benson, these classifying spaces are also very closely related to the Dwyer-Wilkerson space BDI(4). Erratum (11 Feb 2005). An error in the paper was pointed out to the authors by Andy Chernak. The error is corrected in the erratum at the end of version 2, which should be read alongside the paper.

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