We show that the automorphism group of a linking system associated to a saturated fusion system $\mathcal {F}$ depends only on $\mathcal {F}$ as long as the object set of the linking system is $\mathrm {Aut}(\mathcal {F})$ -invariant. This was known to be true for linking systems in Oliver’s definition, but we demonstrate that the result holds also for linking systems in the considerably more general definition introduced previously by the author of this article. A similar result is proved for linking localities, which are group-like structures corresponding to linking systems. Our argument builds on a general lemma about the existence of an extension of a homomorphism between localities. This lemma is also used to reprove a theorem of Chermak showing that there is a natural bijection between the sets of partial normal subgroups of two possibly different linking localities over the same fusion system.

We classify all (saturated) fusion systems on bicyclic 2-groups. Here, a bicyclic group is a product of two cyclic subgroups. This extends previous work on fusion systems on metacyclic 2-groups. As an application we prove Olsson's conjecture for all blocks with bicyclic defect groups.

We prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × × … × for some positive integers n1, …, nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology.

We develop a theory of 2-fusion systems of even characteristic, and use that theory to show that all S3-free saturated 2-fusion systems are constrained. This supplies a new proof of Glauberman's Theorem on S4-free groups and its various corollaries.

We define sparse saturated fusion systems and show that, for odd primes, sparse systems are constrained. This simplifies the proof of the Glauberman–Thompson p-Nilpotency Theorem for fusion systems and a related theorem of Stellmacher. We then define a more restrictive class of saturated fusion systems, called extremely sparse systems, that are constrained for all primes.

In this paper we explore the possibility of defining $p$-local finite groups in terms of transfer properties of their classifying spaces. More precisely, we consider the question, posed by Haynes Miller, of whether an equivalent theory can be obtained by studying triples $(f,t,X)$, where $X$ is a $p$-complete, nilpotent space with a finite fundamental group, $f:BS\to X$ is a map from the classifying space of a finite $p$-group, and $t$ is a stable retraction of $f$ satisfying Frobenius reciprocity at the level of stable homotopy. We refer to $t$ as a retractive transfer of $f$ and to $(f,t,X)$ as a retractive transfer triple over $S$.

In the case where $S$ is elementary abelian, we answer this question in the affirmative by showing that a retractive transfer triple $(f,t,X)$ over $S$ does indeed induce a $p$-local finite group over $S$ with $X$ as its classifying space.

Using previous results obtained by the author, we show that the converse is true for general finite $p$-groups. That is, for a $p$-local finite group $(S,\mathcal{F},\mathcal{L})$, the natural inclusion $\theta:BS\to X$ has a retractive transfer $t$, making $(\theta,t,|\mathcal{L}|^{\wedge}_p)$ a retractive transfer triple over $S$. This also requires a proof, obtained jointly with Ran Levi, that $|\mathcal{L}|^{\wedge}_p$ is a nilpotent space, which is of independent interest.