The transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the $p$-local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are however not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups, a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite $p$-group. As an application of the existence of a punctured group, we show that the subgroup homology decomposition on the centric collection is sharp for the fusion system. We also prove a Signalizer Functor Theorem for punctured groups and use it to show that the smallest Benson-Solomon exotic fusion system at the prime $2$ has a punctured group, while the others do not. As for exotic fusion systems at odd primes $p$, we survey several classes and find that in almost all cases, either the subcentric linking system is a punctured group for the system, or the system has no punctured group because the normalizer of some subgroup of order $p$ is exotic. Finally, we classify punctured groups restricting to the centric linking system for certain fusion systems on extraspecial $p$-groups of order $p^3$.
Bibliographical noteAcknowledgements. It was Andy Chermak who first asked the question in 2011 (arising out of his proof of existence and uniqueness of linking systems) of which exotic systems have localities on the set of non identity subgroups of a Sylow group. We thank him for comments on an earlier version of this paper and many helpful conversations, including during a visit to Rutgers in 2014, where he and the third author discussed the possibility of constructing punctured groups for the Benson-Solomon systems. We are grateful to George Glauberman for communicating to us several comments, corrections, and suggestions for improvement. We are especially grateful to the referee for pointing out that Lemma 4.8 conflicts with a lemma of Levi and Oliver, for alerting us to
errors too numerous to mention, and for other suggestions. We would like to thank the Centre for Symmetry and Deformation at the University of Copenhagen for supporting a visit of the third named author, where some of the early work on this paper took place. Finally, the authors thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Groups, representations and applications”, where work on this paper was undertaken and supported by EPSRC grant no EP/R014604/1.
- 20D20, 20J06, 20D05, 55R35