Partial insights into partial groups

For p a prime number, the study of p-local structures of finite groups, that is, of how a group acts
on its p-subgroups by conjugation, lies at the heart of several branches of mathematics. It first arose in the
classification of finite simple groups (especially for p = 2), but it also plays an important role in representation
theory and algebraic topology. This topological perspective has been developed through strong international
collaborations involving mathematicians from Denmark, such as Jesper Grodal and Jesper Møller, and from
France, such as Bob Oliver, together with researchers from several other countries.
In 2013, Andy Chermak introduced partial groups, a new tool for studying p-local structures. Roughly
speaking, a partial group is a group-like object in which not all products are defined. Although they originate
in group theory, partial groups can also be viewed as simplicial sets, combinatorial objects that play a central
role in algebraic topology. There is nowadays growing interest in partial groups among both group theorists and
algebraic topologists. Recently, while we were both visiting Copenhagen, I happened to meet Philip Hackney,
who had recently become interested in partial groups. During one of his talks, the question arose of whether
a finite group could admit infinitely many partial subgroups. We eventually proved, by introducing a suitable
notion of dimension, that every finite group has only finitely many partial subgroups.
In this talk, after some historical background on the emergence of partial groups, I will try to explain how
they can be understood from the perspective of (symmetric) simplicial sets, and how this topological viewpoint
can be powerful to study them.