Path partial groups
Antonio Díaz Ramos, Rémi Molinier, Antonio Viruel
TL;DR
The paper establishes the universality of the category ${\mathcal{P}}art$, showing that every abstract group $H$ appears as ${\rm Aut}_{{\mathcal{P}}art}({\mathbb M})$ for infinitely many non-isomorphic partial groups ${\mathbb M}$. It constructs a graph-driven family ${\mathbb M}(G,{\mathcal H})$ that encodes graph automorphisms and vertex-group automorphisms, and it highlights the path specialisation ${\mathbb P}(G)={\mathbb M}(G,{\mathbb Z_2})$, for which ${\rm Aut}_{{\mathcal{P}}art}({\mathbb P}(G))\cong {\rm Aut}_{{\mathcal G}raphs}(G)$. A key result is that the graph $G$ can be recovered from the maximal locally finite-subgroup graph of ${\mathbb M}(G,{\mathcal H})$, linking graph-theoretical invariants to algebraic properties of partial groups. Together, these results provide a framework to study universality and rigidity in ${\mathcal{P}}art$ and to translate graph-theoretic data into partial-group invariants via path constructions and colimits.
Abstract
It is well known that not every finite group arises as the full automorphism group of some group. Here we show that the situation is dramatically different when considering the category of partial groups, ${{\mathcal P}art}$, as defined by Chermak: given any group $H$ there exists infinitely many non isomorphic partial groups ${\mathbb M}$ such that $\operatorname{Aut}_{{\mathcal P}art}({\mathbb M})\cong H$. To prove this result, given any simple undirected graph $G$ we construct a partial group ${\mathbb P}(G)$, called the path partial group associated to $G$, such that $\operatorname{Aut}_{{\mathcal P}art}\big({\mathbb P}(G)\big)\cong \operatorname{Aut}_{{\mathcal G}raphs}(G)$.