Path partial groups

TL;DR

This paper demonstrates that in the category of partial groups, any group can be realized as an automorphism group, contrasting with the classical case where not all groups are automorphism groups of groups.

Contribution

It constructs a new class of partial groups called path partial groups, linking their automorphisms to graph automorphisms, and shows any group can be realized as a partial group automorphism.

Findings

Any group can be realized as a partial group automorphism group.

Introduces path partial groups associated with graphs.

Establishes a correspondence between graph automorphisms and partial group automorphisms.

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 $\operatorname{Aut}_{{\mathcal P}art}\big({\mathbb P}(G)\big)\cong \operatorname{Aut}_{{\mathcal G}raphs}(G)$.