On Construction of Global Actions of Finite Partial Group Actions on Sets

TL;DR

This paper develops a method to construct global actions from finite partial group actions on sets, generalizing classical results like the orbit-stabilizer theorem to partial actions.

Contribution

It introduces a way to construct global actions from partial group actions and generalizes the orbit-stabilizer theorem for partial actions.

Findings

Generalization of orbit-stabilizer theorem for partial actions

Explicit calculation of orbit sizes in the global set

Framework for constructing global actions from partial actions

Abstract

A generalization of G-sets, called partial G-sets, are the sets that admit an action of partial maps on their subsets. Partial actions are a powerful tool to generalize many results of group actions. These generalizations are obtained by using global actions when they exist. The main objective of this paper is to construct the global action of a given finite partial group action on a set. For this, first we generalize orbit-stabilizer theorem for partial group actions and use it to know the exact size of the orbits in the global set.