Partial Group Symmetry in Figures I: Semidirect Products and the Six Coins

TL;DR

This paper introduces a new concept of partial symmetry for subsets of Euclidean space, capturing more detailed symmetry information than traditional groups, and develops a framework using partial group actions and semidirect products.

Contribution

It defines a novel partial group F for subsets of Euclidean space, extending symmetry analysis beyond connected sets, and introduces partial group actions and semidirect products to characterize these symmetries.

Findings

The partial group F captures detailed symmetries of disconnected sets.

A new definition of partial group action is proposed.

Construction of semidirect products for partial groups is presented.

Abstract

In this paper, we construct a partial group \(\mathcal{P}(F)\) that represents the "partial symmetry" inherent in a subset \(F\) of \(d\)-dimensional Euclidean space. In cases where \(F\) is not connected, \(\mathcal{P}(F)\) captures more detailed information than the conventional symmetry group \(G(F)\). To establish a stronger connection between \(\mathcal{P}(F)\) and \(F\), we introduce a novel definition of partial group action. Furthermore, to characterize \(\mathcal{P}(F)\) in specific cases, we define partial group actions on other partial groups and present a construction of the corresponding semidirect product.