Weak Cayley table groups of some crystallographic groups

TL;DR

This paper characterizes the group of symmetries of the weak Cayley table for all wallpaper groups and some crystallographic groups, revealing their algebraic structure.

Contribution

It explicitly determines the weak Cayley table groups for all seventeen wallpaper groups and additional crystallographic groups, expanding understanding of their symmetry properties.

Findings

Computed W(G) for all wallpaper groups

Identified structural differences among crystallographic groups

Enhanced understanding of weak Cayley table symmetries

Abstract

For a group $G$, a weak Cayley isomorphism is a bijection $f:G \to G$ such that $f(g_1g_2)$ is conjugate to $ f(g_1)f(g_2)$ for all $g_1,g_2 \in G$. They form a group $\mathcal W(G)$ that is the group of symmetries of the weak Cayley table of $G$. We determine $\mathcal W(G)$ for each of the seventeen wallpaper groups $G$, and for some other crystallographic groups.