---- > [!definition] Definition. ([[Cayley diagram]]) A **Cayley diagram** for a [[group]] consists of nodes that are connected by $\textcolor{Thistle}{\text{arrow}}$s that are $\textcolor{Thistle}{\text{generator}}$-colored (or labeled) where: >- an $\textcolor{Thistle}{\text{arrow}}$ of a particular color represents a specific $\textcolor{Thistle}{\text{generator}}$; >- each $\textcolor{SkyBlue}{\text{action}}$ of the group is represented by a unique $\textcolor{SkyBlue}{\text{node}}$ (sometimes we will label $\textcolor{SkyBlue}{\text{node}}$s by the corresponding $\textcolor{SkyBlue}{\text{action}}$). >- Equivalently, each $\textcolor{SkyBlue}{\text{action}}$ is represented by a (non-unique) $\textcolor{SkyBlue}{\text{path}}$ starting from the *solved state*. >\ >Lines with no tips are two-way arrows. >\ >Multiple paths can lead to the same node; these are 'relations' in our [[group]]. > [!basicproperties] > **Regularity Property.** The algebraic relations of a group (e.g., $yx y^{-1}=x ^{-1}$ in the [[dihedral group]]) give Cayley diagrams a uniform symmetry: every part of the diagram is structured like every other. For example, look at the Cayley diagram below for $D_{3}$ and think about > "Blue-Red-Blue = Backwards Red": ![[CleanShot 2023-09-10 at 11.20.52.jpg]] > \ > ![[CleanShot 2023-09-10 at 11.22.48.jpg]] > \ > ![[CleanShot 2023-09-10 at 11.24.34.jpg]] > \ > ![[CleanShot 2023-09-10 at 11.25.10.jpg]] ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```