 # Introduction to Groups and Geometries

## Section1.5More preliminary topics

### Subsection1.5.1A useful tool: commutative diagrams

A directed graph (or digraph ) is a set $$V$$ of vertices and a set $$E\subset V\times V$$ of directed edges. We draw pictures of digraphs by drawing an arrow pointing from a vertex $$v$$ to a vertex $$w$$ whenever $$(v,w)\in E\text{.}$$ See Figure 1.5.1.
A path in a directed graph is a sequence of vertices $$v_0,v_1,\ldots,v_{n}$$ such that $$(v_{i-1},v_i)\in E$$ for $$1\leq i\leq n\text{.}$$ The vertex $$v_0$$ is called the initial vertex and $$v_n$$ is called the final vertex of the path $$v_0,v_1,\ldots,v_{n}\text{.}$$ Figure 1.5.1. Example of a directed graph with vertex set $$V=\{a,b,c,d\}$$ and edge set $$E=\{(a,b),(c,b),(c,a),(a,d),(d,c)\text{.}$$ The vertex sequences $$c,b$$ and $$c,a,b$$ are both paths from $$c$$ to $$b\text{.}$$
A commutative diagram is a directed graph with two properties.
1. Vertices are labeled by sets and directed edges are labeled by functions between those sets. That is, the directed edge $$f=(X,Y)$$ denotes a function $$f\colon X\to Y\text{.}$$
2. Whenever there are two paths from an initial vertex $$X$$ to a final vertex $$Y\text{,}$$ the composition of functions along one path is equal to the composition of functions along the other path. That is, if $$X_0,X_1,\ldots,X_n$$ is a path with edges $$f_i\colon X_{i-1}\to X_{i}$$ for $$1\leq i\leq n$$ and $$X_0=Y_0,Y_1,Y_2,\ldots,Y_m=X_n$$ is a path with edges $$g_i\colon Y_{i-1}\to Y_{i}$$ for $$1\leq i\leq m\text{,}$$ then
\begin{equation*} f_n\circ f_{n-1}\circ\cdots\circ f_1=g_m\circ g_{m-1}\circ\cdots\circ g_1. \end{equation*}
Figure 1.5.2 shows a commutative diagram that illustrates the definition of conjugate transformations. Figure 1.5.3 shows a commutative diagram that goes with Fact 1.4.4. Figure 1.5.2. A commutative diagram illustrating the definition of conjugate transformations $$f,g$$ given in Exercise Group 1.3.3.3–6. Figure 1.5.3. A commutative diagram showing the relationship $$\overline{f}\circ \pi = f$$ in Fact 1.4.4.

### Exercises1.5.2Exercises

#### 1.

Let $$r$$ be a pure, unit quaternion. Use (1.2.13) to show that the map $$\R^3_\Quat \to \R^3_\Quat$$ given by $$u\to rur$$ is the reflection across the plane normal to $$r\text{.}$$ That is, show that $$rur=u-2(u\cdot r)r\text{.}$$ See Figure 1.5.4. Figure 1.5.4. The reflection of $$u\in \R^3_\Quat$$ across the plane normal to $$r\in \R^3_\Quat\text{.}$$

#### 2.Commutative diagram examples.

1. Draw a commutative diagram that illustrates the results of Exercise 1.3.3.5.
2. The distributive law for $$\Z_n$$ says that
\begin{equation*} [x]\left([y]+[z]\right) = [x][y] + [x][z] \end{equation*}
for all $$[x],[y],[z]\in \Z_n\text{.}$$ Label the maps in the commutative diagram below to express the distributive law. Figure 1.5.5.