Preorders and their special cases

$$\boxed { \begin{array} {} & \text {preorders} & \\ & \text {reflexive:} \; x \leq x & \\ & \text {transitive:}\; x\leq y \;\&\; y\leq z \Rightarrow x\leq z & \\ \boxed { \begin{array} {} \text {partial order} \\ \text {skeletal:}\; x\leq y \;\&\; y\leq x \Rightarrow x=y \\ \text {example: power set } X\mathcal P \\ \end{array} } & \boxed { \begin{array} {} \text{intersection} \\ \text{discrete order: only $x=x$ allowed } \\ \end{array} } & \boxed { \begin{array} {} \text {equivalence relation} \\ \text {symmetric: } x\leq y \Rightarrow y\leq x \\ \text{example: congruence } m \equiv n \; (\text{mod } k) \\ \end{array} }\\ \end{array}  }$$

Groups in linear algebra and elementary geometry

Preliminary Draft 

There are several groups that play an important role in linear algebra and elementary geometry.

The following table shows what they are and the relations between them.

It includes one of the simplest possible examples.

In the table $\bf k$ is an arbitrary commutative field (like $\R$ or $\bf C$).

$$\boxed {\begin{array} {} \text {rotations} & \subset & \text {+ reflections = isometries} & \subset & \text {affine transformations} \\ \hline \text {SO}_n ({\bf k}) \ltimes \text {Trans}_n ({\bf k}) & \subset & \text {Isom}_n ({\bf k}) = \text {O}_n ({\bf k})  \ltimes \text {Trans}_n ({\bf k}) & \subset & \text {Aff}_n ({\bf k}) = \text {GL}_n ({\bf k})  \ltimes \text {Trans}_n ({\bf k}) \\ \cup &&  \cup && \cup \\ \text {SO}_n ({\bf k}) & \subset & \text {O}_n ({\bf k})   & \subset & \text {GL}_n ({\bf k}) \\ \text {SO}_1(\R) = \text {O}_1(\R)^+ = \{1\in\R\} & \subset & \text {O}_1 (\R) = \{1,-1\in\R\}   & \subset & \text {GL}_1 (\R)= \{A\in\R: A\not= 0\}  \\   \end{array} }$$

The $\ltimes$ symbol denotes semi-direct product.

Reference: Andrew Baker, Matrix Groups