Groups in linear algebra and elementary geometry

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 the simplest possible example, the one-dimensional real case.

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

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

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

Reference: Andrew Baker, Matrix Groups