There are five groups that play an important role in linear algebra and elementary geometry.
The following table shows what they are and the relations between them.
Here $\bf k$ is an arbitrary commutative field (like $\R$ or $\bf C$).
\[ \boxed {\begin{array} {} \text {metric-preserving} && \text {not metric-preserving} \\ \hline \text {Isom}_n ({\bf k}) = \text {O}_n ({\bf k}) \ltimes \text {Trans}_n ({\bf k}) & \subset & \text {Aff}_n ({\bf k}) = \text {GL}_n ({\bf k}) \ltimes \text {Trans}_n ({\bf k}) \\ \cup && \cup \\ \text {O}_n ({\bf k}) & \subset & \text {GL}_n ({\bf k}) \end{array} } \]
The $\ltimes$ symbol denotes semi-direct product.
Reference: Andrew Baker, Matrix Groups
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