Isometry Group
For , the isometries of form a subgroup of and therefore a group.
Proposition 1.27 (Isometry Groups In )
Proof
The identity map is an isometry. The composition of two isometries preserves distance, and if is an isometry then the distance-preservation equation applied to shows that its inverse is an isometry. These three facts verify the subgroup conditions.
Stated in
- Proposition 1.27 (Isometry Groups In )§1.7 Geometric Examples of Subgroups