Can someone explain with some simple examples what is meant by the isometry and isometry group? I'm a student of physics and I often come across this term in texts of general relativity for various spaces. In particular, I want to understand the statement why
The isometry group of de Sitter space is the Lorentz group $O(1,n)$.
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As an example, let $\mathbb{E}$ be Euclidean Space. Then every isometry, $A$ of $\mathbb{E}^N$ can be written as $$A: \bf{x} \mapsto B {\bf x} + \bf{v}$$
With $B\in O(N)$ and $\bf{v} \in \mathbb{E}^N$. Then one can define a map $$\phi: \text{Isom}(\mathbb{E}^N) \rightarrow O(N)$$ By $A \mapsto B$. the group $\text{ Isom}$ is a group homomorphism whose kernel is equal to $\text{Trans}(\mathbb{E}^N)$ of all translations of $\mathbb{E}^N$.
An isometry is a shape preserving transformation. Rotations and reflections are two examples. A dilation is not an isometry because it changes the size of the shape. An example of an isometry group would be all the transformations of a say a regular hexagon (rotations and reflection) that would result in no change in the appearance (symmetry). The group would have 12 elements.