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What is the relation between the groups $I_2(ℝ)$ and $GL_2(ℝ)$ ?

I use those definitions:
$I_2(ℝ):= \{ ψ:ℝ^2→ℝ^2:d\left[ψ(x),ψ(y)\right]=d(x,y) \}$

And if I would define $\text{GL}_2(ℝ)$ in this context, then I think you could say the set of all the linear bijections: $\text{GL}_2(ℝ):= \{L:ℝ^2↔ℝ^2:∀x,y∈ℝ^2,α∈ℝ\left[L(x+y)=L(x)+L(y), αL(x)=L(αx)\right]\}$

Is it true that $GL_2(ℝ)\subset I_2(ℝ)$ ? I actually think this is not true, but can anybody help me how I can relate those two groups ?

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Does an arbitrary map from $GL_2(\mathbb{R})$ preserve distances? – Raskolnikov Mar 17 '13 at 13:22
Think of an invertible non orthogonal matrix. – Shahab Mar 17 '13 at 13:26
up vote 3 down vote accepted

$I_2(ℝ) \cap GL_2(ℝ) = O_2(ℝ)$

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Neither group is included in the other. For one direction, pick any matrix which does not preserve distances as @Shabab said. For the other direction $GL_2(\mathbb R)$ preserves the origin but $I_2(\mathbb R)$ contains also translations.

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