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I am trying to do an exercise in the book "Analysis on Lie group" as follows: Let $G$ be the group of all affine transformations in the plane, i.e. $G$ contains all the mapping of form $(x,y)\mapsto (x\cos\theta-y\sin\theta+a,x\sin\theta+y\cos\theta+b)$ where $a,b\in \mathbb{R}$ and $\theta\in [0,2\pi]$. Prove that $G$ is unimodular.

In fact I have found a left Haar measure on $G$ is $d\theta dadb$, but the same method doesn't work for calculating right Haar measure. So I get stuck. Somebody can help me? Thanks a lot!

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The unimodular function restricted to $SO(2)$ must be trivial by compacity, for elements of $\mathbb R^2\subset G$ you can compute explicetely $\det Ad (g) =1$ so the modular functions restricted to $\mathbb R^2 $ is trivial too. every element is a product of an element in each of this subgroups hence $G$ is unimodular.

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