Cosets of subgroup of linear maps from $\mathbb{R}$ to itself

Question

$G=\{\alpha_{ab}|a,b\in \mathbb{R}\ ,a \neq {0}$} where $\alpha_{ab}$ is the mapping from $\mathbb{R} to \mathbb{R} defined as \alpha_{ab}:x\to ax+b $
$H=\{\alpha_{ab} \in G|a \in \mathbb{Q}\ ,a \neq {0} $}
I can show $G$ is a group and $H$ is its subgroup but not able to find the cosets of $H$.
Please help me .
Any help will be appreciated

Answer 1

First of all, you should verify that $H$ is not only a subgroup, but a normal subgroup, otherwise you can't talk about cosets (as opposed to left/right cosets). You can check that $H$ is normal by showing that $gh\in H$ if and only if $hg\in H$. The cosets are the equivalence classes under the relation $g~h$ if and only if $g^{-1}h\in H$. For any $g\in G$, you need to determine the equivalence class of maps $h\in G$ satisfying $g^{-1}h\in H$. It'll be helpful for you to explictly write what the inverse of the map $\alpha_{ab}$ is.