Both, left and right translation , are homeomorphisms .
Proof
Given a group \(G\). For every \(g\in G\) the left translation \(L_g\) is continuous, since it is a composition of continuous functions (\(L_g=p\circ i_g\) with \(i_g(g')=(g,g')\) and \(p\) denotes the group product map
).
Since \((L_g)^{-1}=L_{g^{-1}}\) and \(L_{g^{-1}}\) is also continuous, all left translations are homeomorphisms.
Similarly, right translations are also homeomorphisms.