This question is probably trivial for anyone who is more familiar with rational homotopy theory than me, but anyway:

Let $G$ be a simply-connected topological group. In particular, it is an $H$-space. Therefore the theorem of Cartan and Serre tells us that the rationalization $G_{\mathbb{Q}}$ of $G$ has the homotopy type of a product of Eilenberg-Maclane spaces. Here are my questions:

- Can one choose a functorial rationalization $G \to G_{\mathbb{Q}}$ that also preserves products, thereby turning $G_{\mathbb{Q}}$ into a topological group and $G \to G_{\mathbb{Q}}$ into a group homomorphism? Would this group structure coincide up to homotopy with the $H$-space structure on the product of EM-spaces?
- If the above is true, then the map $(BG)_{\mathbb{Q}}\to B(G_{\mathbb{Q}})$ induced by $G \to G_{\mathbb{Q}}$ should be a homotopy equivalence. Would this imply that $(BG)_{\mathbb{Q}}$ also decomposes into a product of Eilenberg-Maclane spaces?