Suppose $G$ a finitely generated nilpotent topological group and we consider its rationalization $G_\mathbb{Q}$. This space may fail to be a topological group, but it's always a group-like H-space. By the way we will assume $G_\mathbb{Q}$ is a topological group.

We know that for rationalization of groups maps of the form $x\to x^n$ are bijections. My question is: can we determine if $G$ is rational (as topological space) by the maps of the above form? For example is this theorem correct?

Theorem: A topological group $G$ (such that its rationalization is a topological group) is rational, that is $G$ is homotopy equivalent to $G_\mathbb{Q}$, iff all the maps $x\to x^n$ are homotopy equivalences.

Suppose $G$ a finitely generated nilpotent topological group and we consider its rationalization $G_\mathbb{Q}$. This space may fail to be a topological group, but it's always a group-like H-space.

We know that for rationalization of groups maps of the form $x\to x^n$ are bijections. My question is: can we determine if $G$ is rational (as topological space) by the maps of the above form? For example is this theorem correct?

Theorem: A topological group $G$ is rational, that is $G$ is homotopy equivalent to $G_\mathbb{Q}$, iff all the maps $x\to x^n$ are homotopy equivalences.