Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$ using the group law, while $\ast$ denotes the operation in the fundamental group. The way I can check this is rather direct: one lifts the loop (up to based homotopy) to a segment in $V$ and uses the identification of $\pi_1(T)$ with the lattice $\Gamma$.
Is there a more conceptual way to prove this identity that will extend to more general (real or complex) Lie groups, or maybe to linear algebraic groups? Or is this fact false in more generality?
Best Answer
Yay! It's the Eckmann-Hilton argument!
There are two group structures on $\pi_1(G)$ and they commute with each other. It turns out that that is sufficient to show that they are the same structure and that that structure is commutative.
For a proof of this, using interpretative dance, take a look at the movie in this seminar that I gave last semester. There's also something on YouTube by The Catsters (see the nLab page linked above).
(Forgot to actually answer your question!) This only depends on the fact that $\pi_1$ is a representable group functor and that $G$ is a group object in $hTop$. So it will extend to other group objects in $hTop$, such as those that you mention. This also explains why $\pi_k$ is abelian for $k \ge 2$ since $\pi_2(X) = \pi_1(\Omega X)$ and $\Omega X$ is a group object in $hTop$.