A new group operation on the fudamental group

Question

Suppose $G$ is a topological group with operation $\cdot$ and identity element $x_0$. Let $\Omega (G, x_0)$ denote the set of all loops in $G$ based at $x_0$. For $f, g\in\Omega (G, x_0)$ define $f\otimes g$ by the rule $$(f\otimes g)(s)=f(s)\cdot g(s)$$ Then $\otimes$ induces a group operation on the fundamental group $\pi_1(G, x_0)$ as follows, $$[f]\otimes[g]=[f\otimes g]$$ Is this group operation $\otimes$ same as the usual group operation on $\pi_1(G, x_0)$ ? Is $\pi_1(G, x_0)$ abelian under this new group operation ?

Answer 1

Yes.

Look up the «Eckmann-Hilton argument».