Topological group: Multiplying two loops is homotopic to linking these paths?

Let G be a topological group and let $s_1$ and $s_2$ be loops in G (both loops are based at the identity e of G). Is it true that the loop $s_1s_2$ (where the multiplication is the one of the group structure of G) is equal, in $\pi_1(G,e)$, to the loop $s_1*s_2$ where this product is given by first going around $s_1$ and then $s_2$ (i.e., do we have $[s_1s_2] = [s_1*s_2]$)? If yes, what is the proof for this?

Instead of using the Eckmann-Hilton argument abstractly, you can make it explicit, as follows:

We can reparameterize the loop $s_1$ to be constant (and equal to the neutral element) on $[1/2,1]$ and $s_2$ to be constant on $[0,1/2]$. Then $[s_1 \ast s_2] = [s_1 s_2]$ and since $s_1s_2 = s_2s_1$ we also get that the fundamental group is abelian.

Yes and this is known as the Eckmann-Hilton argument. (Wikipedia)