Is this a different proof of the fundamental group being abelian? I have proved the fundamental group of a topological group is abelian. But I've found nowhere the similar proof as mine. Everywhere I looked up, it was done either exploiting categorical properties or something like taking product of two paths. 
My proof goes as follows:
Let $a$ and $b$ be two loops in a topological group $(G,\bullet )$ starting at the identity element $e$. We need to show $ a\ast b \simeq b\ast a$, where "$\ast$" is the fundamental group operation.
Now for each $t,s\in [0,1]$, define
$F_t(s)=a(st)\ast(a(t)\bullet b(s))\ast \bar a(st)$
Now {$F_t$} gives the homotopy between $b$ and $a\ast b \ast \bar a$.

The main idea is at each time $t$, we first go to $a(t)$ along $a$ and then traverse the translated path $a(t)\bullet b$ and then return back along the inverse path of the first one. Continuity of $F$ follows from pasting lemma.

This proof seems correct but why do other proofs avoid this straightforward argument? 
 A: Your idea is very nice, but the definition
$$F_t(s)=a(st)\ast(a(t)\bullet b(s))\ast \bar a(st)$$
is inadequate because $a(st)$, $a(t)\bullet b(s))$, $\bar a(st)$ are single points of $G$ which cannot be composed by $*$ which is the succession of paths. What you mean is
$$F_t = a_t * (a(t) \bullet b) * \overline{a_t}$$
where $a_t(s) = a(st), \overline{a_t}(s) = a_t(1-s) = a((1-s)t)$. Note that $\overline{a_t}$ is the inverse of $a_t$, not part of the inverse $\bar a$ of $a$. In fact, $a((1-s)t) \ne  a(1-st) = \overline{a}(st)$. Explicitly you can also write
$$F_t(s) =  \begin{cases} a(3st) & 0 \le s \le 1/3 \\ a(t) \bullet b(3s - 1) & 1/3 \le s \le 2/3 \\  a((3(1-s)t)  & 2/3 \le s \le 1 \end{cases}$$
A: I really like your argument. This one I found uses the same idea, but it is neither as concise, nor as clear as yours. This proof is also pretty short, but the argument is a lot more subtle, and it took me a while to see what is happening. Finally, these notes describe the ideas of your proof without a formal proof. After reading these, I think that most authors are writing for an audience experienced enough that either (a) the readers are expected to fill in the details equivalent to your proof themselves, or (b) they would consider more subtle arguments just as easy. 
