[증명] Condition for fundamental group to be abelian.

Proposition.

Let $x_0$ and $x_1$ be points of the path-connected space $X$. Show that $\pi_1(X,x_0)$ is abelian if and only if for every  pair of $a$ and $b$ of paths from $x_0$ to $x_1$, we have $\hat{a}=\hat{b}$.

Proof.
Let $h=f*a$.
First, start with $\hat{a}=\hat{b}$. Then $\hat{a}([f]*[g])=\hat{h}([f]*[g])=[\bar{h}]*[f]*[g]*[h]=[\bar{a}]*[\bar{f}]*[f]*[g]*[f]*[a]$$=[\bar{a}]*[g]*[f]*[a]=\hat{a}([g]*[f])$. Because $\hat{a}$ is an isomorphism, $[f]*[g]=[g]*[f]$.

Now, start with $[f]*[g]=[g]*[f]$. Then $\hat{h}([g]*[f])=\hat{h}([f]*[g])=[\bar{h}]*[f]*[g]*[h]=[\bar{a}]*[\bar{f}]*[f]*[g]*[f]*[a]$$=[\bar{a}]*[g]*[f]*[a]=\hat{a}([g]*[f])$, so $\hat{h}=\hat{a}$ Now, by setting $f=b*\bar{a}$, we get $[h]=[b], \hat{h}=\hat{b}=\hat{a}$.

End of proof.