I am trying to show that there is a canonical ismorphism between the abelianized fundamental groups, $\pi_1(X,p)_{ab}$ and $\pi_1(X,q)_{ab}$ of the path-connected space $X$.
I know since $X$ is path connected there exists an isomorphism $f: \pi_1(X,p) \to \pi_1(X,q)$ defined by $f(\lambda) \to \alpha * \lambda * \alpha^{-1}$
Ok, so I think there exists an isomorphism $k : \pi_1(X,p)_{ab}$ and $\pi_1(X,q)_{ab}$ that is defined as $k([g])=[f(g)]$
I have so far shown that $k$ is a homomorphism, but I am stuck trying to show that $k$ is bijective.
Here is where I am stuck:
$$k([g_1]) = k ([g_2])$$ Then,
$$ [ f(g_1)]= [f(g_2)]$$
I know that $f(g_1)= f(g_2)$ implies that $g_1=g_2$. But I don't know how to go from equality on the equivalence classes to $f(g_1)=f(g_2)$
My group theory knowledge is unfortunately rather weak.