Massey Exercise 2.3.1 Let x,y \in X and let \gamma_1 and \gamma_2 be two path classes from x to y. We can characterize the conditions under which \gamma_1 and \gamma_2 give rise to the same isomorphism \pi(X,x) \rightarrow \pi(X,y) by observing the following. Suppose \varphi_1 : \alpha \mapsto \gamma_1^{-1} \alpha \gamma_1 and \varphi_2 : \alpha \mapsto \gamma_2^{-1} \alpha \gamma_2 are the same. Then we have, for all \alpha: % Denote \beta = \gamma_2 \gamma_1^{-1}, and notice that \beta \in \pi(X,x) and \gamma_1 \gamma_2^{-1} = \beta^{-1}. Then we have \beta \alpha \beta^{-1} = \alpha. Since this holds for all \alpha, we have that \beta \in Z(\pi(X,x)). By assuming that \gamma_2 \gamma_1^{-1} \in Z(\pi(X,x)), we can reverse the argument above to conclude that \varphi_1(\alpha) = \varphi_2(\alpha). So we have that the isomorphisms induced by \gamma_1 and \gamma_2 are identical if and only if \gamma_2 \gamma_1^{-1} \in Z(\pi(X,x)).