For two points $x_1$ and $x_2$ in a path-connected component, we know that their fundamental groups are isomorphic. In other words $$\pi_1(X,x_1)\cong\pi_1(X,x_2)$$
Let $\alpha$ be a path from $x_1$ to $x_2$. Also, let $p$ be a loop at $x_1$. Can we say $\alpha*\alpha^{-1}*p*\alpha*\alpha^{-1}$ and $p$ are homotopic?
One would be tempted to say the following:
We can prove that $\alpha*\alpha^{-1}*p*\alpha*\alpha^{-1}$ and $p$ are homotopic by proving $[\alpha*\alpha^{-1}*p*\alpha*\alpha^{-1}]=[p]$. Now $[\alpha*\alpha^{-1}*p*\alpha*\alpha^{-1}]=[\alpha*\alpha^{-1}]*[p]*[\alpha*\alpha^{-1}]$, and clearly $[\alpha*\alpha^{-1}]=[e_{x_1}]$. Hence $[\alpha*\alpha^{-1}*p*\alpha*\alpha^{-1}]=[e_{x_1}]*[p]*[e_{x_1}]=[p]$.
Are $\alpha*\alpha^{-1}*p*\alpha*\alpha^{-1}$ and $p$ homotopic?
I am not convinced with the validity of the above argument. Could someone comment on the validity?
Thanks in advance!