The following is one of Hatcher AT exercise problem.
Show the action of $\pi_1(\Bbb RP^n)$ on $\pi_n(\Bbb RP^n)\simeq\Bbb Z$ is trivial for $n$ odd and nontrivial for $n$ even.
Using the fact that for $n\geq 2$, the action of $\pi_1(\Bbb RP^n)$ on $\pi_n(\Bbb RP^n)$ corresponds to the action of $\pi_1(\Bbb RP^n)$ to $\pi_n(S^n)$, I can show the statement. How about the case $n =1$? Noting $\Bbb RP^1\cong S^1$, for $[\gamma]\in\pi_1(S^1)$, the action of $[\gamma]$ on $\pi_1(S^1)$ is $[\gamma]+[\alpha]-[\gamma] = [\alpha]$ for $[\alpha]\in\pi_1( S^1)$ so that the action is also trivial. Am I correct?