I got a little bit confused with Hatcher's explanation of fundamental group action on the chain complex of the universal cover (Algebraic Topology, A. Hatcher, p. 328).
Assume $X$ is a path-connected space with universal cover $\tilde{X}$. As I unterstood, there supposed to be a right action $*$ of $\pi_1(X)$ on $C_n(\tilde{X})$ deinfed for $[\gamma]\in\pi_1(X)$ and an $n$-singular simplex $\sigma\in C_n(\tilde{X})$ as follows $$ \sigma*[\gamma]=\tilde{\gamma}\circ\sigma $$ where $\tilde{\gamma}:I\rightarrow\tilde{X}$ is a unique lift of the loop $\gamma:I\rightarrow X$ with $p(\tilde{\gamma}(0))=\gamma(0)=\gamma(1)$ (here $p$ is the covering map).
My problem pops out when it comes to verifying group action properties. Namely we should have $\sigma*([\gamma_1][\gamma_2])=(\sigma*[\gamma_1])*[\gamma_2]$. But (denoting by $\centerdot$ the path concatenation) $$ LHS=\sigma*[\gamma_1\centerdot\gamma_2]=\widetilde{\gamma_1\centerdot\gamma_2}\circ\sigma=(\widetilde{\gamma_1}\centerdot\widetilde{\gamma_2})\circ\sigma $$ whereas $$ RHS=(\widetilde{\gamma_1}\circ\sigma)*[\gamma_2]=\widetilde{\gamma_2}\circ\widetilde{\gamma_1}\circ\sigma $$ I don't see how $LHS=RHS$. Should it be the case that $\widetilde{\gamma_1}\centerdot\widetilde{\gamma_2}=\widetilde{\gamma_2}\circ\widetilde{\gamma_1}$? It seems strange to me.