I am reading Algebraic Topology by Allen Hatcher (available online at http://www.math.cornell.edu/~hatcher/AT/AT.pdf) and at line 1 of page 42, it reads:
"... because of the relation $L_{gg'}=L_gL_g'$, ..."
But it seems to me that in the previous page, he only shows that $L_{gg'}=L_gL_g'$ if there exists $G_{\alpha}$ with $g, g'\in G_{\alpha}$. If $g$ and $g'$ are in different $G_{\alpha}$'s, how to prove the relation $L_{gg'}=L_gL_g'$?
The context is quoted below:
If $g$ and $g'$ are in different $G_\alpha$, then by definition of $L$, $L(gg') = L_gL_{g'}$. If $g$ and $g'$ are in the same $G_\alpha$, $L(gg') = L_{gg'} = L_gL_{g'}$ by the formula. So for all $g$, $g'$, $L(gg') = L_gL_{g'}$, i.e. $L$ preserves the group operation.