Corollary 12.2:

If X is a topological group with unit element $e$ then $\pi_1(X,e)$ is abelian. Moreover, if $\gamma_1,\; \gamma_2$ are two loops based at $e$ define the binary operation $\circ$ on $\pi_1(X,e)$ by³

$$[\gamma_1] \circ [\gamma_2] = [\gamma_1(t) \cdot \gamma_2(t)]$$

where $\gamma_1(t)\cdot\gamma_2(t)$ denotes the group multiplication in $X.$ Then

$$[\gamma_1] \circ [ \gamma_2 ]=[\gamma_1] [ \gamma_2 ],$$

the right hand side being the product in $\pi_1(X,e)$. In other words, $\gamma_1(t) \cdot \gamma_2(t) \sim \gamma_1 \ast \gamma_2$.