A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$ $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let f,h be closed paths in G based at the Identity element e of G. Define $f.h$ by $(f.h)(t) = u(f(t),h(t))$ for all $t\in I$. Prove that $f*h$ ~ $f.h$ ~ $h*f$ and deduce that the fundamental group $\pi(G,e)$ is abelian.
Note the notation $f*h$ ~ $f.h$ means $f*h$ is equivalence to $f.h$, that is $f*h$ and $f.h$ are homotopic relative to $\{0,1\}$.
This exercise is proposed in Kosniowski's "A first course in Algebraic Topology" chapter 15. I am having trouble solving this and any help would be appreciated!
Let $c_e$ be the constant loop based at $e$. Then $f\cdot h \sim (f * c_e)\cdot(c_e * h) = f * h$ and $f\cdot h \sim (c_e * f)\cdot (h * c_e) = h * f$.