forgetful functor from Grp to Set as conservative Functor

The forgetful functor from Grp to Set is a conservative functor.

Let \(\varphi: G \rightarrow H\) be a bijective group-homomorphism between groups \(G,H\). Then there exists a unique set theoretic inverse map

\begin{align*} \varphi^{-1}: H \rightarrow G \end{align*}

Then applying \(\varphi\) to \(\varphi^{-1}(h\cdot h'), \varphi^{-1}(h) \cdot \varphi^{-1}(h)\) results in

\begin{align*} \varphi(\varphi^{-1}(h \cdot h')) =& h \cdot h' \\ =& \varphi(\varphi^{-1}(h) \cdot \varphi(\varphi^{-1}(h')) \\ =& \varphi(\varphi^{-1}(h) \cdot \varphi^{-1}(h')) \end{align*}

and hence by injectivity of \(\varphi\) we conclude that

\begin{align*} \varphi^{-1}(h \cdot h') = \varphi^{-1}(h) \cdot \varphi^{-1}(h') \end{align*}