Bijective group homomorphism/Inverse mapping/Homomorphism/Fact

Let ${}G$ and ${}H$ be groups, and let

$\varphi \colon G\longrightarrow H$

be a group isomorphism.

Then also the

inverse mapping

$\varphi ^{-1}\colon H\longrightarrow G,h\longmapsto \varphi ^{-1}(h),$

is a group isomorphism.

Proof, Write another proof