## Inverse Map of a Bijective Homomorphism is a Group Homomorphism

## Problem 445

Let $G$ and $H$ be groups and let $\phi: G \to H$ be a group homomorphism.

Suppose that $f:G\to H$ is bijective.

Then there exists a map $\psi:H\to G$ such that

\[\psi \circ \phi=\id_G \text{ and } \phi \circ \psi=\id_H.\]
Then prove that $\psi:H \to G$ is also a group homomorphism.