# Tagged: Eckmann–Hilton argument

## Problem 268

Let $G$ be a group with the identity element $e$ and suppose that we have a group homomorphism $\phi$ from the direct product $G \times G$ to $G$ satisfying
$\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}$ for any $g\in G$.

Let $\mu: G\times G \to G$ be a map defined by
$\mu(g, h)=gh.$ (That is, $\mu$ is the group operation on $G$.)

Then prove that $\phi=\mu$.
Also prove that the group $G$ is abelian.