Why does a group homomorphism send an identity element to an identity element?

In modern algebra, a homomorphism is a structure-preserving map between two groups, $G$ and $H$, defined by the property $f(g\oplus g^{\prime })=f(g)\otimes f(g^{\prime })$.

To understand why the identity element $e$ of $G$ must map to the identity element of $H$, consider that for any element $g$ in $G$, the identity property dictates $g\oplus e=g$. Applying the homomorphism $f$ to both sides yields:

$f(g\oplus e)=f(g)$

By the definition of a homomorphism, this expands to:

$f(g)\otimes f(e)=f(g)$

In group theory, the cancellation laws allow us to remove $f(g)$ from both sides of the equation. This leaves $f(e)$ equal to the identity element in $H$. Because homomorphisms preserve the fundamental algebraic structure, they necessarily map the unique identity of one group to the unique identity of the other.1