Surjective homomorphism from direct sum $X\oplus Y$ to $X$

Question

Suppose $X$, $Y$ are two abelian groups, and we have a surjective homomorphism $f$ from the direct sum $X\oplus Y$ to $X$, is it always true that $Y\cong \ker(f)$ ?

Thanks!

Answer 1

No. Take $X = \mathbb{Z}_2$ and $Y = \mathbb{Z}_4$. Define $$f:\mathbb{Z}_2\oplus \mathbb{Z}_4 \to \mathbb{Z}_2: (a,b) \mapsto b \mod 2$$ Then $f$ is clearly surjective but $$\ker(f) = \{(0,0),(0,2),(1,0),(1,2)\} \cong \mathbb{Z}_2 \times \mathbb{Z}_2 \ncong \mathbb{Z}_4.$$