Prove that there is no homomorphism from $\mathbb{Z}_{8} \oplus \mathbb{Z}_{2}$ ont0 $\mathbb{Z}_{4} \oplus \mathbb{Z}_{4}$.
My idea for the proof : Let $\phi$ be such homomorphism. Since Ker $\phi$ is a subgroup of $\mathbb{Z}_{8} \oplus \mathbb{Z}_{2}$ , then i must find all possible subgroups of it and then prove that $(\mathbb{Z}_{8} \oplus \mathbb{Z}_{2})/$Ker$\:\phi $ is not isomorphic with $\mathbb{Z}_{4} \oplus \mathbb{Z}_{4}$ where Ker $\phi$ can be any of the subgroup. To prove it isnt isomorphic is by finding elements in $\mathbb{Z}_{8} \oplus \mathbb{Z}_{2}$ that has order such that no elements in $\mathbb{Z}_{4} \oplus \mathbb{Z}_{4}$ have that order.
Is there any better way..?
Best Answer
A surjective function between two finite sets of the same cardinality is also injective, so your homomorphism would be an isomorphism. This is not possible, because the first group has an element of order $8$ and the second one doesn't.