How many homomorphism are there from the cyclic group $\mathbb{Z}_6$ to $\mathbb{Z}_2\times \mathbb{Z_4}$
Consider a homomorphism $\phi: \mathbb{Z}_6 \rightarrow \mathbb{Z}_2 \times \mathbb{Z}_4$
Since all homomorphisms maps the identity of the first group to that of the second,
so there are 8 homomorphisms are there i am right ...some one help me please
A homomorphism of a cyclic group is uniquely determined by where it sends the generator, i.e. which element you choose to be $\phi(1)$. Also, remember that whatever $\phi(1)$ you choose, you must have $$0=\phi(0)\\=\phi(1+1+1+1+1+1)\\=\phi(1)+\phi(1)+\phi(1)+\phi(1)+\phi(1)+\phi(1)$$Not every element in $\Bbb Z_2\times\Bbb Z_4$ fulfills this, but each one of those is a valid candidate. Count them and you have your answer.