TITLE: there is no injective group homomorphism from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$
QUESTION [5 upvotes]: there is no injective group homomorphism from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$
But i don't know why it is true.
should i investigate all group homomorphisms from $\mathbb Z\times\mathbb Z$ into $\mathbb Z$?
Do you have any idea to help me? :(

REPLY [3 votes]: Hint : $\mathbb{Z} \times \mathbb{Z} $ is not cyclic but all subgroups of $\mathbb{Z}$ are cyclic.