Is $(\mathbb Z\times \mathbb Z)/(4\mathbb Z\times 5 \mathbb Z)\simeq \mathbb Z$,
where $4\mathbb Z\times 5 \mathbb Z$ is given as a subgroup of $\mathbb Z\times \mathbb Z$?
My feelings says that I need to find well defined $f:\mathbb Z\times \mathbb Z\to \mathbb Z$ which has a kernel $4\mathbb Z\times 5\mathbb Z$
So using first isomorphism theorem I can show the isomorphism.
I could not find the map and how can I show it in different way?
Best Answer
By definition of the quotients:
\begin{alignat}{1} (\mathbb Z\times \mathbb Z)/(4\mathbb Z\times 5 \mathbb Z) &= \{(m,n)+4\Bbb Z\times5\Bbb Z\mid m,n\in\Bbb Z\} \\ &= \{(m+4\Bbb Z,n+5\Bbb Z)\mid m,n\in\Bbb Z\} \\ &= \{(k,l)\mid k\in\Bbb Z/4\Bbb Z,l\in\Bbb Z/5\Bbb Z\} \\ &= \Bbb Z/4\Bbb Z \times\Bbb Z/5\Bbb Z \\ \end{alignat}
So, your quotient is precisely (not just isomorphic to) $\Bbb Z/4 \Bbb Z\times\Bbb Z/5\Bbb Z $, which is not isomorphic to $\Bbb Z$, if only for reasons of cardinality.