Classify the group (Z4 x Z2)/({0} x Z2).
I know the order of (Z4 x Z2)/({0} x Z2) is 4, and groups with order 4 are either the Klein-4 Group or Z4. I know Z4 is abelian and cyclic and the Klein-4 Group is abelian and noncyclic. I also have a theorem that states since 4 and 2 are not relatively prime, Z4 x Z2 is not cyclic. So would the group be the Klein-4 group then?
On
$\newcommand{\ZZ}{\mathbb{Z}}$ @Not Me's answer is the best way to do this, but if you haven't learned about the first isomorphism theorem yet:
Consider $(a,b) \in \ZZ_4 \times \ZZ_2$, and what it is equivalent to in the quotient group. It is equivalent to $(c,0)$ exactly when $a = c$. So a good conjecture is that the map $\varphi : (\ZZ_4 \times \ZZ_2) / (\{0\} \times \ZZ_2) \to \ZZ_4$ that sends $\overline{(a,b)} \mapsto a$ is a well-defined isomorphism.
Let $\varphi:\Bbb Z_4\times \Bbb Z_2\longrightarrow \Bbb Z_4\times\{0\}$ be the morphism defined by $\varphi(x,y)=(x,0)$. Show it's a surjective homomorphism, and use first isomorphism theorem.