Let $G = Z_4 \times Z_{2}$. Let $H_1 = \langle(2,1)\rangle$ and $H_2 = \langle (2,0)\rangle$. The groups $G / H_1$ and $G / H_2$ are isomorphic either $Z_4$ or the Klein 4-group $V$. Compute which one each is isomorphic to.
First I computed that $\langle(2,1)\rangle$ in $G$ is $\{(0,0),(2,1)\rangle$ and $\langle (2,0)\rangle$ is just $\{(0,0),(2,0)\}$ in $G$. They are groups of order 4. For next step I am confused how can I literally compute $G / H_1$ and $G / H_2$?? Or I should used the fundamental theorem of homomorphism to define some $\phi: G \rightarrow G'$? But how can I go from here?
Do you have two elements of order two? Then you have the Klein Vierergruppe, otherwise you have the cyclic one:
$$u:=(1,1)+H_1\;\implies u\neq2u\neq3u\implies G/H_1\cong C_4$$
Yet
$$v:=(1,1)+H_2\implies2u=\overline 0\;,\;\;\text{since}\;\;(2,2)=(2,0)\in H_2$$
$$w:=(1,2)+H_2\implies 2v=\overline 0\;,\;\;\text{since}\;\;2v=(2,4)=(2,0)\in H_2$$
so $\;H/H_2\cong V\cong C_2\times C_2\;$