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Classify the group $G=(\mathbb{Z}_2\times\mathbb{Z}_4)/(<(1_2,2_4)>)$ using the fundamental theorem finite abelian groups.

The order of $\mathbb{Z}_2\times\mathbb{Z}_4$ is $8$

The order of $<(1_2,2_4)>$ is $2$

Therefore , order of $G$ is $8/2=4$

My question is how do I know if $G\approx\mathbb{Z}_2\times\mathbb{Z}_2$ or $G\approx\mathbb{Z}_4$

Thanks in advance

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The Fundamental Theorem Of Finite Abelian Group tells you that an Abelian Group of order $4$ must be isomorphic to either $\Bbb Z_2 \times \Bbb Z_2$ or $\Bbb Z_4$.

Beyond this if you want to know whether $$\Bbb Z_2 \times \Bbb Z_4/ \langle (1_2,2_4)\rangle \approx \Bbb Z_2 \times \Bbb Z_2$$ or $$\Bbb Z_2 \times \Bbb Z_4/ \langle (1_2,2_4)\rangle \approx \Bbb Z_4$$

you will have to dirty your hands a bit.

Let $H=\langle (1_2,2_4)\rangle$.

By manual calculations we see that each of $(0,0)+H,\;(1,0)+H,\;(1,1)+H,\;(1,3)+H$ constitute to $4$ distinct cosets of $H$ in $\Bbb Z_2 \times \Bbb Z_4$ and hence these are $4$ distinct elements in $G=\Bbb Z_2 \times \Bbb Z_4/ \langle (1_2,2_4)\rangle$.

We observe that $(1,1)+H,\;(1,3)+H$ have order $4 $, so $G$ must be cyclic.

Hence $G\approx \Bbb Z_4$

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    So, because we are in $\mathbb{Z}_2\times\mathbb{Z}_4$ , I have to manually check if any one of the :$(1,0) + H, (1,1) + H , (1,2) + H , (1,3) + H , (0,0) + H , (0,1) +H $ has order $4$?? If one of this have order $4$ I can conclude $G\approx\mathbb{Z}_4$? – 2017-01-31
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    When we say two groups $G_1$ and $G_2$ are isomorphic, it means that they are same in every aspect, except that they are represented with different notations. It means that these two groups have the same structure. If $G_1$ has $10$ elements then $G_2$ also has 10 elements. If $G_1$ has say four elements of order $5$ then $G_2$ must also have four elements of order $5$. What I mean to say is that they are literally the same objects with just a different set of notations. So yes, if $|G|=4$ and if you can find an element of order $4$ in $G$ then the structure of $G$ will be same as $\Bbb Z_4$ – 2017-01-31