Which of the following groups contains a unique normal subgroup of order $4$? 
Which of the following groups contains a unique normal subgroup of order $4$?

*

*$Z_2\oplus Z_4$


*$D_8$


*$Q_8$


*$Z_2\oplus Z_2\oplus Z_2$

$\{0\}\oplus Z_4$ is a normal subgroup of $1$? I have no idea about $2,3$. In $4$, we have $Z_2\oplus Z_2\oplus \{0\}$, will it be normal? I think yes. Thank you for discussion.
 A: Note that options (1) and (4) are out, since they are both abelian and contain multiple subgroups of order 4; $0 \oplus \mathbb{Z}_4$ and $\mathbb{Z}_2  \oplus \langle 2 \rangle$ in the first case, and $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus 0$ and $0 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$ in the second.
Also, $Q_8$ is out, since it contains three subgroups of order 4, all of which are normal (they each have index $2$). Namely, the subgroups are $\{\pm 1, \pm i\}$, and the similar choices for $j$ and $k$. Note that these are also the cyclic subgroups generated by elements of order $4$.
A: This question is from the GATE exam (the PhD entrance exam in India). This particular one was later marked as incorrect, i.e. in the answer keys for that year's GATE exam, the correct answer for this problem was indicated as "Any combination". The reason is that it is assumed that at least one answer is correct (there can be several correct answers in GATE). While this problem has 0 correct answers. From this point of view, this is an exceptional problem (maybe something like 1 problem in 10 years' GATE exams).
The GATE exam usually tests your knowledge of particular facts. You have to solve something like 65 problems and get 100 points in 3 hours, so you literally have about 1-2 minutes for each point. Problems such as this one are 1-point problems, as they do not require one to do any long calculations that would require more than 1-2 minutes. And if you know certain facts, all 1-point problems are quite easy, but if you don't, most of them would take you 10+ minutes to solve, and you simply do not have that time.
In this particular problem, the things to know are:


*

*All presented groups are of order 8. The notations are given on one of the first pages of each exam, so $D_8$ is unambiguous. 

*A subgroup of order 4 of a group of order 8 has index 2.

*Every subgroup of index 2 is normal, so from this point we simply ignore the normality requirement, and look for unique subgroups of order 4.

*If H_i are subgroups of G_i then the direct sum of H_i is a subgroup of the direct sum of G_i, and its order is the product of the orders of H_i. Hence, in (1) and (4) we can construct several subgroups of order 4.

*In (3), $Q_8$ has several "imaginary units", each generating a subgroup of order 4: $i^2=-1 \implies \{i,-1,-i,1\}$, and similarly for $j$ and $k$.


At this point, on a typical GATE exam you should stop and mark (2) as the only possible answer. So, you can see that 1-2 minutes is more than enough, provided you know all these facts. And, if you do not know, for example, that a subgroup of index 2 is necessarily normal, you are in big trouble.
However, and this is the problem with this question, (2) is also incorrect.


*

*In $D_8$ elements are the identity, the three (non-zero) rotations, and the four reflections about the four symmetry axes (the two diagonals and the two middle lines). Two rotations are of order 4, and they generate the same subgroup of order 4. The other rotation (by 180 degrees) is of order 2, so are the reflections, so we can combine at least some of them (for example, the 180-rotation and a diagonal reflection) to form a subgroup of order 4 (in this particular case both the 180-degree rotation, and the chosen diagonal reflection can move each vertex to the opposite one only, so we have 4 possible combinations of the arrangement of the vertices).


Just wanted to mention where this problem comes from, what kind of skills and time to solve such problems require, and also the specific problem with this one.
