Finite $2$-group with derived subgroup of order 8 Does there exist a finite non-abelian $2$-group $G$ such that $G^{\prime}\cong D_8$ or $G^{\prime}\cong Q_8$? By an easy inspection with GAP, I could not find any example!  Any answer or comment will be greatly appreciated!
$D_8$ as a derived subgroup Every undergraduate student knows that there are (exactly) two non-abelian groups of order 8: Dihedral ($D_8$) and Quaternion ($Q_8$). The group $Q_8$ has many interesting properties; simple of them are: it has unique element of order $2$, all subgroups are normal, etc. A property, which is strange and not obvious, the group $Q_8$ has, is that there is no group $G$ such that $G/Z(G)$ is isomorphic to $Q_8$. On the other hand, $D_8$ has no such property; there are (infinitely many) groups $G$ such that $G/Z(G)$ is isomorphic to $D_8$; example - $D_{16}$.   The question, I would like to ask, is related to a property of $D_8$, which is not shared by $Q_8$. There are (infinitely many) groups whose derived subgroup is $Q_8$; example- $SL_2(3)$. For groups $G$ of order &lt;100, one can check on GAP that $D_8$ is not a derived subgroup of any group. The natural question is then:  Question 1: Does there exist a finite/infinite group whose derived subgroup is equal to $D_8$?  Question 2: Does there exist a finite/infinite group whose derived subgroup is equal to (general) dihedral group $D_{2n}$ of order $2n$ ($n&gt;2$)?
SQL Server: Examples of PIVOTing String data Subdivision surface modifier issue I'm following . After using Boolean I cleaned up the extra vertices. Edge flow looks fine on my mesh.   However, when I apply a Subdivision Surface modifier my mesh becomes really messy as if something is wrong with the edge flow.   Can you guys help spot the problem?  Mesh topology:    Subdivision issue: Can I use an inner tube that's too narrow for the tyre? I recently got a puncture and when I went to replace the inner tube I discovered that I'd bought the wrong size tube. My tyre is 700 x 40c, the tube is 700 x 28-35c.  What is likely to happen if I use this tube? Will the tube explode when I attempt to inflate it? Implement a Selective Comment Ban How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$? JavaScript adding decimal numbers issue $f_n(x) = x - x^n$ for $x\in [0,1]$. Does the sequence converge pointwise or uniformly on $[0,1]$? $f_n(x) = x - x^n$ for $x\in [0,1]$. Does the sequence converge pointwise or uniformly on $[0,1]$?   We also have to prove why it is or isn't pointwise or uniformly convergent.   I have worked out that on $x\in[0,1)$, the limit as $n$ approaches infinity is $x$. And when $x=1$ the limit is $0$.   So this means its pointwise convergent right? But how do I prove this?  Also how on earth do I prove it is not uniformly convergent? (I'm just assuming it's not). Taylor series $\frac{z^2+z}{(z-1)^2}$ at $z=-1$ How many of Trelawney's prophecies have come true? Throughout the Harry Potter series, Professor Trelawney makes many different prophecies and predictions. Although most of them are implied to be nonsense, often enough they still get fulfilled. Which ones have come true? Which ones have been proven false? Which ones are still possible?  How many of Trelawney's prophecies have come true?  I'm not asking if Divination in Harry Potter is real, or what it is. I am asking for a finite list of which of Trelawney's prophecies came true, and which ones didn't. Using "answer your own question" to advertise completed open source project? A square with side $a$ is given. What is the average distance between two uniformly-distributed random points inside the square?  For more general "rectangle" case, see . The proof found there is fairly complex, and I am looking for a simpler proof for this special case. I expect it could be significantly simpler.  See also . Suppose that:  $0 \rightarrow M_{1} \rightarrow M \rightarrow M_{2} \rightarrow 0$  Is an exact sequence of $R$-modules (for $R$ commutative integral domain). Show that $\mathrm{rk}(M) = \mathrm{rk}(M_{1})+\mathrm{rk}(M_{2})$    I have shown the inequality $\mathrm{rk}(M) \geq \mathrm{rk}(M_{1})+\mathrm{rk}(M_{2})$, however for the opposite direction I'm lost. I tried to take $N$ as a linear independent subset and tried to decompose it into $N_{1}$ (the elements in $\mathrm{Ker}(g)$) and $N_{2}$ (the elements not in $\mathrm{Ker}(g)$), but I haven't got far, since $N_{1}$ gives us a linearly independent set in $M_{1}$, however $N_{2}$ can give us a linearly dependent set in $M_{2}$.   We define $\mathrm{rk}(M)$ in this case as the supremum over all cardinalities of the linearly independent subsets of $M$ (where the only finite linear combination equaling $0$ implies trivial coefficients for $r \in R$).  Would this be the proper way to show the result? I think I'm missing something important. "How are" or "How is" the wife and kid? How are the wife and kid?   How is the wife and kid?      Which is more correct? Could gravity accelerate light? Solving linear congruences by hand: modular fractions and inverses Simultaneous collaborative editing of a LaTeX file I have a vocabulary like this:   Parent 1  Child 1 Child 2 Child 3   Parent 2  Child 1 Child 2    Now I would like to use a view to create a list of a Parent term with its Child terms. I try to do this with Views UI, but only show the term parent... No the child term's. It is very confusing for me.  Can anyone help with this case? Thanks :D This is a continued discussion from () by return count only.     Problem      Given four lists A, B, C, D of integer values, compute how many tuples   (i, j, k, l) there are such that A[i] + B[j] + C[k] + D[l] is zero.      To make problem a bit easier, all A, B, C, D have same length of N   where \$0 \le N \le 500\$. All integers are in the range of \$-2^{28}\$ to \$2^{28} - 1\$   and the result is guaranteed to be at most \$2^{31} - 1\$.      Example:      Input:  A = [ 1, 2] B = [-2,-1] C = [-1, 2] D = [ 0, 2]       Output:  2       Explanation:      The two tuples are:         (0, 0, 0, 1) -> A[0] + B[0] + C[0] + D[1] = 1 + (-2) + (-1) + 2 = 0   (1, 1, 0, 0) -> A[1] + B[1] + C[0] + D[0] = 2 + (-1) + (-1) + 0 = 0      I'm wondering if there are any ideas to have a solution less than \$O(n^2)\$ time complexity.  Source code in Python 2.7,  from collections import defaultdict def four_sum(A, B, C, D):     sum_map = defaultdict(int)     result = 0     for i in A:         for j in B:             sum_map[i+j] += 1     for i in C:         for j in D:             if -(i+j) in sum_map:                 result += sum_map[-(i+j)]      return result  if __name__ == "__main__":     A = [1, 2]     B = [-2, -1]     C = [-1, 2]     D = [0, 2]     print four_sum(A,B,C,D) Can I pack a cardboard box as checked luggage? As mentioned in , I'm planning on traveling with a desktop computer as checked baggage. After considering the options, I've decided to take it in its original cardboard box (used for shipping), making sure to pad everything inside very well.  What should I do to securely prepare the box before checking it in? Some suggestions others have given me so far:   Spray paint the box black so it's not obvious there's a computer inside. Wrap the entire box with duct tape to reinforce it. Use tape to seal the "hand holes" on either side used for carrying the box. Run packing tape around all the edges to reinforce the box. Female main character in an ice world earning gems/runes Here's what I remember:   The main character's female (99% sure) There is one part where she has to cross a big tundra and she has to kill a big ice creature The kids in the story go to a school-like place where they train for something. I'm pretty sure they earn gems or runes or something like that There's one part where the main character has to play a piece of music perfectly to earn her gem/rune thing It's definitely a series; young-adult (I read them in middle school)   Thanks!