If $G$ is a group and $G'$ is generated by $\{xyx^{-1}y^{-1}|x,y\in G\}$, then $G'\trianglelefteq G$ and  $G/G'$ is Abelian.  At first, I thought this is easy because I thought $g[x,y]g^{-1}=[gxg^{-1},gyg^{-1}]$. But it was a mistake. I could not prove that it is also subgroup of G and $G/G'$ is abelian. I need help.
Trying to get my head around the commutator subgroup. This is an excercise from Artin's Algebra:  Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian.  Here is what I've done:  Let $xC,yC \in G/C$ then $xyx^{-1}y^{-1}C = C$ since $xyx^{-1}y^{-1}$ is a commutator hence belongs to $C$. But then $xyC = yxC$ so $xC$ and $yC$ commute in $G/C$. This can be done for any elements, so $G/C$ is abelian.  This seems somewhat surprisingly short. Is that all there is to it?   Regards
I'm using IEEEtran to format an article, and I wanted to include an image. I included the graphicx and hyperref packages, and used the following code, like normal, to include an image.  \begin{figure}[!t] \centering \includegraphics[width=3in]{fetch} \caption{Pipeline: Fetch Instruction} \label{fig:fetch} \end{figure}   What appears, however, is this:    Fetch.png is a file in the directory, and if I remove it, pdflatex errors instead of producing an output. I'm using MiKTeX 2.9 on Windows 7. Any ideas? Some faces of a model don't respond to UV Mapping I can't seem to get these particular faces to take the UV Mapping. I selected the entire object, entered edit mode, hit U>Unwrap.   The rest of the mesh takes the uv wrapping, but not those faces for some reason.    It looks normal in solid mode.     Does anyone know what could be causing this? Understanding MongoDB BSON Document size limit Is there any short way to save the pipe output to the same file that it's processed. For example, this is what I am actually doing  $ cat filename | sort | uniq &gt; result $ rm -f filename $ mv result filename   I was wondering if there was a way to do it in just one line (not appending those commands using &amp;&amp;)  This is not the way, but to get an idea  $ cat filename | sort | uniq &gt; filename The screen of my phone is physically broken, so it doesn't display anything, but the touch aspect does seem to be working, as I can unlock my phone, and get haptic feedback.  Before I go to get it replaced, I'd like to back up some personal data from the built-in memory to the SD card. I'm fairly certain that if I can get adb to recognize the device, then I can use  to see where I am touching the screen and do what I need to do. Problem is, when I plug it in, it's not recognized, while my emulator is.  I suspect that I might not have USB debugging enabled, which I think is necessary for adb to work. If this is the case, can someone give me screenshots of their Droid X (running 2.3.X) going from the home screen, through all the menus, to activating debug mode? Hopefully I can do it by dead reckoning and adb from there.  Edit: I haven't changed anything about the ROM or launcher; it's all stock. Intuitive understanding of the derivatives of $\sin x$ and $\cos x$ One of the first things ever taught in a differential calculus class:   The derivative of $\sin x$ is $\cos x$. The derivative of $\cos x$ is $-\sin x$.   This leads to a rather neat (and convenient?) chain of derivatives:   sin(x) cos(x) -sin(x) -cos(x) sin(x) ...   An analysis of the shape of their graphs confirms some points; for example, when $\sin x$ is at a maximum, $\cos x$ is zero and moving downwards; when $\cos x$  is at a maximum, $\sin x$ is zero and moving upwards.  But these "matching points" only work for multiples of $\pi/4$.  Let us move back towards the original definition(s) of sine and cosine:  At the most basic level, $\sin x$ is defined as -- for a right triangle with internal angle $x$ -- the length of the side opposite of the angle divided by the hypotenuse of the triangle.  To generalize this to the domain of all real numbers, $\sin x$ was then defined as the Y-coordinate of a point on the unit circle that is an angle $x$ from the positive X-axis.  The definition of $\cos x$ was then made the same way, but with adj/hyp and the X-coordinate, as we all know.  Is there anything about this basic definition that allows someone to look at these definitions, alone, and think, "Hey, the derivative of the sine function with respect to angle is the cosine function!"  That is, from the unit circle definition alone.  Or, even more amazingly, the right triangle definition alone.  Ignoring graphical analysis of their plot.  In essence, I am asking, essentially, "Intuitively why is the derivative of the sine the cosine?" A short story about finding an inhabitable planet and meeting aliens So the story goes, humans in a spaceship search for an inhabitable planet, being surprised they found none in the vicinity of Earth. They find one, and it seems uninhabited. At the same time are observed by the aliens, who go down on the planet to meet them. The aliens sit them down and tell them a story about war and conflict that ruled the galaxy, and how all the sentient races in the galaxy got together and got rid of the race that tried to conquer all by blowing up their planets.  In the end you realise man is this terrible conqueror the aliens are talking about. Insert Login IP Ranges using apex in salesforce Is it possible to insert to Login IP ranges via apex? Counting the occurrences / frequency of array elements How do I remove "Ubuntu" in the bios boot menu? (UEFI) Help with a proof. Countable sets. When I install Selenium I get the following error:  Shubham@Shubham-To-be-filled-by-O-E-M:~$ sudo apt-get update     Get:1 http://security.ubuntu.com/ubuntu xenial-security InRelease [107 kB]     Hit:2 https://repo.skype.com/deb stable InRelease                            Hit:3 http://in.archive.ubuntu.com/ubuntu xenial InRelease                          Get:4 http://in.archive.ubuntu.com/ubuntu xenial-updates InRelease [109 kB]     Get:5 http://in.archive.ubuntu.com/ubuntu xenial-backports InRelease [107 kB]       Fetched 323 kB in 8s (38.6 kB/s)                                                    Reading package lists... Done     Shubham@Shubham-To-be-filled-by-O-E-M:~$ sudo pip install selenium     Traceback (most recent call last):       File "/usr/bin/pip", line 9, in &lt;module&gt;         from pip import main     ImportError: cannot import name main   How should I proceed? Error when using endfloat with unicode characters Complex zeros of the polynomials $\sum_{k=0}^{n} z^k/k!$, inside balls Clean install of Windows 10 OEM: license key issues? Why does a 95% Confidence Interval (CI) not imply a 95% chance of containing the mean? It seems that through various related questions here, there is consensus that the "95%" part of what we call a "95% confidence interval" refers to the fact that if we were to exactly replicate our sampling and CI-computation procedures  many times, 95% of thusly computed CIs would contain the population mean. It also seems to be the consensus that this definition does not permit one to conclude from a single 95%CI that there is a 95% chance that the mean falls somewhere within the CI. However, I don't understand how the former doesn't imply the latter insofar as, having imagined many CIs 95% of which contain the population mean, shouldn't our uncertainty (with regards to whether our actually-computed CI contains the population mean or not) force us to use the base-rate of the imagined cases (95%) as our estimate of the probability that our actual case contains the CI?   I've seen posts argue along the lines of "the actually-computed CI either contains the population mean or it doesn't, so its probability is either 1 or 0", but this seems to imply a strange definition of probability that is dependent on unknown states (i.e. a friend flips fair coin, hides the result, and I am disallowed from saying there is a 50% chance that it's heads).  Surely I'm wrong, but I don't see where my logic has gone awry... Who was flying the Y-Wing when the Death Star exploded? Near the end of Star Wars IV: A New Hope, we see four spacecraft flying from the Death Star just moments before it explodes.     As you can see, there is the Millennium Falcon crewed by Han and Chewbacca, a pair of X-Wings which has to be Luke and Wedge, and then in the bottom corner, we see a Y-Wing. Who is this pilot and did he have any screen time to justify him actually existing in this moment? For as far as I can tell during the first trench runs, they were conducted by the Y-Wings and all of them seemed to have been destroyed during their run. So as I asked, who is flying this ship and how did survive to this moment? I'm looking for the method by which the partial derangement formula $D_{n,k}$ was derived. I can determine the values for small values of N empirically, but how the general case formula arose still alludes me.  Any links/books or an explanation will be appreciated.  The formula is:  $D_{n,k} = {n \choose k}!(n-k)$  Links: Why would a static nested interface be used in Java? Prove that a graph with the same number of edges and vertices contains one cycle