sorry if this is a basic question, but I was reading about the permutation group $S_3$, and I read that it has three subgroups: $\langle y\rangle$, $\langle xy\rangle$, $\langle x^2y\rangle$ of order 2 and $\langle x\rangle$ of order 3.   Can someone mention which subgroups these are, and how to determine their order? are any of these groups normal?
I'm being asked to find 2 subgroups of $S_3$, one of which is normal and one that isn't normal. I guess, to find the non normal subgroup is easier. I would do this by trial and error, but since the group is $S_3$ and easily visualizable, I guess that there should be a geometrical property that makes it easy to find a normal subgroup. Any of you guys know one? Or should I try this by brute force?
Can I update an Mac App purchased by a different Apple ID? How to integrate Tikz code in SW? I'd like to know how to insert Tikz code in Scientific Workplace tex file. Since SW doesn't support Tikz directly, now I just create Tikz code in another tex editor and export as pdf. Then I convert pdf to images, and import images into SW. I have an ASUS K55VM and it came with DOS, and then Windows 7 was installed. I am planning to install Ubuntu on it. My computer had 4 partitions: 240GB, 230GB, 230GB, and 230GB, and I changed this to 5 partitions: 243GB, 230GB, 230GB, 150GB, and 80GB. I installed Ubuntu 12.04 to the 80GB partition through a USB flash drive. It installed fine, however the problem is during the Boot option in Windows Boot Manager. When I choose Ubuntu, this what is displayed:  Windows failed to start. A recent hardware or software change might be the cause. To fix the problem:   1. Insert your Windows installation disc and restart your computer.   2. Choose your language settings, and then click "Next,"   3. Click "Repair your computer." If you don't have this disc, contact your system administrator or computer manufacturer for assistance.    File: \ubuntu\winboot\wubildr.mbr   Status: 0xc000000e   Info: The selected entry could not be loaded because the application is missing or corrupt.   I tried to change the boot option for the OS and made Ubuntu as my first option. It worked, however Windows 7 does not show up in GRUB. In this fear I did the following:     Restarted my PC changed boot option 1 to Windows.   Formatted the drive to which Ubuntu was installed (80GB one) through Windows.   Restarted again.     The problem persists. Windows Boot Manager still shows Ubuntu, and when I choose this it gives two lines one like: grub rescue&gt;, but my Windows 7 is working perfectly fine.  Now I want a solution to remove Ubuntu from the Windows boot option and reinstall completely. Outline around text on CMYK black background in Photoshop I'm trying to make a document with one color black  (c:0,m:0,y:0,k:100) background with red text on it. However in CMYK the text has this weird grey white-ish edge. In RGB it doesn't show but in CMYK it shows. I'm wondering will this show when printing or do I need to use different methods? ie. using rich black etc.  Here's the RGB and CMYK comparison Should reputation from SO carry over to Meta? What's your recommendation on drive partitioning schemes for a desktop and home server? What partitioning scheme do you recommend for a desktop? I've always created three or four primary partitions -- root, swap, home, and sometimes a separate boot partition. Ubuntu's default install offers LVMs. I've never had to add additional drives or space, so it never seemed like a big deal. Whenever I do a fresh install, though, I always think there might be a better way. So I have an old unlocked phone and I'd like to load up my own OS.  I've seen some mods out there.  CyanogenMod seems to be the most popular.  However, I was curious if I could just install a pure untouched version of the OS -- as distributed by Google. I seem to remember being able to download a clean compiled version of the Android OS.    Does Google distribute any binaries like this, and if not, is there someone out there that does?    Additionally, is this even worth trying?  These days is it necessary for Android to be packaged with special drivers and such for all of the different hardware out there? Error term in equally spaced nodes Convert .mov to .mp4 I want to convert .mov to .mp4! I saved the video from iMovie but it is in .mov format! How do I convert it to .mp4?  I'm using a MacBook and I want to share it to my phone, but my phone can't open .mov format video. Applying MVT and IVT (?) to show something about f'''? (edit: actually, Taylor's Theorem using Lagrange remainder) I believe that the correct answer to this question will involve applying both the Mean Value Theorem and the Intermediate Value Theorem, perhaps several times.  However, I can't see how to proceed.  I went down what I thought were the typical paths and just reached dead ends.  Does someone maybe have a hint?  I think that with the appropriate nudge I should be able to solve this question.  If not, I'll edit and ask if someone knows how to solve the whole thing.  Here's the question:  Let $f: \mathbf R \to \mathbf R$ be such that $f$, $f'$, $f''$, and $f'''$ exist and are continuous on $\mathbf R$ and satisfying $f(-3)=-1$, $f(0)=0=f'(0)$, and $f(3)=8$.  Prove that there exists $\xi \in (-3, 3)$ such that $f'''(\xi) \ge 1$. I have a question about the following problem from a Putnam review:  Let $n\in \mathbb{N}$. Find how many pairs of natural numbers $(x, y)\in \mathbb{N}\times \mathbb{N}$ solve $$ \frac{xy}{x+y}=n. $$  I have found some solutions for all $n$, such as $(2n, 2n)$ and $(n+1, n(n+1))$, but I feel as though this is the wrong approach, as the question is only asking for the number of solutions to the equation.  I don't really want a complete solution, but any hints would be greatly appreciated. Default behavior of "git push" without a branch specified Skype app in my Ubuntu system work perfectly. But today when I run the app, I see this:    How Can I fix this? As you can see in the picture, I have 13 out of 5. But it's been like this for about two months! What's the reason for that? Extended support for Ubuntu 12 Not sure if there is a better communication mechanism for asking this question. Ubuntu 12 is EOL April 2017. Do corporations have any option to extend support on 12?  Thanks Convergence of $\sum_{n=1}^\infty{\left(\sqrt[n]{n}-1\right)}$ Selecting specific row in attribute table using ArcPy (without iterating through whole table)? "Not with a whimper, but with a roar of triumph!" Looking for the title of an old sci-fi book, pretty sure it is a Baen or Del Rey imprint. Basically, mankind has reached the stars and expanded throughout the galaxy. Suddenly, an unknown race appears and begins a war of extermination against man. Finally, the galaxy is desolate, and one man remains. He is in a starship with an AI and becomes injured in the battle.  The ship's AI takes over and speeds out into the unknown, followed by the last remaining member of the unknown alien race. As all speeds are at tremendous multiples of the speed of light, the man cannot get away, but the alien cannot catch up.  Finally, communication is opened, at which point the man asks "why?" The alien tells him that man had discovered immortality, but that it required the death of a member of a race known as the Flower People for every human who achieved immortality. The alien says that Man would have found the answer eventually, but that genocide was so intolerable that the aliens came from another dimension, to which they could never return, to stop Man, which resulted in the war of extermination.  Finally, the light speeds become so great that the two remaining living beings in the universe approach the Big Crunch. As the Man begins to dive into the singularity, with the alien hot on his heels, he says that maybe there will be room in the next universe for both races to live in peace, and the final line is "not with a whimper, but with a roar of triumph"! Inequality $\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1} \ge 1$ Let $R$ be a commutative ring with $1$. Suppose that all ideals $I \neq R$ are prime. Prove that $R$ is a field.   Help me some hints.  Thanks a lot.