Let $f:G_1 \to G_2$ be a homomorphism between multiplicative groups.  Suppose that $N_1$ is a normal subgroup of $G_1$. Is the image $f(N_1)$ of $N_1$ a normal subgroup of $G_2$?
I want to find an example of a homomorphism $f:G\to H$ such that $A$ is a normal subgroup of $G,$ but $f(A)$ is not so in $H.$  I know that if $f$ was onto $f(A )$ must be normal , but otherwise i want to find an example!
Is there a way to render an object with a subdivision surface modifier with a wireframe which is the original geometry as well as the wires that make the subdivision surface?  Example:   This wireframe here is curved. What if there was displayed the straight wireframe. The wires that make are made by the subdivision surface. Thats what I want to achieve. How to do it?  What I mean is - How to render the curved wireframe (which is the original geometry with the straight wireframe (which is the wireframe the is made by the subdivision surface)? What ubuntu applications provide support? (guides, how to's, status) Also, specifically, details on support for syncing music to iPhone's and iPod's with the latest OS version:   Can you sync music to apple devices using >iOS4? (iPhone &amp; iPod) What applications allow you to do this and how? (guides, how to's, status)    What has already been established is:   Apple does not support Linux and has no plans to.  provides some support, however it currently does not support music/video synchronization with devices &gt;iOS4. Go to  and scroll to the section titled "" to see if your device version is supported. Undergraduate classical mechanics introduces both Lagrangians and Hamiltonians, while undergrad quantum mechanics seems to only use the Hamiltonian. But particle physics, and more generally quantum field theory seem to only use the Lagrangian, e.g. you hear about the Klein-Gordon Lagrangian, Dirac Lagrangian, Standard Model Lagrangian and so on. Why is there a mismatch here? Why does it seem like only Hamiltonians are used in undergraduate quantum mechanics, but only Lagrangians are used in quantum field theory? X Windows special programs: do some of these still exist? What is a *.pkinfo file?   Google Search results talk about Package Info files and how they contain map documents and layers, so I don't see how this is different from a .mpk file.    I also have not found anything about how to create a *.pkinfo file. Are there integer solutions to $9^x - 8^y = 1$? This came up in proving non-regularity of a certain language (powers of 2 over the ternary alphabet). Any clue to the above equation could help me move forward.  Edit: Of course, $x = 1, y = 1$ is a solution. I am looking for non-trivial solutions. Schengen area 90 days limit: Can I leave on the 91st day? Can anyone tell me if the 90 days is in fact 90 nights? Can you leave the Schengen area on the 91st day? Or are you only allowed 89 nights and you are required to leave on the 90th day? Should I test private methods or only public ones? What is the difference between Unix, Linux, BSD and GNU? What is the very fundamental difference between Unix, Linux, BSD and GNU?  Unix was the earliest OS, so the term 'Unix like' is understandable, since they have kernel, file system structure, most of the commands, users etc are same as Unix.  Still why are they different? What made them set apart? Is it the kernel? On  (but I remember seeing it in other cases) the diff shows the same tag being both added and removed. How do you distinguish between magic effects that bypass immunity and those that don't? This question was triggered by .  Basically, it goes like this: If you have a weapon with a certain magical effect, let's say the minor property unbreakable (can not be broken, requires special means) or temperate (no harm in temperatures between -20 to 120 F), it can bypass the immunity of something like a Werewolf because it is magical instead of mundane. This goes for any minor property because those properties are under the Magic Items portion of the DMG on pg.143.  How does this differ from a pair of gauntlets with the exact same effect being used as an improvised weapon (emphasis added for clarity) from dealing damage in the same manner to the same Werewolf? Or something that's clearly not a weapon like a Shield of Missile Snaring being used to bash an opponent? I have a shapefile with contours and their heights.   Is it possible to make a DEM from contours in QGIS without using GRASS?   I found  but the answer is for ArcGIS Desktop. Broken screen while debug mode was disabled. How can I re-enable adb? 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. Induction on Real Numbers How can I improve Ubuntu overall system performance? What are your tips for improving overall system performance on ubuntu? Inspired by this  I realized that some default settings may be rather conservative on Ubuntu and that it's possible to tweak it with little or no risk if you wish to make it faster.  This is not meant to be application specific (e.g. make firefox load pages faster), but system wide.  Preferably 1 tip per answer, with enough detail for people to implement it.  A couple of mine would be:   Install  (via Software Center or sudo apt-get install preload); Change  value - "which controls the degree to which the kernel prefers to swap when it tries to free memory";   What are yours?  PS: Since this is not intended to have a unique answer but rather, several useful tips, I'm making this community wiki out-of-the-box. How can we know that all of the results for entangled photons are not chosen when the pair is created? expected true to respond to true? How to understand the Ubuntu file system layout? In Windows there’re perhaps only a couple of important folders (by important I mean important in my logical picture of the Windows file system) in the installation drive (in my case C:\). Namely Program Files and Windows. I simply stay away from Windows folder and the “add remove program files” is good enough to handle the program files folder of Windows. Of course there’s a folder named Users where the users (who are not admins) can access only their folders.  Thus there’s a clear picture at some level in my mind of the Windows file system. In Ubuntu, when I reach the location /, there’s a huge list of folders, most of which I have no clue as to what they contain. The /bin folder seems to be the equivalent of the Windows folder in windows. The /usr folder seems like it’s the equivalent of the Users folder in Windows. But even the /home folder looks like it can fit the bill.  Please understand that I do understand, that Ubuntu (Linux) has a different character than that of Windows, i.e., there need not be exact equivalent of Windows functions, in Ubuntu. All I am looking for is a bit more clearer picture of the Ubuntu file system.   This question is a part of a bigger question which I am splitting up to make it more answerable. The original question can be found here: Is it possible to run the Windows 10 Command prompt on Windows 7? Why does a circle enclose the largest area?