Is every subgroup of a finitely generated group also finitely generated? And if so does it have the same amount of generators?
I was wondering this today, and my algebra professor didn't know the answer.      Are subgroups of finitely generated groups also finitely generated?    I suppose it is necessarily true for finitely generated abelian groups, but is it true in general?  And if not, is there a simple example of a finitely generated group with a non-finitely generated subgroup?  NOTE:  This question has been merged with another question, asked by an undergraduate.  For an example not involving free groups, please see Andreas Caranti's answer, which was the accepted answer on the merged question.
How can I center a too wide table? Where to place my local website starting with the 2.4.7 version of apache2? I recently installed Ubuntu 14.04, then I installed lamp-server and placed my webpages in /var/www directory, but when I opened localhost in the browser there was nothing. I think that happened because Zend updated Apache.  Whatever the reason may be, I want to know where to keep my php files so that I can access them from my browser. I'm interested: What is C#'s analog of std::pair in C++? I found System.Web.UI.Pair class, but I'd prefer something template-based.  Thank you! How to display Trash and Home folder icons? How to get the keyboard to input Japanese? I need an approach to allow delegated admins (i.e., non-Administators) to add members to Public Groups and Collaboration Groups (the GroupMember and CollaborationGroupMember objects).  I added a custom object (UserGroupRequest__c) to capture which Users should be added/removed to which Groups.    I've also built a batchable class to process the requests.  However, GroupMember is a Setup object, and CollaborationGroupMember is not.  So this doesn't work if I combine the operations in a single batchable class.  Should I instead be using 2 batch classes, calling one from the finish method of the other?    Any recommendations on how best to proceed?  Am I way off base?    Would a trigger on the Request object perhaps be better? I'm guessing that there may be a permission problem if this is attempted via a trigger as the delegated admins won't be able to add users to public Groups.  Also, I'm guessing the trigger approach would run into the same mixed DML issue - on Insert of the Custom object, I would need to Insert/Delete from GroupMember and from CollaborationGroupMember, and then perform a 2nd update on the Custom object to indicate that processing was completed... How to appeal to undelete an answer deleted by moderator? How can someone make an appeal to un-delete an answer that was deleted by a moderator?    There should at least be an appeal/vote process, if the deletion was unfair. Moderators are humans like rest of us, and they can be unjust. What's difference between shadowing and overriding a method in C#? How can I bind the meta key (a.k.a.  / Windows key) to a left click? Hitting the meta key should trigger a left click, and pressing down on it should allow me to drag and drop (just like it would do when holding down left click).  I tried using System Preferences -> Shortcuts and Gestures:    but I cannot set "meta" alone as being a shortcut.  I use Kubuntu 14.10 Plasma 4 and Kubuntu 14.04 LTS x64.   (On Windows 7 SP1 x64 Ultimate I can achieve that using the program .) Booking a flight without passport info I have been looking to book a bundle (hotel and flight) with Expedia to Puerto Vallarta MX. When I get to the booking section it does not ask for my passport number. Do they ask for my passport after I book? Or later when the date gets near? My passport is currently being renewed and I want to book my hotel and flight before they sell out but I am afraid it'll ask me for my passport number. Can anyone inform me about this situation? Proving a simple graph is a connected graph Does any proof exist that a simple graph with $n$ vertices such that the least vertex degree is $\geq \frac{n-1}{2}$ is a connected graph? (i.e. does such a proof have a name?) How do I get the source code for eglibc in a format that will allow stepping through the source with gdb?    I have tried using apt-get source libc6-dev and that works for getting the source code, but I think that since the pre-processor directives were not applied that the line numbers that gdb has does not match with the source I downloaded. This makes it less useful when trying to step through the source and look at how functions in libc are working. In calculating work done by a constant force over a constant distance, why doesn't the subject's initial velocity matter? Assume a point-mass $m$ is travelling in a straight line, and a force $F$ will act on $m$ (in the same direction as $m$'s velocity) over a constant distance $d$; why doesn't $m$'s velocity matter to the calculation of work done on $m$ by $F$? Work is defined such that, in this example, the work done by $F$ on $m$ is equal to $Fd$, but it seems that if $m$ were moving slower, it would spend more time in the field, allowing $F$ more time to act on $m$, thereby doing more work. In fact, if $m$'s velocity were very great, it would hardly spend time in $F$'s field at all (so very little work done). Maybe I misunderstand work; can someone address this confusion of mine? How to reverse the pagination numbers of an archive? I have archive pages that each contain titles and excerpts from 10 blog entries. Both entries and archive pages are in reverse chronological order:   when a new blog entry is written its title and excerpt go to the top of the front page; when you are on one page and you click through to "Next page", you get the titles and excerpts for the 10 previous blog entries.   This is pretty standard for blogs.  But the archive pages are numbered by default in the same order. Thus Page 1 contains excerpts from the most recent 10 entries, Page 2 contains excerpts from the next 10 most recent, etc.  To be clear: at the moment, "Next page" refers to the page with the next higher number. Thus the front page, which contains excerpts from the most recent entry and the nine next most recent entries, is labelled Page 1, and the 10 entries posted before those 10 entries are on Page 2.  I need the page-numbering the other way round, namely in direct chronological order, so that for example if there are 97 entries then   entries 28-37 would be excerpted on Page 4, and the PREVIOUS page from there, containing excerpts from entries 18-27, would be called page 3.   How can I achieve this, whether by amending functions.php in the child theme (preferably), by writing a "Snippet" (next most preferred), or by amending archive.php in the Astra parent theme directly (least preferred, because of possible update problems)?  Edit  The following isn't a solution but it may be part of one. This re-numbers pages into direct chronological order, but unfortunately it also reverses the order in which they are shown (from reverse chronological to direct chronological). So it causes the front archive paget to contain excerpts from the earliest posts rather than the latest ones!  &lt;?php  add_filter('pre_get_posts', 'posts_by_forward_date');  function posts_by_forward_date($query) {   if(is_archive() || is_home()) {     $query-&gt;set('order', 'ASC');   } } ?&gt; Can X.org Load Two Different Drivers? (Triple-Head Setup with Intel and NVidia Card) We want to set up a 3 screens with an onboard intel graphics adapter and an nvidia card.  Both cards are recognized in lspci   01:00.0 VGA compatible controller: NVIDIA Corporation GT218 [GeForce 210] (rev a2) 00:02.0 Display controller: Intel Corporation 2nd Generation Core Processor Family Integrated Graphics Controller (rev 09)   And bascially work: If I switch to the console, I see output on the Intel card, X is only running on the Nvidia card.  But when I type xrandr | grep conn I get only one connected port:   DVI-I-0 disconnected (normal left inverted right x axis y axis) VGA-0 disconnected (normal left inverted right x axis y axis) DVI-I-1 connected 1920x1200+0+0 (normal left inverted right x axis y axis) 518mm x 324mm HDMI-0 disconnected (normal left inverted right x axis y axis)   I am on Ubuntu 12.10 (and XFCE) and installed the proprietary nvidia driver.  I was assuming that X maybe has a problem to autodetect both cards, so I thought the following xorg.conf could help:   Section "ServerLayout"     Identifier     "Layout0"     Screen      0  "Screen0"     Screen      1  "Screen1"     InputDevice    "Keyboard0" "CoreKeyboard"     InputDevice    "Mouse0" "CorePointer" EndSection  Section "Files" EndSection  Section "InputDevice"     # generated from default     Identifier     "Mouse0"     Driver         "mouse"     Option         "Protocol" "auto"     Option         "Device" "/dev/psaux"     Option         "Emulate3Buttons" "no"     Option         "ZAxisMapping" "4 5" EndSection  Section "InputDevice"     # generated from default     Identifier     "Keyboard0"     Driver         "kbd" EndSection  Section "Monitor"     Identifier     "Monitor0"     VendorName     "Unknown"     ModelName      "Unknown"     HorizSync       28.0 - 33.0     VertRefresh     43.0 - 72.0     Option         "DPMS" EndSection  Section "Device"     Identifier     "Device0"     Driver         "nvidia"     VendorName     "NVIDIA Corporation"     BoardName      "GeForce 210" EndSection  Section "Screen"     Identifier     "Screen0"     Device         "Device0"     Monitor        "Monitor0"     DefaultDepth    24     SubSection     "Display"         Depth       24     EndSubSection EndSection   Section "Monitor"     Identifier     "Monitor1"     VendorName     "Unknown"     ModelName      "Unknown"     HorizSync       28.0 - 33.0     VertRefresh     43.0 - 72.0     Option         "DPMS" EndSection  Section "Device"     Identifier     "Device1"     Driver         "intel"     VendorName     "Intel"     BoardName      "Internal Intel Graphics" EndSection  Section "Screen"     Identifier     "Screen1"     Device         "Device1"     Monitor        "Monitor1"     DefaultDepth    24     SubSection     "Display"         Depth       24     EndSubSection EndSection   But it is still not working.   Is this generally possible to use two different drivers in X ? Is it necessary to create an xorg.conf manually then? (because Ubuntu 12.10 normally works without the xorg.conf) Are there any errors in my xorg.conf? Group of $r$ people at least three people have the same birthday? What is the probability that in a randomly chosen group of $r$ people at least three people have the same birthday?   $\displaystyle 1- \frac{365\cdot364 \cdots(365-r+1)}{365^r}$   $\displaystyle \frac{365\cdot364         \cdots(365-r+1)}{365^r} +{r\choose 2}\cdot \frac{365\cdot364\cdot363 \cdots         (364-(r-2) +1)}{364^{r-2}}$  $\displaystyle 1- \frac{365\cdot364             \cdots(365-r+1)}{365^r} +{r\choose 2}\cdot \frac{365\cdot364\cdot363 \cdots (364-(r-2) +1)}{364^{r-2}}$  $\displaystyle\frac{365\cdot364 \cdots(365-r+1)}{365^r}                 $     My attempt :  (May be typo in option $(3)$ of question !)  $$P(\text{at least 3 persons have same birthday})$$  $$= 1 - \{P\text{(no one has same birthday) + P(any 2 have same birthday)\}}$$  So, option $(3)$ is true.     Can you explain it, please?   It  before,  but I'm not satisfied by explanation. How can one set the executable bit on files using umask? According to documentation, a umask of 000 should work, but in practice this only works on directories.  Also open to other methods of automatically setting +rwx on a particular user's created files if you have any. We just had a scenario where the enemy set up a 'death trap' with arrow slits covering a corridor, allowing them to pop in, shoot, then pop out -- all without actually entering the corridor.  Which was fine in and of itself, until my cleric ran through the corridor at top speed, with Spirit Guardians up.  The DM and I discussed it, and as far as we can tell the spell doesn't require a line of sight and isn't blocked by walls -- just being 'in range' gets you the damage.  So I wiped out an entire encounter with one well placed third level spell.  In and of itself, that doesn't seem too OP -- at level 5/6 a third level spell is a pretty big investment!  But it still seems a bit odd that I can do it without even LOS.  Are there any rules I'm missing? For example, rigid bodies and fluid. I know it's kind of possible by setting the rigid bodies as fluid obstacles, but it's not "true" interaction in that the fluid does not affect the rigid bodies...  By "true" interaction I mean where two (or more) physics systems affect each other simultaneously. For example, with rigid body and cloth, the rigid body affects the cloth but the cloth doesn't affect the rigid body (at least the way I'm doing it), so the rigid body just eventually stretches the cloth to an extreme and falls through...  Are there any ways to achieve this? (for any interaction, not just rigid bodies and fluid)  Or, if not, does anyone know whether there are any plans to implement this sort of feature?  Thanks Sort Order on the User Filter is Surprising To Me When filling in the "Type to Find Users" , it seems to sort by exact match and then by reputation when I am on the reputation tab.   For example, please consider the following search results:    This threw me off since I thought that the sort order is reputation. I guess this often might give people what they want, but when picking a popular name is shows a lot of 0 rep users. Maybe adding a filter for low rep to the exact match might give more expected results, or just sorting on Rep?