Suppose G is an finite group. Let N be a normal subgroup of G and A be an arbitraty subgroup. Prove that $$\mid AN\mid=\frac{\mid A\mid\mid N\mid}{\mid A\cap N\mid}$$  My initial approach is that I should somehow invoke one of the isomorphism theorems that involves quotient group and then apply another theorem to grasp about its size. Yet I've no idea how to actually approach it, especially with the given condition that $N$ is normal.
Let $H$, $K$ be subgroups of $G$. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.  I need this theorem to prove something.
Opening files from version 2.8 in 2.79 Having recently switched to Google Chrome, I was wondering what files and directories I should be including in my backups to make sure that bookmarks and history are included.  Along the same lines, is there a process to restore these settings on a new install?  This is on Windows Vista. I know that symlinks have a separated inode from the original file and that they only point for the inode containing the data. My question is, where is the hardlink name stored if there's no field in the inode structure refering to "name". I know that this could be a a relatively easy question, but as all hardlinks have the same inode number i assume that no inode is created when creating a hard link. How can I get file extensions with JavaScript? What would be a good term for continuous testing post-deployment? I'm thinking of live production sites here, where there is a continuous community or much content being updated, sort of like a newspaper site or community site (the one I work at is a healthcare improvement company) and we have news articles, programs and so on.  While the underlying code is tested and released there are often production issues that come up, maybe certain pages stop working, style sheets suddenly don't appear like they did due to a service patch or some tweak done to improve production security or performance.  There is Acceptance Testing for when a deployment goes out to make sure things work right, Regression Testing to make sure nothing has been broken since previous releases.  Considering we've done these types of tests and things are working fine what would you call testing, or checking, the site to make sure things still work fine?  To be sure new pages are being crawled for search indexing?  To check that newly added pages are using plug-ins correctly?  Or so on.  My manager and I were discussing some issues that have cropped up and decided we needed to do something like this, but then to sort of note it in our status reports we didn't really have a name for it.  Anyone done this and have a term that might be good?  Or would you consider it covered by something like Continuous Acceptance Testing or some such term? How to easily build your own Linux Distro? I wanted to have a go at creating my very own Linux Distribution. Could you suggest some nice and easy-to-follow tutorials (preferably text based and not videos). I have heard something about Arch Linux but I don't know how to go from there. What do I need? What is the difference between a URI, a URL and a URN? People talk about URLs, URIs, and URNs as if they're different things, but they look the same to the naked eye.  What are the distinguishable differences between them? Installing Maya on Ubuntu Linux Installing Autodesk Maya on Ubuntu is possible, but can be a bit challenging.  Assuming that one already has the required specs and a solid graphics card driver installed, how can it be done? I want to monitor the output of a command and whenever it contains a certain string I want to run another command.  Example: when command1 outputs 1234 I want to run command2 Most instructions for equipment and software make a habit of shortening sentences by dropping words of certain parts of speech - usually definite and indefinite articles.  For example:     PULL (THE) LEVER TO OPERATE      CARS MAY BE TOWED WHEN (THEY ARE) PARKED ILLEGALLY      PLEASE USE (THE) OTHER DOOR   Is there a name for this custom? And what is the best way to describe it? How do I install Ubuntu 16.04 from an SD card? Could someone explain how to install Ubuntu from an SD card? I don't have any CDs or USB sticks around. My current operating system is Windows 10 64 bit. Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$ Suppose that $S_0$ and A are positive numbers, let $$S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right)$$ with $n \geq 0 $.      (a)Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$       (b)Show that $S_{n+1} \leq S_n $ , if $n \geq 1$      (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists      (d) find s   (a) Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$  Given $$P_n:  S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \geq \sqrt{A}$$  $$P_0: S_{1} = \frac{1}{2}\left(S_0 +\frac{A}{S_0}\right) \geq \sqrt{A}  $$  We assume that $P_n$ is true  $$P_{n+1}: S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right)$$ by assumption $$S_{n+2}= \frac{1}{2}\left(S_{n+1}\left(1 +\frac{A}{(S_{n+1})^2}\right)\right) \geq \frac{1}{2}\left(\sqrt{A}\left(1 +\frac{A}{(\sqrt{A})^2}\right)\right)$$ $$ S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right) \geq \sqrt{A}  $$  It follows that $S_{n+1} \geq \sqrt{A} $    (b) Show that $S_{n+1} \leq S_n $ , if $n \geq 1$  $$S_{n+1} \leq S_n$$ $$\frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \leq S_n $$ Dividing by $S_n$ $$\frac{1}{2}\left(1 +\frac{A}{S_n^2}\right) \leq 1 $$ $$\frac{A}{2S_n^2} \leq \frac{1}{2}$$ $$A \leq S_n^2$$ $$S_n \geq \sqrt{A}$$ As  $S_{n+1} \leq S_n$ yields a true statement, it follows $S_{n+1} \leq S_n$ is true.    (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists  Since $S_{n+1} \leq S_n$, the sequence is non-increasing,   using the non-increasing theorem stating that      if $\{S_n\}$ is non-increasing then $$\lim\limits_{n \rightarrow &gt; \infty} S_n = \inf\{S_n\} $$     (d) find s    Is the argumentation in (a) and (b) appropriate? Also, I have to admit I m getting less confident in my argumentation (c) and (d). How to proceed in (c) and (d)?  Much appreciated for your input or help. PHP regular expressions: No ending delimiter '^' found in I need a bit of help with this.      She is a photographer but that camera is her brother's-in-law     or:        She is a photographer but that camera is her brother-in-law's   I am aware that I could change the construction to make things easier, like:      She is a photographer but that camera belongs to her brother-in-law       However, I would really like to find out what's the correct way to write the sentence. Prove $P(x,y)$: If $x$ and $y$ are odd integers, then the product  $xy$ must also be odd.   I need a direct proof of this.  I know that $ x $ and $y$ both have to equal to $2n+1$ in order for them to be odd. But that's all I have. Any help will be appreciated. Code formatting - tag/block rather than indentation - code block more obvious How can anything ever fall into a black hole as seen from an outside observer? The event horizon of a black hole is where gravity is such that not even light can escape. This is also the point I understand that according to Einstein time dilation will be infinite for a far-away-observer.  If this is the case how can anything ever fall into a black hole.  In my thought experiment I am in a spaceship with a powerful telescope that can detect light at a wide range of wavelengths.  I have it focused on the black hole and watch as a large rock approaches the event horizon.  Am I correct in saying that from my far-away-position the rock would freeze outside the event horizon and would never pass it? If this is the case how can a black hole ever consume any material, let alone grow to millions of solar masses.  If I was able to train the telescope onto the black hole for millions of years would I still see the rock at the edge of the event horizon?  I am getting ready for the response of the object would slowly fade.  Why would it slowly fade and if it would how long would this fading take?  If it is going to red shift at some point would the red shifting not slow down to a standstill? This question has been bugging me for years!  OK - just an edit based on responses so far.  Again, please keep thinking from an observers point of view.  If observers see objects slowly fade and slowly disappear as they approach the event horizon would that mean that over time the event horizon would be "lumpy" with objects invisible, but not passed through?  We should be able to detect the "lumpiness" should we not through? The + vowel letter I've been told that when "the" is proceeded by a vowel sound, like "apple" or "hour", it's pronounced as "thee" and not as "thu".  But after listening to a couple of songs, I noticed that sometimes this "rule" is not followed. Take for example the two Katy Perry's songs, "Roar" and "The one that got away". In the first she sings "I got thee eye of thu tiger", but in the second she sings "Thu one that got away".  I don't know if it was sang this way to better suit the song melody (I understand nothing about those techniques), but I got confused. What's the correct pronunciation?  Thanks in advance. Very large log files, what should I do? ( deals with a similar issue, but it talks about a rotated log file.)  Today I got a system message regarding very low /var space.   As usual I executed the commands in the line of sudo apt-get clean which improved the scenario only slightly. Then I deleted the rotated log files which again provided very little improvement.  Upon examination I find that some log files in the /var/log has grown up to be very huge ones. To be specific, ls -lSh /var/log gives,   total 28G -rw-r----- 1 syslog            adm      14G Aug 23 21:56 kern.log -rw-r----- 1 syslog            adm      14G Aug 23 21:56 syslog -rw-rw-r-- 1 root              utmp    390K Aug 23 21:47 wtmp -rw-r--r-- 1 root              root    287K Aug 23 21:42 dpkg.log -rw-rw-r-- 1 root              utmp    287K Aug 23 20:43 lastlog    As we can see, the first two are the offending ones. I am mildly surprised why such large files have not been rotated.  So, what should I do? Simply delete these files and then reboot? Or go for some more prudent steps?  I am using Ubuntu 14.04.  UPDATE 1  To begin with, the system is only several months old. I had to install the system from scratch couple of months back after a hard disk crash.  Now, as advised in ,  I first checked the offending log files using tail, no surprise there. Then, for deeper inspection, I executed this script from the .  for log in /var/log/{syslog,kern.log}; do    echo "${log} :"   sed -e 's/\[[^]]\+\]//' -e 's/.*[0-9]\{2\}:[0-9]\{2\}:[0-9]\{2\}//' ${log} \   | sort | uniq -c | sort -hr | head -10 done   The process took several hours.  The output was in the line of,   /var/log/syslog : 71209229  Rafid-Hamiz-Dell kernel:  sda3: rw=1, want=7638104968240336200, limit=1681522688 53929977  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device 17280298  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device    1639  Rafid-Hamiz-Dell kernel:  EXT4-fs warning (device sda3): ext4_end_bio:317: I/O error -5 writing to inode 6819258 (offset 0 size 4096 starting block 54763121030042024)        &lt;snipped&gt;  /var/log/kern.log.1 : 71210257  Rafid-Hamiz-Dell kernel:  attempt to access beyond end of device 71209212  Rafid-Hamiz-Dell kernel:  sda3: rw=1, want=7638104968240336200, limit=1681522688    1639  Rafid-Hamiz-Dell kernel:  EXT4-fs warning (device sda3): ext4_end_bio:317: I/O error -5 writing to inode 6819258 (offset 0 size 4096 starting block 954763121030042024)    (/dev/sda3 is  my home directory. As we can find,   lsblk /dev/sda NAME   MAJ:MIN RM   SIZE RO TYPE MOUNTPOINT sda      8:0    0 931.5G  0 disk  ├─sda1   8:1    0 122.1G  0 part / ├─sda2   8:2    0   7.6G  0 part [SWAP] └─sda3   8:3    0 801.8G  0 part /home    Why a process will want to write beyond the limit is actually outside the scope of my comprehension. Perhaps I will want to ask a different question in this forum if this continues even after a system update.)  Then, from  (you may want to check  for a deeper understanding), I executed,   sudo su - &gt; kern.log &gt; syslog   Now, these files have zero sizes. The system is running fine before and after a reboot.   I will watch these files (along with others) in the next few days and report back should they behave out-of-line.  As a final note, both  the offending files (kern.log and syslog), are set to be rotated, as inspection of the files (grep helped) inside  /etc/logrotate.d/ shows.  UPDATE 2  The log files are actually rotated. Looks like the large sizes were attained on a single day. Intuition behind the Definition of Conditional Probability (for 2 Events)