The title essentially says it all.  Is there a way of efficiently finding the commutator subgroup of $A_4,$ and groups in general?  I know that the commutator subgroup must be normal in $A_4.$  I know the conjugacy classes are classes of cycle type:  $1$  $2, 2$  $3, 1$  The normal subgroups must be a union of the conjugacy classes. Does this mean that the commutator is:  e+conjugacy class of cycle type $(2, 2)$+ conjugacy class of cycle type $(3, 1),$  where e is the identity element of $A_4$?  Please make the explination as simple as possible, I'm having difficulties with abstract algebra.
Find the derived subgroup of $A_4$.   Since it is $A_4$, for a permutation $\sigma$ to be in $A_4$, $\sigma$ must have a cycle structure of $2$ cycles. Therefore, $\sigma=(ab)(cd)$.  The commutator of such elements, would be obviously another permutation of $2$-cycle (of length $2$).  Does it mean that $A'_4=A_4$?
How to safely remove proprietary Intel driver I recently installed the proprietary Intel drivers for my Toshiba Satellite A105 s4284 running Ubuntu 14.10. Nouveau was working just fine, I was just attempting to make it so that I was able to run CS:S (no need for explanation on that.)  Long story short, I went and downloaded the Intel Graphic Drivers for Linux* from . After downloading and installing the .deb, I ran intel-linux-graphics-installer which opened up a window that downloaded &amp; installed the open Intel drivers. I then rebooted, and it's slow. Very slow, especially on 3D effects and when I hit Alt+Tab.  Now, I don't know if the previously installed Nouveau drivers are conflicting with these new Intel drivers (if that's the issue, what should I do about fixing it?) What I do know is that my system isn't behaving right and I want it to (obviously.) If removing the Nouveau drivers isn't the solution, how can I safely remove the just-installed Intel drivers?  Does anyone have any suggestions? What is a magic number, and why is it bad? Automatically rename files when they are placed in a specific directory Is it possible to automatically rename a file when it's placed in a specific directory?  For example I have a directory named "dir0".I move or copy a file named "file1" to "dir0".then "file1" should rename to "file1_{current timestamp}" Will Stack Overflow's engine be (someday, or ever) open sourced? I lost a couple hundred lines of a Vim plugin I wrote and continue back up every so often because I extracted a tar file containing an older version. I decided not to use source control management when I wrote the plugin, since I only update it periodically.  Why doesn't tar ask before overwriting existing files by default? I'm guessing that there's some rationale behind why the default behavior is to overwrite files silently, since we usually want safety over speed and convenience. For example, we usually use #ifdef NDEBUG, not #ifdef DEBUG; similarly, I would expect tar ask me before overwriting any files, and for it to have some flag which I can use to tell it to overwrite files without asking me. Also, how can make it so that tar does ask me by default, without me having to give it any flags? I'm fine with using tools other than tar, if they support this safer default behavior.  Thank you for your time! Why no file-name in Inode information? 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. Consider this example from the Rogue class traits:     Trained Skills: Stealth and Thievery. From the class skills list below, choose four more trained skills at 1st level.   &nbsp;&nbsp;&nbsp;&nbsp;Class Skills: Acrobatics (Dex), Athletics (Str), Bluff (Cha), Dungeoneering (Wis), Insight (Wis), Intimidate (Cha), Perception (Wis), Stealth (Dex), Streetwise (Cha), Thievery (Dex)   The Rogue is trained in Stealth and Thievery. However, they're also included in the Class Skills section, and it invites you to: 'choose four more trained skills' from that list. This seems to be similar for all the classes.  So can you train Stealth and Thievery as a choice as well have it initially trained or the equivalent for other classes? divisor of a product of integers is a product of divisors I read this recently on the web and can't manage to understand it. Not homework -- I haven't done math homework for years.  If $d\mid ab$ then $d=d_1 d_2$, that $d_1|a$, that $d_2|b$ and, further, if $(a,b)=1$ then $(d_1,d_2)=1$.  I know it's the same question... if $a = a_1 a_2 ... a_n$, why can't $d$ divide some $a_i$ and only $a_i$? Replacing strings in files based on certain search criteria is a very common task. How can I    replace string foo with bar in all files in the current directory? do the same recursively for sub directories? replace only if the file name matches another string? replace only if the string is found in a certain context? replace if the string is on a certain line number? replace multiple strings with the same replacement replace multiple strings with different replacements What is the physical significance of dot & cross product of vectors? Why is division not defined for vectors? I get the physical significance of vector addition &amp; subtraction. But I don't understand what do dot &amp; cross products mean?   More specifically,    Why is it that dot product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB\cos\theta$? Why is it that cross product of vectors $\vec{A}$ and $\vec{B}$ is defined as $AB\sin\theta$, times a unit vector determined from the right-hand rule?   To me, both these formulae seem to be arbitrarily defined (although, I know that it definitely wouldn't be the case).   If the cross product could be defined arbitrarily, why can't we define division of vectors? What's wrong with that? Why can't vectors be divided? Limiting number of lines in textarea I'm looking for a javascript that can limit the number of lines (by line I mean some text ended by user pressing enter on the keyboard) the user is able to enter in textarea. I've found some solutions but they simply don't work or behave really weird. The best solution would be a jquery plugin that can do the work - something like , but it should be able to limit text line count not character count. It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.  Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".   Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?  I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism. I am a beginning indie developer, and I want to get the word out about my game. What sites can I go to to ask about reviews of my game, and where might I be able to talk about and get feedback on my game from players? Prove that if a function $f$ has a jump at an interior point of the interval $[a,b]$ then it cannot be the derivative of any function. Prove that if a function $f$ has a jump at an interior point of the interval $[a,b]$ then it cannot be the derivative of any function.  I know that for $f$ is differentiable in $(a,b)$ and that it has one-sided derivative $f_+' (a)≠f_-' (b)$ at the endpoints. If $C$ is a real number between $f_+' (a)$  and $f_-' (b)$, then there exists $c∈(a,b)$ such that $f' (c)=C $. How can I use this to prove the above? What is the double slash (//) directory? I mistakenly typed cd // instead of cd /. To my surprise, current directory showed up as //.   What is that directory? Why does it exist?  apple@snipped $ pwd /home/apple apple@snipped $ cd / apple@snipped $ pwd / apple@snipped $ cd // apple@snipped $ pwd // apple@snipped $ cd /// apple@snipped $ pwd / Cannot compile Glasgow Haskell Compiler How can I find the version of Ubuntu that is installed? Wi-Fi not working on Lenovo ThinkPad E570 (Realtek RTL8821CE) We just bought a Lenovo  (which is supposed to have the Intel 8265 for wireless). This laptop was bought in part because it is certified to run Ubuntu and as such to me at least it would make sense for things to work out of the box... but no such luck.  First, I installed Ubuntu 17.10 and after installation realised Wi-Fi was not working. I also tried Ubuntu 16.04 from a USB drive since that is the version of Ubuntu it is certified for, but no luck there either.  The wireless chip is not detected by lshw (or even lspci by the looks of it), ip link does not show a wireless interface, and the iwlwifi kernel module is not loaded at boot (I can load this myself using modprobe iwlwifi but this does not make Wi-Fi work). I think it is a combined Wi-Fi/Bluetooth chip, and Bluetooth does appear to work (or at least Bluetooth shows up in Gnome and rfkill list).  modinfo iwlwifi | grep 8265 shows me a file ending in -34 is supposedly loaded but only a file ending in -33 (and some other lower numbers) is present in /lib/firmware, if that helps).  I have also installed the Linux 4.14.9 kernel in an attempt to get things to work, but this did not resolve the issue either.  Further information  root@ThinkPad-E570:~# modinfo iwlwifi | grep 8265 firmware:       iwlwifi-8265-34.ucode root@ThinkPad-E570:~# rfkill list all 0: tpacpi_bluetooth_sw: Bluetooth     Soft blocked: no     Hard blocked: no 1: hci0: Bluetooth     Soft blocked: no     Hard blocked: no root@ThinkPad-E570:~# dmesg | grep iwl root@ThinkPad-E570:~# lspci -nnk | grep 0280 -A3 05:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. Device [10ec:c821]     Subsystem: Lenovo Device [17aa:c024]   Update  I just noticed that the laptop does not have an Intel wireless chip at all, but in fact has a Realtek one... which means that the  of the shop I bought this laptop at were not accurate... Sigh. Same problem stands, however, just with a crappier wireless chip. finding and replacing elements in a list Open Mapping Theorem: counterexample