How to show effectively that every finite group is isomorphic to a subgroup of symmetric group
This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. Can someone explain this proof step by step? Thank you so much! Here is the proof:  Let $G$ be a group. We show that $G$ is isomorphic to a subgroup of $S_G$. Define a one-to-one function $\phi: G \to S_G$ such that $\phi(xy)=\phi(x) \phi(y)$ for all $x,y \in G$. For $x \in G$, let $\lambda_x: G \to G$ be defined by $\lambda_x (g) = xg$ for all $g \in G$. The equation $\lambda_x (x^{-1} c) = x(x^{-1} c) = c$ for all $c \in G$ shows that $\lambda_x$ maps $G$ onto $G$. If $\lambda_x (a) = \lambda_x (b)$, then $xa=xb$ so $a=b$ by cancellation. Thus $\lambda_x$ is also one to one, and is a permutation of $G$. We now define $\phi: G \to S_G$ by defining $\phi(x) = \lambda_x$ for all $x \in G$. To show that $\phi$ is one to one, suppose that $\phi(x) = \phi(y)$. Then $\lambda_x = \lambda_y$ as functions mapping $G$ into $G$. In particular $\lambda_x (e) = \lambda_y (e)$, so $xe=ye$ and $x=y$. Thus $\phi$ is one to one. It only remains to show that $\phi(xy) = \phi(x) \phi(y)$, that is, that $\phi_{xy} = \lambda_x \lambda_y$. Now for any $g \in G$, we have $\lambda_{xy} (g) = (xy)g$. Permutation multiplication is function composition, so $(\lambda_x \lambda_y)(g) = \lambda_x (\lambda_y (g)) = \lambda_x (yg) = x(yg)$. Thus by associativity, $\lambda_{xy} = \lambda_x \lambda_y$.
What is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$  I tried to factorize $n^3+100$, but $100$ is not a perfect cube. I wish it were $1000$. When to use “that” and when to use “which”, especially in relative clauses When is it appropriate to use that as opposed to which with relative clauses? I think the title says all I want: I want to add context menu items for a specific file type like the yellow marked ones in the screenshot. Which registry changes do I have to make? In my case I want to assign ir.exe to .rb files. I have a 4GB DDR3 800Mhz recently I installed another 4GB. It turns out it's  DDR3 667MHz. Does it really matter I should own both rams 800MHz?    Slot1 shows recent RAM installation. ALSO I don't know if this is 1600MHz (it shows 800x2 and 667x2 for other) or just 800MHz ram. When I slam my hand onto my keyboard, only a maximum of 6 keys will be registered, I've tested this for the past few minutes and was unable to get more than 6. Is this the same on all keyboards? If not, what would be the distinguishing factor, If yes: why 6 specifically?  For clarification: I do not have a usecase for this, I am simply curious Every planar graph has a vertex of degree at most 5. I am trying to prove the following statement, any help!?     Prove that every planar graph has a vertex of degree at most 5. Virtual Host not working as expected when I'm using .dev domain extension I'm doing a Laravel project and I need to set up a virtual host example.dev in my system. For that I've created a copy of 000-default.conf and named it as example.dev.conf and placed in /etc/apache2/sites-available. The contents of the file is given below (comments removed):  &lt;VirtualHost *:80&gt;     &lt;Directory /var/www/html/example/public/&gt;         Options Indexes FollowSymLinks MultiViews         AllowOverride All         Require all granted     &lt;/Directory&gt;      ServerName example.dev     ServerAlias www.example.dev      ServerAdmin admin@example.dev     DocumentRoot /var/www/html/example/public      ErrorLog ${APACHE_LOG_DIR}/error.log     CustomLog ${APACHE_LOG_DIR}/access.log combined &lt;/VirtualHost&gt;   Then I've enabled the site using a2ensite command and disabled 000-default.conf using a2dissite command. Then it will open my project when I try localhost (/var/www/html) in my browser and not example.dev. The virtual host example.dev always produce "The site can't be reached" error in browsers.  Any suggestions? How can I center a too wide table? Let $\alpha,\beta,\gamma$ be the roots of $x^3-3x^2+1 = 0$. Prove that $$\alpha^n+\beta^n+\gamma^n \equiv 4^n+5^n+(-6)^n \pmod{17}$$ where $n$ is an integer.   It sort of makes sense why they are congruent since we can say $x^3-3x^2+1 \equiv (x-4)(x-5)(x+6) \pmod{17}$, but since $\alpha,\beta,\gamma$ aren't integers, how do we prove this? Is there any simple (IE: right click in Nautilus) way to password protect a particular folder/file in Ubuntu? I've got a few files containing sensitive info and I'd much prefer that if/when I leave my computer alone, they aren't accidentally accessed by someone else.  The secruty does not have to be extremely tight. My only concern is that when family/friends come over, I don't really like the idea of them looking at my bank details, accounts or, you guessed it, porn collection.  A simple, effective way to let me put my machine in the hands of someone else knowing that said machine can not cause me embarresment is the sole reason why I'd like to see this in Ubuntu. Print composer custom grid label problem In answer to  Klas Karlsson puts forward a solution to the problem of formatting print composer grid labels for projected maps.  I'm very weak on scripting, but in QGIS 2.14 I've tried to enter the text as shown in Klas's illustration, which generates error messages. In my case I'm working with a coordinate system that has a six-digit Easting and a six-digit Northing and I'm trying to get a result similar to that shown in Klas's picture (but with a six-digit Northing). Can anyone help me with this? During The Great Battle of the Last Alliance, there were an estimated 100,000 elves that fought with the armies of Isildur. This was one of the first scenes in the first LOTR movie. However, in the trilogy, it appears that only a small number of elves show up to Helms Deep and overall left the majority of the fighting to the humans. Why did the elves not participate during the rest of the war? The only solution that I have been able to come up with is $d=1$, but because $d&gt;2$, I can't seem to figure out how to find a solution. How to prove the property of the Lemoine point of a triangle? From Wolfram , I know there is a Lemoine point of triangle, also called symmedian point, the sum of squared distances of this point to all the three sides is algebraically minimum.    How to prove such a property? And now I am unable to chmod it back.. or use any of my other system programs. Luckily this is on a VM I've been toying with, but is there any way to resolve this? The system is Ubuntu Server 12.10.   I have attempted to restart into recovery mode, unfortunately now I am unable to boot into the system at all due to permissions not granting some programs after init-bottom availability to run- the system just hangs. This is what I see:  Begin: Running /scripts/init-bottom ... done [   37.062059] init: Failed to spawn friendly-recovery pre-start process: unable to execute: Permission denied [   37.084744]  init: Failed to spawn friendly-recovery post-stop process: unable to execute: Permission denied [   37.101333] init: plymouth main process (220) killed by ABRT signal   After this the computer hangs. What is the difference between using IDisposable vs a destructor in C#? When would I implement IDispose on a class as opposed to a destructor?  I read , but I'm still missing the point.    My assumption is that if I implement IDispose on an object, I can explicitly 'destruct' it as opposed to waiting for the garbage collector to do it.  Is this correct?  Does that mean I should always explicitly call Dispose on an object?  What are some common examples of this? @class vs. #import It is to my understanding that one should use a forward-class declaration in the event ClassA needs to include a ClassB header, and ClassB needs to include a ClassA header to avoid any circular inclusions. I also understand that an #import is a simple ifndef so that an include only happens once.  My inquiry is this: When does one use #import and when does one use @class? Sometimes if I use a @class declaration, I see a common compiler warning such as the following:     warning: receiver 'FooController' is a forward class and corresponding @interface may not exist.   Would really love to understand this, versus just removing the @class forward-declaration and throwing an #import in to silence the warnings the compiler is giving me. How is Mono magical? I'm learning C#, so I made a little C# program that says Hello, World!, then compiled it with mono-csc and ran it with mono:  $ mono-csc Hello.cs $ mono Hello.exe Hello, World!   I noticed that when I hit TAB in bash, Hello.exe was marked executable. Indeed, it runs by just a shell loading the filename!   Hello.exe is not an ELF file with a funny file extension:  $ readelf -a Hello.exe readelf: Error: Not an ELF file - it has the wrong magic bytes at the start $ xxd Hello.exe | head -n1 00000000: 4d5a 9000 0300 0000 0400 0000 ffff 0000  MZ..............   MZ means it's a Microsoft Windows statically linked executable. Drop it onto a Windows box, and it will (should) run.  I have wine installed, but wine, being a compatibility layer for Windows apps, takes about 5x as long to run Hello.exe as mono and executing it directly do, so it's not wine that runs it.  I'm assuming there's some mono kernel module installed with mono that intercepts the exec syscall/s, or catches binaries that begin with 4D 5A, but lsmod | grep mono and friends return an error.   What's going on here, and how does the kernel know that this executable is special?    Just for proof it's not my shell working magic, I used  (aka sh) to run it and it still runs natively.    Here's the program in full, since a commenter was curious:  using System;  class Hello {   /// &lt;summary&gt;   ///   The main entry point for the application   /// &lt;/summary&gt;   [STAThread]   public static void Main(string[] args) {     System.Console.Write("Hello, World!\n");   } } Triggered send status queued, need to get an alert How do I put big brackets under different parts of an equation so that I can write, e.g., text specific to these parts? For example: f =  x^3 +  2     |___|  |__|        |     |    text 1  text 2