I have a small question about simple groups: When we're decomposing a group $G$ we take a non-trivial normal subgroup of maximum order $G_1$ so that $\frac{G}{G_1}$ is simple and so on. But why is $\frac{G}{G_1}$ simple? I thought it had to do with the fact that $G_1$ is of maximum order, so there is no 'bigger' normal subgroup in $G$?
I need to prove that $H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.  My proof to the $(\Rightarrow)$ direction seems too much trivial:  Let us assume there exist $A$ so that $A/H\lhd G/H$. Then by definion, $H$ must be normal in $A$. Because $H$ is maximal, we get $H=A$ and therefore $A/H={1}$  Is it correct?  Update:  Now I see that I need to prove that not only $A\lhd H$ but also $A\lhd G$. Assumig I have proven that, is the proof correct?
I am trying to achieve something like this:     This image is linked from .  also provides a solution and seems simpler. Both answers deal with the algorithmicx package however. I am interested in the solution for the algorithm2e package.   I read the documentation of the algorithm2e package but didn't find an answer.   How can I achieve the colored effect?  EDIT:  Here is sample code to create a minimum working example.  \documentclass{article}  \usepackage[ruled]{algorithm2e} % http://www.ctan.org/pkg/algorithm2e  \begin{document}  \begin{algorithm}[h]  \caption{Evolutionary algorithm}   initialize population \;  \For( \emph{Evolutionary loop}){$g := 1$ to $G_{max}$}  {     do things \;     evolve population \;  }  celebrate \; \end{algorithm}  \end{document} Does "singular" have a common meaning across matrices, decomposition, vectors and values? Get e-mail addresses from Gmail messages received Get polynomial interpolation formula Permutation module of $S_n$ Let $G=S_n$ and let $V$ be the permutation module of $G$ with basis $\{x_1,\ldots,x_n\}.$  Let $\lambda, \mu \in \mathbb{C}$ to allow one to define a $\mathbb{C}G$-homomorphism $\rho:V \to V$ by $$\rho(x_j):=\lambda x_j+\mu\sum_{i \neq j}x_i.$$  By using the above fact or otherwise, how can we prove that $V$ is the direct sum of two non-isomorphic irreducible $\mathbb{C}G$ -submodules?  I tried to prove this by construction. A familiar irreducible submodule in this case is the $1$-dimensional space $U:=\operatorname{span}\{x_1+\cdots+x_n\}$. I intend to find another $(n-1)$-dimensional submodule $W$  which makes $V=U\oplus W$ hold, but it's hard to do so. Is there a way to use the fact instead of a random construction? Why is it possible to answer a question twice? You normally answer a question and you have the opportunity to answer again. The second time, a message is shown suggesting the use of the edit tool, but I think it's kind of wrong. I mean, if you answer something and then you want to say anything else you know that you must use edit.  Why not disable the answer button if you already have an answer? Proving $9^n - 8n - 1$ is divisible by $8^2$ for $n\ge 0$? Why don't you have to press CTRL+ALT+DEL to logon anymore in Windows 8? In previous versions of Windows, you had to press Ctrl + Alt + Delete to logon, or to unlock a locked workstation.  This was because this key sequence was recognized only by the OS and thus other software couldn't intercept it and display a spoofed logon screen to capture passwords.  Starting with Windows 8, you now just have to press Enter to get to the logon screen.  What is to stop someone from writing a fake logon screen?  Did Windows 8 add some sort of new security mechanism to mitigate this security issue? Batch file to delete files older than N days I am looking for a way to delete all files older than 7 days in a batch file.  I've searched around the web, and found some examples with hundreds of lines of code, and others that required installing extra command line utilities to accomplish the task.  Similar things can be  in just a couple lines of code. It seems that something at least remotely easy could be done for batch files in Windows. I'm looking for a solution that works in a standard Windows command prompt, without any extra utilities. Please no PowerShell or Cygwin either. FoxTrot Bill Amend Problems So I found this on the Wolfram website today:     So I was wondering about how one might be able to (if possible) solve those four problems by hand. Here are the problems, $\LaTeX$ed:    $ \lim_{x \to +\infty} \dfrac {\sqrt{x^3-x^2+3x}}{\sqrt{x^3}-\sqrt{x^2}+\sqrt{3x}} $ $ \displaystyle\sum_{k=1}^{\infty} \dfrac {(-1)^{k+1} k^2}{k^3+1} $  $ \dfrac {\mathrm{d}}{\mathrm{d}u} \left[ \dfrac {u^{n+1}}{(n+1)^2} \cdot \left[ (n+1) \ln u - 1 \right] \right] $  $ \displaystyle\int_0^{2\pi}\displaystyle\int_0^{\frac{\pi}{4}}\displaystyle\int_0^4 \left( \rho \cos \phi \right) \rho^2 \sin \phi \, \mathrm{d}\rho \mathrm{d}\phi \mathrm{d}\theta$     Ideas   The degree of the numerator is $\frac{3}{2}$. The degree of the denominator is $\frac{3}{2}$. The expression is of an indeterminate form, namely $\frac{\infty}{\infty}$, so we use l'Hoptial's Rule.  No ideas, really.  Derivatives are always pretty easy, although this one is a bit bashy. Basically bash Product and Chain Rules, etc.  Integration by Parts bash? The $\mathrm{d}\theta$ part is very trivial. I mean for the $\rho$ and $\phi$. When I have these facts,   French coins were not found in the ruin. German coins were not found in the ruin.   which of these sentences are grammatically correct?  1.     No French and German coins were found in the ruin.   2.     No French or German coins were found in the ruin.   3.     No french nor German coins were found in the ruin.   I think 1 is correct but I'm not sure.  By the way, I can also say     Neither German nor French coins were found in the ruin.   Also, when you have an additional fact   Italian coins were not found in the ruin   which of these statements are grammatically correct?  1.     No French, German and Italian coins were found in the ruin.   2.     No French, German or Italian coins were found in the ruin.   3.     None of French, German and Italian coins were found in the ruin.   4.     None of French, German or Italian coins were found in the ruin.   Thanks, Lost space after removing bootcamp So I removed bootcamp and now there is a partition called "free space" that I can't remove.  Here is my output from diskutil list:    Here is the screenshot from disk utility showing the partitions:    And here is the error I get when I try to remove "free space":    Heres the output when I run   sudo diskutil apfs resizeContainer disk0s2 0 Geometry with Trigonometry If $(a,b)=1$ then prove $(a+b, ab)=1$. I have a song that Windows can't play because there is a question mark in the name of the file.  "Where Have All the Cowboys Gone?.ogg" // as an example   So I try to rename it and Windows complains whether I try it in Explorer or from command prompt.  Error I get when trying to copy, rename, or move is:     The Filename, directory name, or volume label syntax is incorrect   Is there a Windows way to force a rename in this case?   Update  I'll keep an eye on this question, but after 13 answers and many attempts (aside form 3rd party solutions) it seems that Windows can't do this (or at least my windows can't, no short names). So I'm accepting the answer which was my original solution anyway of using Linux. It would be nice to see Windows handle this somehow, so don't stop just because I've accepted this answer, the question still stands! How to change the theme of Snap Apps? I am enjoying the new "snap" applications, but apps installed using this method doesn't seem to be using the system gtk theme I have set.   Some use Adwaita or worse, Raleigh.   How can I have them use the standard Ambience theme I'm using across the whole system? changed primary email in google account. how to log in? I added gmail to my google account. Now when i try to log into stack exchange using google it prompts me to create a new account instead of using the existing.   The google account is still the same but google changed the primary email adress.  How can i log in again? Let's say that I have a folder, within my Documents folder, that has files I want no one to access without a password.  Is there a way to lock down that folder so that it's password protected / encrypted?  In fact, is it possible to password-protect any individual folder? I am looking for an animated series, shown in the 90s. I remember an episode where the boy went into VR and fought a medusa-like monster.  It may be that this was the premise for the whole series: That the children heroes go into VR every time to fight the evil guys.   I am not sure if the whole series were computer-animated but surely there were such elements together with classic animation.  The style of the series was quite dark and cool for teenage boys. I'm wondering something. Whenever i right click on minecraft and click open with another application, i try to find Java JDK. I have installed it for minecraft itself, but i just cant find where java is on the list of other applications. Any help?