Prove a certain group $G\lesssim A_m$ I'm trying to solve this problem from my group theory course:  Let $H$ be a subgroup of a simple nonabelian group $G$ of index $m\geq  5$. Prove that $G$ is isomorphic to a subgroup of $A_m$ ($G\lesssim  A_m$, being $A_m$ the alternate group of degree $m$).  The things I've noticed so far are:  $|G|=|G:H|\cdot |H|=m|H|$ From Cayley's Theorem I then get that $G\lesssim S_{m|H|}$  I don't know if these are useful things in this problem. I know the simplicity of $G$ is required, since $A_m$ is simple for $m\geq 5$, but I don't know how to follow. How can I solve this? Any help will be appreciated, thanks in advance.
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$ Let $G$ be a simple group and there exists a subgroup $H$ of index $n \in \mathbb{N}_{\ge 3}$. Show that: $|G|$ divides $n!/2.$ I saw a similar question .   There was written: "The proof is really simple. Let $X$ be the set of left cosets of $H$. Consider $\phi: G \to \text{Sym}(X)$ given by $\phi(x)(aH)=(xa)H$. Then $\phi$ is a homomorphism. Consider now $K=\ker \phi$. Then $K$ is a normal subgroup of $G$ contained in $H$. Finally, $G/K$ is isomorphic to a subgroup of $\text{Sym}(X)$, which has order $n!$, where $n=[G:H]$. Thus, $[G:K]$ is finite and divides $n!$."  In fact, I don't understand the last sentence. "Thus, $[G:K]$ is finite and divides $n!$." Also did I understand it right that $ker(\phi)=H$?  Can you also give me an advice how to proof it for $n \ge 3$. (maybe induction)  Thank you for taking your time.
Software was blocked from opening because it is not from an identified developer I went to run LibreOffice on my OS X 10.9 and it would not let me install it. It said it was from an unidentified developer. I'm aware of the security features of the Mac and went to System Preferences to allow the software to run.   When I opened this setting I saw a button that said, '"LibreOffice" was blocked from opening because it is not from an identified developer.' and there was a button next to it that said "Open Anyway". I clicked this button and nothing happened. I clicked it multiple times, like a hundred, out of frustration. Is there something else I need to do? Could someone verify this is not a bug?   UPDATE: I've included the photo below of what my System Preferences screen looked like and the button I pushed that didn't work.  UPDATE II: So I tried opening the application again and went into the system preferences and pressed the "Open Anyway" button again and nothing happened at first but a few minutes later while I was typing an email something flashed across my screen, possibly another dialog (or more likely a chewpacabra), but it quickly disappeared and the next thing I know LibreOffice launched! Number reference headline I try to number the header of my references. I succeeded to add the number to the table of contents, but I can't get it to work for the header.  To add the number to the toc I used:  \addcontentsline{toc}{section} \protect\numberline{\thesection}References}   If I try the same for \refname, I get 5References without space.   \renewcommand{\refname}{\protect\numberline{\thesection References}}   When I put the references title outside the curly brackets, it's shown in a new line. What am I missing here? The full bibliography call looks like this:  \renewcommand{\refname}{\protect\numberline{\thesection References}} \addtocounter{section}{1} \cleardoublepage \phantomsection \addcontentsline{toc}{section}{\protect\numberline{\thesection}References} \bibliographystyle{apacite} \bibliography{bachelor}   EDIT 1:  As @TeXnician stated, a tilde would solve the problem. It is not very elegant, though, as I have to use approximatly three of them to get the same space as in the other headlines.   \renewcommand{\refname}{\protect\numberline{\thesection~~~References}}   EDIT 2: @Werner suggested a shorter solution, but it throws multiple errors, when I try to integrate it into my .tex file. Here's a short reproducable example which replicates the errors.  \documentclass[12pt]{article} \usepackage[utf8]{inputenc} \usepackage[margin=1.1in]{geometry} \usepackage{listings} \usepackage{apacite} \usepackage{natbib}  \begin{filecontents}{mybib.bib}  @Article{draxler2010sample,     author    = {Draxler, Clemens},     title     = {Sample size determination for Rasch model tests},     journal   = {Psychometrika},     year      = {2010},     volume    = {75},     number    = {4},     pages     = {708--724},     publisher = {Springer}, }  \end{filecontents}   \begin{document}  \tableofcontents      \section{First Chapter}  \citep{draxler2010sample}  \let\oldsection\section% Store \section in \oldsection \renewcommand{\section}[2]{\oldsection{#2}}% Update \section to gobble * \bibliography{mybib} \bibliographystyle{apacite}  \appendix  \section{R Code}  \subsection{Greedings Function}  \begin{lstlisting}     print("Hello World") \end{lstlisting}  \end{document} Will Unity allow making custom launcher icons from .desktop files or via menu editing system? (Right now the launcher doesn't give the option to "keep in launcher" on all programs.  For some programs I use, I have to make custom launchers or .desktop files.  For instance, daily blender builds are generally just folders with an executable.  In basic Gnome or KDE, I can make a new menu entry with the menu editing system. Then, I can also add it to Docky either from the menu or by dragging a .desktop file to it. Unity launcher doesn't support drag and drop, so thats not a bug or anything, but when I open a .desktop file, it has unpredictable results. Most time it will not have "keep in launcher". Sometime it will have a pinnable item without the .desktop's icon, and if I pin the item to the launcher, it will not call upon the program again after closing it. I've also gotten it to just work with a .desktop file for celtx. I have a very large codebase (read: thousands of modules) that has code shared across numerous projects that all run on different operating systems with different C++ compilers. Needless to say, maintaining the build process can be quite a chore.   There are several places in the codebase where it would clean up the code substantially if only there were a way to make the pre-processor ignore certain #includes if the file didn't exist in the current folder. Does anyone know a way to achieve that?  Presently, we use an #ifdef around the #include in the shared file, with a second project-specific file that #defines whether or not the #include exists in the project. This works, but it's ugly. People often forget to properly update the definitions when they add or remove files from the project. I've contemplated writing a pre-build tool to keep this file up to date, but if there's a platform-independent way to do this with the preprocessor I'd much rather do it that way instead. Any ideas? Is there a difference between DR/evil and DR/unholy? Why do gaps appear in my mesh when I animate it? As I attempt to complete a "Walk cycle" with my model, the mesh at the pants line separates from the skin as the model moves or as I rotate the controls. This happens in many places on my model and it reveals the model interior. It is a one piece mesh, though, so I don't understand why this happens. Is this caused by weight painting problems? How can I fix this? Moving all files to a new computer, my concern is links I'm moving all files to a new computer, via an external hard drive. I am certain that broken links will be a concern. What can I do to fix this? Is there a 'Best Practices' kind of way to prevent this problem beforehand? I have many hundreds of .blend files, and of course, I fear losing them. How to only perform LTS upgrades? I have a website the seems to load very slowly. When I run a speed test on it I see there seems to be a 6 secound gap before the HTML loads. Images and JS scripts load very fast after that point.  You can see in the pic below the yellow 'wait time' bars:    This seems to be consistant no matter what the HTML content on the page is.  This site is using a CMS (ModX Revo) so HTML is actually stored in an SQL database and brought forth by PHP, but I've never experienced this issue before.  Would anyone know what is causing this and how I could speed it up? Why an abstract class implementing an interface can miss the declaration/implementation of one of the interface's methods? A curious thing happens in Java when you use an abstract class to implement an interface: some of the interface's methods can be completely missing (i.e. neither an abstract declaration or an actual implementation is present), but the compiler does not complain.  For example, given the interface:  public interface IAnything {   void m1();   void m2();   void m3(); }   the following abstract class gets merrily compiled without a warning or an error:  public abstract class AbstractThing implements IAnything {   public void m1() {}   public void m3() {} }   Can you explain why? When I gain control of a creature, do I also gain control of any auras or equipment attached to it? Number of possible permutations of n1 1's, n2 2's, n3 3's, n4 4's such that no two adjacent elements are same? Given $n_1 $ number of  $1 $'s,  $n_2 $ number of  $2 $'s,  $n_3 $ number of  $3 $'s,  $n_4 $ number of  $4 $'s.  form a sequence using all these numbers such that two adjacent numbers should not be same.  I have tries lot of things but nothing worked.  can somebody tell me how to solve this problem? Computing $ \int_0^\infty \frac{\log x}{\exp x} \ dx $ Why can't I buy the Windhelm house (Hjerim)? Optimizing logical sector size for physical sector size 4096 HDD With many new hard drive disks the physical sector size is 4096. Would it be possible to make the system use a logical sector size of the same size, rather than the default logical sector size of 512?  Will it speed up bulk reads and writes? Where can it be configured? Where are the evolutionary "inbetweeners"? The problem I've always had with evolution is the actual lack of variation between animals. More specifically, the lack of observable gradual change between species.  Take for example the hammerhead shark. It obviously has a strong connection with a normal shark, but where's the "half-hammerhead" shark? Where's the species that only has a SLIGHTLY wider head?  I really doubt that a single mutation on a normal shark resulted in a head shaped like that, it had to happen gradually. Right? As far as I understand, evolution works by random mutation in a child being somehow slightly beneficial to that animal so it get's to live longer and reproduce more, eventually resulting in it's genetic material in dominating the inferior ones before it. But if it was for whatever reason beneficial to have a slightly wider head, why don't all the sharks have slightly wider heads now? On the other hand, if it wasn't beneficial, how come the hammerhead still exists?  Furthermore, I feel like some characteristics of a certain animal can't really have gradual benefits. There's a lizard that shoots blood out of it's eyes. The original ancestor certainly didn't, and I'm sure that if it's child only bled slightly from it's eyes it wouldn't really help if much in life. So for it to actually become beneficial, it seems to me that there had to be a ton of lizards that had a disadvantage of bleeding slightly. In fact, there had to be so many that they evolved even further in the same direction and kept doing that until they bled so much it's an actual benefit (sorry for this turning so gross). So why did the inbetweeners survive long enough to evolve even more, and why did they THEN die out? Replacing the core of a solenoid to increase the pulling force Exporting Data From Salesforce - REST or SOAP? I have now almost completed my family tree, having each branch going back to approximately the French revolution, discovering dozens of previously unknown ancestors.  I would like to share the family tree with my family, the problem is that I cannot have a tree in a nice format that could be printed on a standard A4 sheet (maybe using both sides or whathever).  I have a SVG tree that use graphics, but it is unconventionally large and if scaled down to a A4 sheet, the text would be so small that it would not be legible. The maximum width is of about 50 ancestors of a generation, then it goes sparse again.  I'd like to know if a format exist where the family tree (with direct ancestors only) would be printable, with names but also other complementary information, and the format should be printable as not all my family users are particularly skilled with computers. We are asked the following:      let $f(x)$ be a continuous function on $[0, 1]$ such that   $$\int _ {0} ^ {1} \ f(y) dy = 1$$   prove that $f(x) = 1$ for some $x ∈ [0, 1]$   The question asks to take into account Rolle's Theorem and Mean Value Theorem.  I'm aware of the definitions, but I am really unsure how to answer this question and my tutors are not really responding till after easter.    Could someone please help? Why is there a "y" involved?  Thanks you