I was reading Robinson's book "A Course in the Theory of Groups". In section 9.1 (Hall $\pi$-subgroups) it is commented that, in an insoluble group, the Hall $\pi$-subgroups may not be conjugate. As an example, the author says that the simple group $PSL(2, 11)$ has subgroups isomorphic with $D_{12}$ and $A_{4}$, which are non isomorphic Hall {2, 3}-subgroups and therefore not conjugates.   I would like to know why $PSL(2, 11)$ has subgroups isomorphic with $D_{12}$ and $A_{4}$.   I never heard before of the subgroup $PSL$, so I tried reading about it a bit. I found the following link: , which points to "Section II.8 of Huppert's Endliche Gruppen I" book. Unfortunately, I don't know German, so it did not help me.   EDIT: by following the link suggested as an answer (), I found Suzuki's Group Theory(I) book. In the section 3.6 the author find all the subgroups of $PSL(2, q)$. Problem solved, thanks.
This material is covered in detail in Dickson's "Linear Groups with an exposition of the Galois Field Theory", chapter XXII and Huppert's "Endliche Gruppen", chapter II, paragraph 8. Since I don't speak german and Dickson's treatment often requires deciphering, I was wondering if there is a "modern" account of this somewhere.
Run bash script at login stored in the home folder? When I try to load a LaunchAgent plist from launchctl I can't find out how to run a script in the home directory.  My code is:  &lt;?xml version="1.0" encoding="UTF-8"?&gt; &lt;!DOCTYPE plist PUBLIC "-//Apple//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd"&gt; &lt;plist version="1.0"&gt; &lt;dict&gt;     &lt;key&gt;ProgramArguments&lt;/key&gt;     &lt;array&gt;         &lt;string&gt;bash&lt;/string&gt;         &lt;string&gt;~/script.sh&lt;/string&gt;     &lt;/array&gt;     &lt;key&gt;RunAtLoad&lt;/key&gt;     &lt;true/&gt;     &lt;key&gt;Label&lt;/key&gt;     &lt;string&gt;com.tyilo.test&lt;/string&gt; &lt;/dict&gt; &lt;/plist&gt;   I have tried both with and without bash and also replacing ~ with $HOME. I have also tried using bash -c without it working.  The error code is:   `com.tyilo.test: bash: ~/script.sh: No such file or directory` Object-orientation in C What would be a set of nifty preprocessor hacks (ANSI C89/ISO C90 compatible) which enable some kind of ugly (but usable) object-orientation in C?  I am familiar with a few different object-oriented languages, so please don't respond with answers like "Learn C++!". I have read "" (beware: PDF format) and several other interesting solutions, but I'm mostly interested in yours :-)!    See also How are parameters evaluated for a Plot in Manipulate James Paterson in his novel writes, “she was shrink by trade, a forensic profiler, and Jack Morgan’s number two at Private “.   What does whole sentence mean? How do I start Chrome using a specified "user profile"? I want to know where I can find updated information about all the stable and in-development versions of Ubuntu, where I can find this information?     This question is meant to be a canonical question. It expects to answer the following questions:         Is X version stable yet?   Is X version in beta?   Is X version still supported?   What versions are still supported?   What are the versions in Beta/Alpha?   Until where X version will be supported?   Have X version ceased support?   When will be X version released?   Where I can download any release?   When will the next LTS version of Ubuntu be released?         The list is not exhaustive. Does Linux have the capability to use "junctioned" symbolic links? I'm not sure if this is an actual term or not, so let me explain the concept.  I have a git repository containing all my configuration dot files in ~/dotfiles. I use symbolic links into this directory in order to "activate" them. For example, by executing ln -s ~/dotfiles/bash/bash_profile ~/.bash_profile to produce the link:  ~/.bash_profile -&gt; ~/dotfiles/bash/bash_profile   However, I find myself in a situation where I want to combine the contents of multiple files. For example, I want the ~/.bash_profile symbolic link to point to two separate files, one for each project. E.g.:  ~/.bash_profile (1) -&gt; ~/dotfiles/bash/bash_profile (2) -&gt; ~/dotfiles/proj/bash_profile   I know I can simply concatenate the two files (e.g., cat ~/dotfiles/{bash,proj}/bash_profile &gt; ~/.bash_profile), but if I can do the same thing with symbolic links, I would prefer to.  I imagine that if such a feature exists (Nix is pretty huge), then under the hood it would have to map the two different files together, hiding all sorts of complexities under the hood (mapping file offsets of all non-first "mapped" files, locking all files when writing to the junctioned symbolic link, etc, etc).  If such a feature doesn't exist, are there are plans to implement it? How to start itemize on same line as text? I've created a new environment called 'Example' like this:  \newenvironment{example}     {\textbf{Example:} \begin{itemize}}     {\end{itemize}}   The environment works fine, but the items are starting after the Example: text. Is it possible to start with the first item on the same line as the Example: text and vertically aline the rest of the items with the first item? If in a topological space only finite subsets are compact sets, is it then the discrete topological space?   Thank you. This is a Q&amp;A that I recently have figured out, so I have posted the answer below.  After installing Windows 10, every time I get a kernel update or I run the update-grub2 it always shows Windows 7 or Windows Recovery Environment instead of Windows 10.  How do I fix this permanently? We are in the combat phase:   I declared my attackers  My opponent declared his blockers, stating which attackers he was blocking I cast an instant, which automatically kills his blocker. There is nothing blocking my attacker now   Now we're in the damage phase. I said that my attacker now does damage and  reduces my opponent's hit points. My opponent said that because my attacker was blocked no damage could be done. I pointed out that the blocker was destroyed before the damage phase, so therefore he wasn't blocking, but he wasn't having it. (My attacker does not have trample)  Is this covered by a specific section in the rules? Is losing privileges after placing bounty OK? My rep went over 2,000 today which meant I could edit other people's posts. Great! Then I added a bounty to a previous question, which brought me back down to 1,984...and I can no longer edit other posts. Should this changed? Or is this ideal? Where can I get an Ubuntu Installation CD/DVD/USB? Do positive semidefinite matrices have to be symmetric? Can you have a non-symmetric matrix that is positive definite? I can't seem to figure out why you wouldn't be able to have such a matrix, but all my notes specify positive definite matrices as "symmetric $n \times n$ matrices."  Can anyone help me with an example of a non-symmetric positive definite matrix, or some insight into a proof for why it would need to be symmetric should that be the case? Thanks! How to prove that $p-1$ is squarefree infinitely often, where $p$ is prime?  I had thought of using Dirichlet's theorem on arithmetic progressions.  The number of squareful values of $p-1$ less than or equal to $n$ is bounded by $\sum_{q}{\pi_{q^2,1}(n)}$, each term $\pi_{q^2,1}(n) \approx \frac{\pi(n)}{q \cdot (q-1)}$, and $\sum_{q}{\frac{1}{q \cdot (q-1)}} \lt 1$.  But using for example , the possibility is open that the error in each approximation for $\pi_{q^2,1}(n)$ may continue to increase with $q$, and since the sum runs up to $\sqrt{n}$, this argument is invalid.  Is there a more refined version of Dirichlet's theorem that can be used to circumvent this issue?  Or an entirely different way to prove this statement? Move the Add Favorite / Ignored Tags boxes to the top It seems that most people who use the ignored tags feature end up with a fairly long list of those tags.  Although , it is sometimes necessary to add new ignored tags manually (typically when adding some-tag*).  It would be a lot easier if the add box was above the long list of ignored tags rather than below. I suppose the same would apply to the favorites tags for consistency's sake. I'm having trouble figuring out how to iterate over map layers. I can access the currently highlighted layer in the table of contents via iface.activeLayer() but I want my code to operate on all layers.  I don't see anything in the API that provides easy access to this functionality and couldn't find a good online example but maybe I missed something? Is it possible to upgrade the graphics card in modern laptops? As there's no definitive answer to this, and I would prefer not to be too specific about particular models, what are the things that determine whether this is possible or not?   What expansion slots would the laptop require to do this? How can you find out if a motherboard/chipset supports upgrades?  Is heat an issue? Could it void the warranty? How is battery life affected? Why is Android File Transfer necessary? Apparently newer versions of Android require the use of Android File Transfer. I have an older machine that doesn't require this, and just mounts as a USB mass storage device. I can't see a technical reason why Android File Transfer is now required when it used not to be.  What changed in order to make this necessary? Why is it necessary? I've just had to write a string reverse function in C# 2.0 (i.e. LINQ not available) and came up with this:  public string Reverse(string text) {     char[] cArray = text.ToCharArray();     string reverse = String.Empty;     for (int i = cArray.Length - 1; i &gt; -1; i--)     {         reverse += cArray[i];     }     return reverse; }   Personally I'm not crazy about the function and am convinced that there's a better way to do it. Is there?