Find subgroup of $S_5$ isomorphic to the quaternion $Q$ I'm trying to solve this problem from my abstract algebra course:  Find a subgroup of $S_5$ (Symmetric group of order 5) isomorphic to the quaternion group $Q$.  I started writting down the elements of $Q$ to start trying some examples and see if the quaternion properties where verified: $$Q=\{\pm 1,\pm i,\pm j,\pm j\}.$$ Obviously the element $1$ is $(1)(2)(3)(4)(5)$ in $S_5$. Then I tried with: $$i=(1234)(5)\ \ , \ \ -i=(1432)(5).$$ Both verify they have order $5$, and from them I get $-1=(13)(24)(5)$. Now, I'm stuck, since I think I pick wrong elements because I can't find good elements for $\pm j$ and $\pm k$. What is the easiest way to do this kind of problem where you're asked to find a subgroup isomorphic to a certain group? Any help will be appreciated.
$Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic to a subgroup of $S_n$ for $n\leq 7$. Question is to prove that :     $Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic to a subgroup of $S_n$ for $n\leq 7$.   I see that $Q_8$ is isomorphic to subgroup of $S_8$ by left multiplication action.  Hint given was to prove that stabilizer of any point contains $\{\pm 1\}$.  To prove Cayley's theorem, stating any group is isomorphic to a subgroup of $S_n$ we take action of given group on a set $A$ having same cardinality.  with that motivation I want to check if there is a Isomorphism then there is a map from $G\times A \rightarrow A$. i.e., $G$ gives a permutation group $S_A$.  I tried in same manner. Suppose $Q_8$ is isomorphic to subgroup of $S_n$ with $n\leq 7$  then it should come from a group action of $Q_8$ on a set of cardinality atmost 7.   Suppose $Q_8$ acts on a set $A$ with possible cardinality at most 7.  $stab(a)=\{g\in Q_8 : g.a=g \forall a\in A \}$  $cl(a)=\{g.a : g\in Q_8\}$  I know number of elements in class of $a$ equals to index of stabilizer.  As $cl(a)=\{g.a : g\in Q_8\}\in A$ i.e., $cl(a)\subseteq A$ and as $|A|\leq 7$ i see that $|cl(a)|\leq 7$.  But, $|cl(a)|=|Q_8:stab(a)|$ for any element $a\in A$.  So, $|Q_8:stab(a)|=|cl(a)|\leq 7$ for all $a\in A$.   So, $stab(a)$ should be non trivial subgroup of $Q_8$ if not then $|Q_8:stab(a)|=8$  non trivial subgroup (proper) of $Q_8$ contains $\{\pm1\}$.  So, In the worst case, $\{\pm 1\}\subseteq stab(a)$ for all $a\in A$.  As $Ker(\eta)=\cap_{a\in A}stab(a)$ (where $\eta$ is the action of $Q_8$ on $A$)   we see that $\{\pm 1\}\subseteq Ket(\eta)$ which means that $Ker(\eta)$ is non trivial.  thus there is no isomorphism (coming from $\eta$) between $Q_8$ and any subgroup of $S_7$.  I would be thankful if someone can check whether my approach is correct or if there is any other simple possible way.  P.S : Usually what i do to see whether two groups are isomorphic or not is to check for cardinality, abelian property, no of elements with same order and so on. But I was having no idea when i fail in all these ways. With this Group actions i could see possibility for getting a precise conclusion on Isomorphisms.I would like to Thank Mr. Jyrki Lahtonen (a user of Math.SE) who made me to get used to Group actions.  P.S $2$: If any thing is wrong in my idea, it is entirely my fault, and if anything is correct in this whole credit should go to Mr. Jyrki Lahtonen
$11$ out of $36$? I got this by writing down the number of possible outcomes ($36$) and then counting how many of the pairs had a $6$ in them: $(1,6)$, $(2,6)$, $(3,6)$, $(4,6)$, $(5,6)$, $(6,6)$, $(6,5)$, $(6,4)$, $(6,3)$, $(6,2)$, $(6,1)$. Is this correct? Problem with the header/ footer width I have recently started using the geometry package to format the margins of my documents. Although, I've noticed that the width of the header/ footer remains the same. How can I correct this?    Here's a MWE:  \documentclass{article}  \usepackage{geometry} \usepackage{fancyhdr}     \pagestyle{fancy} \usepackage{blindtext}  \lhead{} \chead{Header} \rhead{} \lfoot{} \cfoot{Footer} \rfoot{\thepage} \renewcommand{\headrulewidth}{1pt} \renewcommand{\footrulewidth}{1pt}  \begin{document}  \newgeometry{margin=0.5in, bottom=1in, top=1in} \Blinddocument  \end{document} Does walking fast prevent eggs hatching? I have a couple of eggs that I've picked up from some Pokestops (I'm lucky enough to live above one).   I walk a lot. I get to work through a 2 mile walk / train journey every day, so I should be picking up loads of steps for the eggs I have. However, because I'm normally always running late... for everything, I have to rush everywhere, so I walk at an above average pace.   Today I've walked around 5 miles, but the game has registered 100 metres. Is this because I walk too fast, or because of something else?  It's worth pointing out that I have the app open when walking... mostly because I have to catch them all. (This made me miss a train this morning... luckily I was early :D) So far I've been using public void run() {} methods to execute my code in Java. When/why might one want to use main() or init() instead of run()? How do I remove duplicates from a C# array? I have been working with a string[] array in C# that gets returned from a function call. I could possibly cast to a Generic collection, but I was wondering if there was a better way to do it, possibly by using a temp array.  What is the best way to remove duplicates from a C# array? Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$ I can't figure out an algebraic proof for the following identity:  $$\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$$  Combinatorical solution:  We can see that as choosing some from $n$ and the rest of $k$ from $m$, thus $k$ in total.  Or we could just choose $k$ from the union. I ran into a problem today when I decided to use npm update -g in a nodejs app directory. This was a bad decision, because it caused me to run into a .   The solution is to reinstall nodejs and npm. Unfortunately when I go to install npm using sudo apt-get install npm it doesn't work.  The full error in the terminal is here:  Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that  some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation: The following packages have unmet dependencies:  npm : Depends: nodejs but it is not going to be installed    Depends: node-abbrev (&gt;= 1.0.4) but it is not going to be installed    Depends: node-ansi but it is not going to be installed    Depends: node-archy but it is not going to be installed    Depends: node-block-stream but it is not going to be installed    Depends: node-fstream (&gt;= 0.1.22) but it is not going to be installed    Depends: node-fstream-ignore but it is not going to be installed    Depends: node-github-url-from-git but it is not going to be installed    Depends: node-glob (&gt;= 3.1.21) but it is not going to be installed    Depends: node-graceful-fs (&gt;= 2.0.0) but it is not going to be installed    Depends: node-inherits but it is not going to be installed    Depends: node-ini (&gt;= 1.1.0) but it is not going to be installed    Depends: node-lockfile but it is not going to be installed    Depends: node-lru-cache (&gt;= 2.3.0) but it is not going to be installed    Depends: node-minimatch (&gt;= 0.2.11) but it is not going to be installed    Depends: node-mkdirp (&gt;= 0.3.3) but it is not going to be installed    Depends: node-gyp (&gt;= 0.10.9) but it is not going to be installed    Depends: node-nopt (&gt;= 2.1.1) but it is not going to be installed    Depends: node-npmlog but it is not going to be installed    Depends: node-once but it is not going to be installed    Depends: node-osenv but it is not going to be installed    Depends: node-read but it is not going to be installed    Depends: node-read-package-json (&gt;= 1.1.0) but it is not going to be installed    Depends: node-request (&gt;= 2.25.0) but it is not going to be installed    Depends: node-retry but it is not going to be installed    Depends: node-rimraf (&gt;= 2.2.2) but it is not going to be installed    Depends: node-semver (&gt;= 2.1.0) but it is not going to be installed    Depends: node-sha but it is not going to be installed    Depends: node-slide but it is not going to be installed    Depends: node-tar (&gt;= 0.1.18) but it is not going to be installed    Depends: node-which but it is not going to be installed E: Unable to correct problems, you have held broken packages.   What can I do to get npm properly installed and running again?  Thanks. How do I suppress the whitespace above an equation block at the page top? How do I suppress the whitespace above the equation block at the page top ? It is added to the top margin, and I don't want that to happen. I know that \abovedisplayshortskip controls this for the whole document, but how can I change this for this particular equation block and NOT for any others ? Negative vspace doesn't work.  \documentclass[pdftex,12pt,a4paper,english,dutch,leqno]{article} \usepackage[top=1cm,right=1cm,bottom=1cm,left=1.5cm,noheadfoot]{geometry}         \usepackage{babel} \usepackage{amsmath} \usepackage{amssymb} \usepackage{empheq}  \begin{document}  \abovedisplayskip=6pt \abovedisplayshortskip=3pt % This should NOT be changed \belowdisplayskip=6pt \belowdisplayshortskip=3pt  \begin{equation*} \left.\begin{array}{r} a_{31}'=a_{11}\cdot a_{31}-a_{31}\cdot a_{11}=0\\ a_{32}'=a_{11}\cdot a_{32}-a_{31}\cdot a_{12}\\ a_{33}'=a_{11}\cdot a_{33}-a_{31}\cdot a_{13}\\ a_{34}'=a_{11}\cdot a_{34}-a_{31}\cdot a_{14}\\ b_{3}'=a_{11}\cdot b_{3}-a_{31}\cdot b_{1} \end{array}\!\!\right\} \Longrightarrow\; \begin{matrix*}[l] \text{Pseudo-code :}\\ a_{11}\cdot b_{3}-a_{31}\cdot b_{1}\mapsto b_{3}\text{.}\\ \forall\;i\in\{4,3,2,1\} : a_{11}\cdot a_{3i}-a_{31}\cdot a_{1i}\mapsto a_{3i}\text{.} \end{matrix*} \end{equation*}  \end{document} Need to print out iPhone text messages I have a court case and have text messages on my iPhone I need to transfer to my iPad email for evidence.  Every time I try it doesn't work, it will send them to my iMessage but not my email.  I need to print them out How do I do this? Assume you have some objects which have several fields they can be compared by:  public class Person {      private String firstName;     private String lastName;     private String age;      /* Constructors */      /* Methods */  }   So in this example, when you ask if:  a.compareTo(b) &gt; 0   you might be asking if a's last name comes before b's, or if a is older than b, etc...  What is the cleanest way to enable multiple comparison between these kinds of objects without adding unnecessary clutter or overhead?   java.lang.Comparable interface allows comparison by one field only Adding numerous compare methods (i.e. compareByFirstName(), compareByAge(), etc...) is cluttered in my opinion.   So what is the best way to go about this? Getting a FILE* from a std::fstream Is there a (cross-platform) way to get a C FILE* handle from a C++ std::fstream ?  The reason I ask is because my C++ library accepts fstreams and in one particular function I'd like to use a C library that accepts a FILE*. Exponential of the differential operator I am not sure whether this question is even well-posed. But today I learnt that $e^Df(x) = f(x+1)$ where $D$ is differential operator and  $$e^D \triangleq \sum_{i=0}^{\infty} \frac{D^i}{i!}.$$  (ref. )  So I was curious as to whether the differential equation  $$\frac{df(t)}{dt} = e^Df(t) = f(t+1)$$  has any solutions apart from $f = 0$? Countries with no visa requirements on its visitors I tried running the following statement:  INSERT INTO VOUCHER (VOUCHER_NUMBER, BOOK_ID, DENOMINATION) SELECT (a.number, b.ID, b.DENOMINATION)  FROM temp_cheques a, BOOK b WHERE a.number BETWEEN b.START_NUMBER AND b.START_NUMBER+b.UNITS-1;   which, as I understand it, should insert into VOUCHER each record from temp_cheques with the ID and DENOMINATION fields corresponding to entries in the BOOK table (temp_cheques comes from a database backup, which I'm trying to recreate in a different format). However, when I run it, I get an error:  Error: Operand should contain 1 column(s) SQLState:  21000 ErrorCode: 1241   I'm running this in SQuirrel and have not had issues with any other queries. Is there something wrong with the syntax of my query?  EDIT:  The structure of BOOK is:  ID  int(11) START_NUMBER    int(11) UNITS   int(11) DENOMINATION    double(5,2)   The structure of temp_cheques is:  ID  int(11) number  varchar(20) I was doing a package removal with apt-get remove but then realized I should have done a --purge along with it to remove the configuration files.  Can I remove the packages configuration files easily or do I need to reinstall the package and then remove with a --purge? What is the distinction being made with regards to truth and falsity in these two sources? How do I downgrade Minecraft to play on a server running an older version? To apply for a Schengen business visa, you need a letter from the company in the Member state inviting you. My company is based in the UK, owns properties in Schengen countries, and I have to inspect their condition.  What letter do I get? How to extend summations to fractional orders? There has been a trend now two persons downvote it then it goes on like that. So let there be down voting but do not display the downvotes till the user votes on it. Else the user reads it with an impression that the question is not good.  Am I missing something?