Prove that $S_5$ has no elements of order $120$.  I know that by Lagrange's theorem, elements can have order $1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120$.  I feel like I need to use some sort of contradiction argument here but I'm not sure where to go with that. Any suggestions?
Given the following groups, what is the maximum possible order for an element for   (a) $S_5$ (b) $S_6$ (c) $S_7$ (d) $S_{10}$ (e) $S_{15}$  My book justifies the answer as          (a) The greatest order is $6$ and comes from a product of disjoint cycles of length 2 and 3          (b) The greatest order is $6$ and comes from a cycle of length $6$      The other answers were justified exactly the same way, that is (c) 12, (d) 30, (e) 105  I do not understand how in (a) we even got the number "6" from $S_5$ and what disjoint cycles they are referring to. Could someone at least justify one for me?
I am writing a document using LaTeX and generating PDF.  Can anybody tell me how I can generate the PDF metadata, e.g. author or title, from LaTeX?  On Windows XP, I am using MikTex's texify to generate the PDF.  On linux, I am using pdflatex to generate the PDF.  Any answers for either platform would be much appreciated! I have a huge amount of 2D-coordinates, associated with a value, e.g.:    x   |   y   | value 27.50   52.15   12.51 61.83   13.32   57.56 36.23   21.83   41.73 40.46   85.67   25.20 ...   The data is not tabular and I Want the points between two data-points to be interpolated in some way (which way is not really clear, yet)  I want to preset the data as heatmap like this:   Is there any ready-to-use package for PSTricks or TikZ to do it? Proof of continuity $f(x+y) = f(x) + f(y)$ Do neutrinos not couple to the Higgs field? I was reading the CernCourier, my favorite source of message on Higgs and friends. I was rather shocked, when I saw this:     "The mechanism by which neutrino mass is generated is not known."   What? Not known?  says: In Higgs-based theories, the property of 'mass' is a manifestation of potential energy transferred to particles when they interact ("couple") with the Higgs field, which had contained that mass in the form of energy.  Does this mean that neutrinos don't couple to the Higgs field? Iterating through a Collection, avoiding ConcurrentModificationException when removing objects in a loop We all know you can't do the following because of ConcurrentModificationException:  for (Object i : l) {     if (condition(i)) {         l.remove(i);     } }   But this apparently works sometimes, but not always. Here's some specific code:  public static void main(String[] args) {     Collection&lt;Integer&gt; l = new ArrayList&lt;&gt;();      for (int i = 0; i &lt; 10; ++i) {         l.add(4);         l.add(5);         l.add(6);     }      for (int i : l) {         if (i == 5) {             l.remove(i);         }     }      System.out.println(l); }   This, of course, results in:  Exception in thread "main" java.util.ConcurrentModificationException   Even though multiple threads aren't doing it. Anyway.  What's the best solution to this problem? How can I remove an item from the collection in a loop without throwing this exception?  I'm also using an arbitrary Collection here, not necessarily an ArrayList, so you can't rely on get. Is there a way to make the following return true?  string title = "ASTRINGTOTEST"; title.Contains("string");   There doesn't seem to be an overload that allows me to set the case sensitivity.. Currently I UPPERCASE them both, but that's just silly (by which I am referring to the  issues that come with up- and down casing).  UPDATE This question is ancient and since then I have realized I asked for a simple answer for a really vast and difficult topic if you care to investigate it fully. For most cases, in mono-lingual, English code bases  answer will suffice. I'm suspecting because most people coming here fall in this category this is the most popular answer.  answer however brings up the inherent problem that we can't compare text case insensitive until we know both texts are the same culture and we know  what that culture is. This is maybe a less popular answer, but I think it is more correct and that's why I marked it as such. I'd like to install a  client for Xubuntu (12.04).  I'm getting non-English Google results, and I didn't find  in the Xubuntu repositories.   Does anyone know of a client that works well for Ubuntu or it's supported derivates (or installation instructions for grive?) Why do positive and negative variants of the same question elicit the same answer? Another question I am struggling with:  Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be defined in a neighborhood of $a$ (except maybe $a$ itself) and suppose that, $\lim_{x \rightarrow a}\left(f(x)+\dfrac{1}{\left|f(x)\right|}\right)=0$. Find $\lim_{x \rightarrow a}f(x)$, and prove by definition, that this is indeed the limit.    Thank you all for your answers!!  But.. How can I prove by definition that this is indeed the limit? take $\epsilon&gt;0$, I have to show that there exists a $\delta&gt;0$ such that, $|x-a| \leq \delta$ implies $|f(x)+1| \leq \epsilon$. now, I do know that there exists an $\delta_1&gt;0$, such that $|f(x)+\frac{1}{|f(x)|}|&lt; \epsilon$.Or, $\frac{\epsilon}{k}$ for some constant $k$ But, how can I continue from there? Why don't we define division by zero as an arbritrary constant such as $j$? Why don't we define $\frac 10$ as $j$ , $\frac 20$ as $2j$ , and so on? I know that by following the rules of math this eventually leads to $1=2$ , but we could make an exception and say that $j$ is the only number such that $0*j \not= 0$ , and put other restrictions necesary so that we don't get contradictions. We do this for $i$ , so why can't we do it here? For example, $i^2$ , is defined as $-1$ , but you could also say $i^2=\sqrt {-1} *\sqrt {-1}=\sqrt {(-1)(-1)}=\sqrt{1}=1$ , but we make an exception for this. Execute task if tasker profile has been active for some time How can I execute a tasker task only after a certain condition has been true for a certain time?  Normally a profile executes its task as soon as all conditions are met. I want to delay this activation, until the conditions have stayed true for some time.  E.g.    do something after the screen has been continuously off or on for 10 minutes do something after wifi xy has not been near for 10 minutes do something after cell towers x,y,z have not been near for 10 minutes. Return vs. Enter? I noticed that my MacBook's keyboard has the word enter above the return key. I also noticed that on Mac OS X, there's a special symbol for return ↩, and another for enter ⌤. How would I send the enter key if my MacBook has no numpad?  Is there any difference between these two keys? Why are duplicates not being merged? I see more and more exact duplicates on SO. Instead of being merged they are being simply closed. I think this is very big disservice to SO. Why don't moderators merge duplicates? Why don't owners of the site do anything about this?  EDIT  It's disservice because this way we have more and more junk. Duplicates are duplicates and should be merged so that information in in one place.     If the duplicate has no answers then there is no benefit to merging.   Of course there is. You don't need to waste your time with dupes.     It also means that the search routes to the main question are   diminished which makes the posting of new duplicates more likely   There's easy solution for this which I gave in  back in march.     If both questions have answers that have been accepted, which one   should be unaccpeted?   There are plenty of solutions. You accept most voted accepted answer or the first accepted answer or both (I think this is be the best). Yes, there will be possibility of many accepted answers to one question. So what?     When a question is closed as a duplicate, there is a link to the   duplicate question. What more do you want?   I don't want yet another link. I want to have all answers to some problem in one place. Is this that hard? Isn't this natural expectaion?     Removal of the question means that the text in it is no longer searchable - that's a disservice.   See my answer above to Scrooge's comment. What does ':' (colon) do in JavaScript? I'm learning JavaScript and while browsing through the jQuery library I see : (colon) being used a lot. What is this used for in JavaScript?   // Return an array of filtered elements (r) // and the modified expression string (t)    return { r: r, t: t }; $L^p$ spaces and counting measure currently I am working on the following two exercises as a revision for my exam.  Let $\mu$ be the counting measure on $\mathbb N$.   Show that if $1 \le p &lt; s &lt; \infty$ then $f \in L^p$ implies $f \in L^s$ and $\|f\|_s \le \|f\|_p$. Show that if $f \in L^p$ for some $p \ge 1$ then $\lim_{s \to \infty} \|f\|_s = \|f\|_\infty$.   My attempts so far:   It is $f \in L^s$ iff $\sum_{k = 1}^\infty |f(k)|^s$ converges. Since $f \in L^p$ we know that $\sum_{k = 1}^\infty |f(k)|^p$ converges. Thus for any $\varepsilon &gt; 0$ there is $N \in \mathbb N$ s.t. \begin{align*} k \ge N \Longrightarrow |f(k)|^p &lt; \varepsilon \Longrightarrow |f(k)| &lt; \varepsilon^{1/p} \end{align*} We have $\forall k \ge N$ that $|f(k)|^s &lt; \varepsilon^{s/p}$. Originally I wanted to use the comparison test now, showing that \begin{align*} \sum_{k = N}^\infty \varepsilon^{s/p} &lt; \infty, \end{align*} but that does not work since \begin{align*} \sum_{k = N}^\infty \varepsilon^{s/p} = \varepsilon^{s/p} \underbrace{\sum_{k \ge N} 1}_{= \infty},  \end{align*} so I don't know how to go on here. Let $K := \|f\|_\infty$. We need to show that $\forall \varepsilon &gt; 0$ $\exists M \in \mathbb N$ s.t. \begin{align*} s \ge M \Longrightarrow \left|\left(\sum_{k = 1}^\infty |f(k)|^s\right)^{1/s} - K\right| &lt; \varepsilon. \end{align*} Choose $M$ s.t. $M &gt; p$. Then $s &gt; p$ and from part (i) we know that \begin{align*} \left(\sum_{k = 1}^\infty |f(k)|^s\right)^{1/s} \le \left(\sum_{k = 1}^\infty |f(k)|^p\right)^{1/p}. \end{align*} But how can I use this? fitting a smaller range when we bought our house, it had a very used 40" gas range. now that range has died. replacing with a new 40" range is very pricey ($2k+), and our cooking needs don't justify the size anyhow.  i'm looking for ideas on how to fit a smaller range. i would imagine that extending the counter top and cabinets would be a pretty extensive change. we've found we can get a 36" range for around 1/2 the cost. would it be feasible to add some sort of storage along side the smaller range to fill in the space?  thanks. i know this is an open-ended question with no specific answer, i'm just looking for ideas. Trying to put a footnote in subsection title:  \subsection{Effect of Lateral Confinement on SW Band Structure\footnote{Parts of this sub--section are based on \citet{Venkat2013}}}   produces the following error:  Argument of \@sect has an extra } ...section are based on \citet{Venkat2013}}}   I am using:   \documentclass[paper=A4, fontsize=12, toc=listof, toc=index, toc=bib]{scrreprt}   Please advise. Removed Windows 7 and installed Ubuntu Server 12.04, but Windows Boot Manager still appears How to fix slow wireless on machines with Intel wireless cards? I have looked through all the previous questions but turning power management off for the Wi-Fi card didn't work. This problem seems to be with the Lenovo Thinkpads. I switched to Ubuntu 11.10 from Windows 7 3 days ago so I'm new to the operating system. I came across a map and when I was playing I found some coloured and bolded item names. I thought it isn't possible as at most you can only use the custom name tag to rename things! Is there a way to colour and bold item names in vanilla Minecraft? External programs are OK. Oh, and the wiki page of formatting codes doesn't really do much.