So I, more or less, need help understanding this problem.   The first thing I have tried is to pick a specific group: $\mathbf{Z}_5$, under multiplication.  To demonstrate, say we pick element $2$ and $|2| = 4$, and then the only other possible orders are $\{1,4\}$.   Indeed, $4 = LCM(1,4)$.  Not really quite sure, how to generalize this, yet. Any hints? What shall I think about? Definitely prime factorization of each element, hmm... I think!
I will quote a question from my textbook, to prevent misinterpretation:     Let $G$ be a finite abelian group and let $m$ be the least common multiple of the orders of its elements. Prove that $G$ contains an element of order $m$.   I figured that, if $|G|=n$, then I should interpret the part with the least common multiple as $lcm(|x_1|,\dots,|x_n|)=m$, where $x_i\in G$ for $0\leq i\leq n$, thus, for all such $x_i$, $\exists a_i\in\mathbb{N}$ such that $m=|x_i|a_i$. I guess I should use the fact that $|x_i|$ divides $|G|$, so $\exists k\in \mathbb{N}$ such that $|G|=k|x_i|$ for all $x_i\in G$. I'm not really sure how to go from here, in particular how I should use the fact that $G$ is abelian.
Can any one explain me what is the use of derivatives in real life. When and where we use derivative, i know it can be used to find rate of change but why?. My logic was in  real life most of the things we do are not linear functions and derivatives helps to make a real life functions into linear.  eg converting parabola into a linear function $x^{2}\rightarrow 2x$  but then i find this; derivation of $\sin{x}\rightarrow \cos{x}$  why we cant use $\sin{x}$ itself to solve the equation. whats the purpose of using its derivative $\cos{x}$. Please forgive me if i have asked a stupid questions, i want to improve my fundamentals in calculus. View vote count button on mobile On the mobile site, can there be a button directly below the vote up/down buttons for viewing vote counts? When I am using the mobile site and I want to view vote counts, I have to switch to the full site to do it which is very inconvenient.   It could be a small link that says "vote counts," because I don't know what the button would be (the letter "V?" a "+/-" thing? I have no idea).  Related:  , but I am requesting an additional button below the vote buttons. That way you won't accidentally click a vote button. Programmatically combining images in PHP Split comma-separated strings in a column into separate rows msiexec is command prompt software that installs an  program. But I have found that you can install an MSI file from the command line by just typing in the name of the MSI file on the command line.  But in order to uninstall the MSI file, it seems you have to call the msiexec program and give it a /x or /uninstall.  How can I uninstall an MSI from the command line without using the msiexec routine? Single word describing something "outside of oneself" Use Unity API from another Thread or call a function in the main Thread Breaking of diffeomorphism invariance after fixing a background metric Following minimal document:  \documentclass{report}  \begin{document} \renewcommand*\thechapter{\Roman{chapter}} \tableofcontents \setcounter{chapter}{6} % Problem starts at chapter 7 \chapter{Seven} \chapter{Eight} \section{Eight-one} \section{Eight-two} \end{document}   yields this output (cut to size for your convenience):    Clearly, the chapter (and section) headings hang into the chapter (section) numbers. For the previous chapter numbers, this isn’t a problem since all chapter titles are aligned to the same column and the numbers I–VI are sufficiently narrow.  (The same happens regardless of document type, I’m actually using scrreprt with the same result.)  How can I prevent this? Prettify OS X console by coloring output? OS X terminal is not using colors for highlighting output from different commands, for example ls.   When compared with other systems like Ubuntu, this is a big loss of accessibility/productivity.  What solutions do I have to improve this experience? For the moment I found only this guide for  but I am looking for a more complete solution. Why don't players get extra xp for large encounters? Based on the table Encounter XP Multipliers from the Basic Rules' DM section, Page 57, when you're building encounters in 5E, you include an XP modifier depending on the number of creatures your PCs will be fighting: 1.5x for a pair, 2x for 3-6 and so on. However, the PCs only get the base XP as a reward.     This doesn't change the actual xp reward ... just your calculations of how difficult the encounter is.   Why is this? After acknowledging that battles vs multiple opponents are more difficult, why not reward them as such? How do I beat the Reaper on Rannoch? I'm having trouble with the last of the Rannoch missions, where Shep wakes up and takes on a Reaper.   Shepard has a targeting laser which will call in an orbital strike with the combined might of the Quarian Migrant Fleet, but there's two major problems standing between the commander and victory.   The Targeting Laser takes a while to power up. The Reaper is trying to kill you.   I can consistently avoid the Reaper's attacks for the first volley, but each time you hit it, the Reaper moves forwards, giving you less time to dodge future assaults. Is there some key to avoiding the Reaper's attacks? Strafing doesn't seem to be enough. Why are the dineutron and diproton unbound? Custom Button on Lightning Experience I created a custom button Send Email in Lightning Experience. The problem is that button in visible in feed section layout, but I want to put it in the upper right corner of page layouts near Edit, Clone, Delete ecc....  How Can I do that? Thanks  I add an image that explain the situation.    EDIT: I would like to explain you better my problem.  I have created three custom quick action:   I have added them to the case page Layout as you can see    I would like to understand the reason why I don’t see these three custom action in the Page Layout in the section in the top right I used to be able to go to Accounts and Import → Add another email address you own and add an email account that's only a forwarding address. That is, it's not a mailbox and I don't have SMTP access to it.  Gmail would send me a confirmation email to that address, I'd confirm it's mine, and done. Now it only gives me the option to add SMTP credentials.  Did they stop supporting the previous method? Is there any other way I can get that option back? Equation mode, labelling equations I have the following   \[K^{-1}\left\|\sum\limits_{n=1}^N a_ny_n\right\|    \leq \left\|\sum\limits_{n=1}^N a_nx_n\right\|    \leq K\left\|\sum\limits_{n=1}^N a_ny_n\right\| (\dagger)\]   But the dagger doesn't appear at the right hand end of the page. How can I Get it so that it does appear on the end of the page? the \hfill command doesn't seem to work. Thank you. Proving the inequality $\frac{n^k}{k^k}\leq\binom{n}{k}\leq\frac{n^k}{k!}$ involving binomial coefficient. What is the best way of proving inequalities including binomial coefficients in general? Lets look at this example:     $$\frac{n^k}{k^k}\leq\binom{n}{k}\leq\frac{n^k}{k!}$$   What I started with is:  $$\binom{n}{k}\frac{1}{n^k}\leq\frac{1}{k!}$$ $$\frac{n!}{k!(n-k)!}\frac{1}{n^k}\leq\frac{1}{k!}$$ $$\frac{(n-1)!}{(n-k)!n^{k-1}}\leq1$$ This is where I got stuck. How can I prove that is is lower than or equal to 1? Could you give me a hint? Check Constraint with Case Statement Windows 7 Aero Snap two windows horizontally As many people already know, there are several ways to trigger Windows 7 Aero Snap feature. A user can either press Win+{Left,Right,Up,Down} to snap the window to the left or right or maximize or restore. One can also drag the title bar to all the corner except bottom one.  However, is there any keyboard or mouse shortcut to place one window on top half the screen, and another window on bottom half the screen? Please do not tell me to use "Show Windows Stacked" because if I have more than two windows, it will place ALL the windows horizontally, which is not what I want. I want to be able to select the window I want and place it automatically either on top or bottom half the screen.  Thanks!  Edit: Thanks for Moab's suggestion, both on recommend WinSplit and updating my question.  I should make it more clear that I'm looking for a software that will do the functions WinSplit provided, but with on screen input, such as touch screen or pen. How do I use 'chmod' on an NTFS (or FAT32) partition?