If G be a non-trivial group with no non-trivial proper subgroup, show that G is a group of prime order.  Attempt:  Given $G\neq \{e\}$ and let $a\in G, a\neq e$. Then $H=&lt;a&gt;$ is a subgroup of G. Then by the hypothesis, $H= \{e\} ~or~ G$. But $H\neq &lt;a&gt;, ~as ~a\neq e$. So G is generated by $a$.  I can show also that G is finite (if G is infinite, considering group $&lt;a^2&gt;$).     Let if possible, $o(G)=n=composite=kl, ~k,l&gt;1$ are intergers. As $G=&lt;a&gt;$, $o(a)=n$. As $&lt;a^k&gt;$ is a subgroup of G, $(a^k)^l=a^{kl}=a^n=e\implies o(a^k)=l&lt;n$, i.e $&lt;a^k&gt;$ is a subgroup of order $&lt;n$, but G cannot have a non-trivial subgroup. So $l=1 ~or~ l=n$. How to conclude the rest?   The problem is in the shaded portion only. Please correct it.
This is something I'm supposed to be able to prove for an upcoming test, but I can't find anything to help me prove this in my notes or the chapter, which is on cosets and Lagrange's theorem. If all I start with is the group having no proper subsets, then that means any subset is the whole group. By Lagrange's theorem, that means the index is $1$, but that doesn't get me anywhere. Where do I start?
Find an example where Markov's inequality is tight in the following sense: For each positive integer $a$, find a non-negative random variable $X$ such that $P(X\ge a)=E(X)/a$.   How to do this problem, I am really confused. Also, what is the definition of Markov's inequality? Sort file by group of lines If I have a file with contents similar to:  FirstSection     Unique first line in first section     Unique second line in first section  SecondSection     Unique first line in second section     Unique second line in second section  ...  NthSection     Unique first line in Nth section     Unique second line in Nth section   Is it possible to use unix commands (e.g. sort, awk) to sort the file alphabetically by the first non-indented line in each three line group, whilst keeping the indented lines under their existing group? Do messages from feeds prevent freezing/deleting a chat room? My two main questions:  Are all URLs that include your User Id tracked?  Can the visitors who use the tracked URLs have Stack Exchange accounts?   Specifically: Recently the ability to earn badges for sharing links to posts on Stack Exchange sites . I'd like to know when and how this tracking works. It's clear that if you go to a question, and make use of the link button, you will get a URL with your User ID that will be tracked if a visitor uses it to get to SO. For example, let's use  as an example. It's regular URL is:  https://stackoverflow.com/questions/309300/defend-php-convince-me-it-isnt-horrible  If you go to that page and use &quot;save link as&quot; in the link section of the question you will be given a tracked url with your User ID:  https://stackoverflow.com/q/309300/[USER ID]  My question is, are there other URLs that would be tracked? All of the following URLs take you to the Question / Answer, but are they tracked? For example what if I just tack my User ID onto the long url?  https://stackoverflow.com/questions/309300/defend-php-convince-me-it-isnt-horrible/[USER ID]  What if I add my User ID to an answer URL?  https://stackoverflow.com/questions/309300/defend-php-convince-me-it-isnt-horrible/309622#309622/[USER ID]  or even  https://stackoverflow.com/q/309622/[USER ID]   Is there a way to know that a URL is being tracked? Is there some sort of format to follow? Are the only tracked URLs the ones provided in the Question links? Finally, does this sentence from the blog post mean that visitors cannot have an account?  Also, the tracked IPs must originate from outside our existing network. Benefits & Practicality of Virtual Machines I have been living under a rock for the past 5 - 8 years, and am just now getting up to speed with virtualization, virtual machines and platforms like VMware or VirtualBox.  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I don't care how clever virtual machines are, if my computer only has 8GB of RAM, and I have 4 VMs running on it, those VMs only get 8GB of RAM to share between them. Putting 4 VMs on my server doesn't endow it with 4 times that amount of RAM! So, as far as I see it, VMs save you money on hardware, but just end up bogging down that hardware as you have multiple machines competing over the same pool of resources. What am I missing here? Can VMs communicate with each other over your network using high-level protocols like TCP/IP, HTTP, FTP, etc.? Otherwise, it just doesn't make sense to implement all these VMs if you have to feed them separate and independent chunks of data/requests to process. In polar coordinate how we can get $dx\;dy=r\;dr\;d\theta$?  with these parameters:    $r=\sqrt{x^2+y^2}$ $x=r\cos\theta$ $y=r\sin\theta$  Tanks. Using the Limit definition to find the derivative of $e^x$ Get Currently Executing Class/Method Name? 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What is the most practical way to implement camera/viewport to a 2D-game?  I've read, that I should store the object world position instead of position relative to the screen?  Current situation:  I have implemented a simple 2D-game where I load objects and levels from XML-files. Currently the level XML-file looks like this:  &lt;map&gt;    &lt;tile obj="ground" x="0" y="555" /&gt;    &lt;tile obj="ground" x="16" y="555" /&gt;    &lt;tile obj="ground" x="32" y="555" /&gt;    ... &lt;/map&gt;   All objects have a 2d-vector "position" storing their current location on the screen.  What I want it to be:    In the picture:   Camera is either 800x600 or 640x480 Blocks and sprites are 16x16 pixels. World size may vary The coordinates probably should be normalized relative to the world, not to the screen? Viewport position relative to player's x, y and moves when the player reaches camera dead zone (similar to ).   I'm asking pseudo examples / articles, but if you need to know what I'm using for the development: SDL &amp; C/C++. Block IPs if they access a resource Where can I download Windows 8 legally, from Microsoft? How to install hp scanjet 200 in Ubuntu 12.04? Unable to scan with hp scan-jet 200. When I type lsusb command in terminal it shows the scanner details.  I opened the simple scan which displays no scanners found, and x sane image scanners also displays the same error.