I'm trying to get some intuition on why order of a subgroup divides order of a group. I know Lagrange's Theorem and its proof (every element of a group is an element of some coest, cosets are disjoint and each one has the same number of elements as subgroup), but the proof is more like magic than any intuition.  So I've been trying to prove this fact (divisibility of order) without Lagrange theorem. Let's say we take a group $H$ and try to extend it, by taking $g \notin H$ and considering group $G$ generated by set $H \cup {g}$. Such a group would have to contain $g^{-1}$ and every word over alphabet $\{g, g^{-1}, H\}$. Is it possible to prove intuitively that such construct is of size $|H|\cdot n$ for some $n \in \mathbb{N}$? This would essentially prove that any group $G$ containing $H$ has order divisible by $|H$|, because we would be able to take $g \in G$ such that $g \notin H$, extend $H$ by it just like above and continue this process until $H$ is extended into $G$.
I am beginning with Abstract Algebra and I'm trying to understand Lagrange's Theorem. The theorem reads      For any finite group $G$, the order of every subgroup $H$ of $G$ should divide the order of $G$.   It seems simple and I've used it to solve some exercices but I believe I'm missing the essence of it. Is there an example that can help me understand it better, or visualize it? Is there a geometric interpretation?
It was either a short story or a novel. I read it about 20 years ago, and some vague memory of it stuck.  It's set in a future where everyone's allotted role is decided by a series of tests once they reach a certain age. They are then classified into a group which decides what jobs they will do.  Our hero rather controversially gets a perfect score in the tests. Nobody had done that in a long time. and at first he worries about what this might mean. He is sent to work for the mysterious scientists who maintain the computer that runs society. Most people don't know they exist. He is disappointed to learn he has to be their tea boy, and learn a tea making ceremony perfectly.  In the process he learns how to fool the machine, and eventually to circumvent the controls society has placed on him. He makes several visits to an area outside the control of the machine - where people are living in poverty and lawlessness.  Can't remember the author, or the name, just that basic idea, and the fact that early in the book is a very grand "tea ceremony" where he must make the tea perfectly. How to determine the value of a domain name? I'd like to register a second-level domain with the .com extension and I noticed that it has already been registered and the owner put it for sale and let the possibility to make an offer in their site or by contacting them.  I saw that domains are generally sold for extremely different prices, so I'm confused.  Are there any general criteria to decide how much I should offer to purchase a domain? How do I remove an OpenID or login? How do I remove login credentials (emails or OpenIDs) from my account?     For more information, see "" in the . Most people with a degree in CS will certainly know what . It helps us to measure how well an algorithm scales.   But I'm curious, how do you calculate or approximate the complexity of your algorithms? The ring $\Bbb Z\left [\frac{-1+\sqrt{-19}}{2}\right ]$ is not a Euclidean domain How to Async.AwaitTask on plain Task (not Task)? I have a couple of floats in my document, which contain large Images (it consumes the half of the page). In this case, the figure get's its own page and no text goes on this page.  How can I configure this? I'd like to have text before and after the floats until the float doesn't use more then 75% of the page height. Gimp: how to remove background noise/artifacts and enhance handwritten text I have a lots of scanned handwritten lecture notes (PDFormat) and I want to improve their readability (I have uploaded an example). In particular I'd like to remove the "squares" on the back. I would appreciate any advice you may have. Thanks in advance. How to modify itemize vertical spacing within blocks in beamer? I have noticed that when I have "itemize" environments defined within blocks in beamer, that they introduce more vertical spacing between the top and the bottom of the blocks, than vertical text. Since I have a slide with some text filled blocks and directly after, a slide with only "itemize" holding blocks, I would like the spacing between the itemized text and the block to be the same as for regular text.   On  I have read about "squeeze" option, but it doesn't seem to work...   Additional info:  The problem is in the additional vertical space that is included by the "\itemize" environment. I have found  on how to remove vertical spacing from the itemize list, and I am using this approach. However, it is a brute force approach. What I would like to know if it is possible to have the same vertical space between the end of the itemize list and the bottom of a block, as the one set between the normal text and the bottom of the block.   MWE:   \documentclass[10pt]{beamer} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{exscale} \usepackage{epsfig} \usepackage{fancybox} \usepackage{lmodern} \usepackage[T1]{fontenc} \usepackage{overpic}  \usepackage{multimedia}  \usetheme{default} \usecolortheme{crane}  % -- Modifications that enable the footline and set black color for the fonts  \setbeamertemplate{footline}{%     \begin{beamercolorbox}[wd=\paperwidth,ht=2.25ex,dp=1ex,right, rightskip=1mm, leftskip=1mm]{titlelike}         \textcolor{black}{             %\inserttitle\hfill\insertauthor\hfill\insertframenumber%             Something\hfill Something Else \hfill\insertframenumber%         }     \end{beamercolorbox} }  \setbeamertemplate{itemize items}[square]  \setbeamercolor{item}{fg=black} \setbeamercolor{frametitle}{fg=black} \setbeamercolor{block title}{fg=black} \setbeamercolor{title}{fg=black} \setbeamercolor{block body}{bg=white}  % END Modifications  \begin{document}   \begin{frame}   \frametitle{text blocks}     \begin{block}{block}         block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   blockk     \end{block}{block}      \begin{block}         block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block     \end{block}     \begin{block}{block}         block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block     \end{block} \end{frame}  \begin{frame}   \frametitle{itemize blocks}     \begin{block}{item}         \begin{itemize}             \item item              \item item              \item item         \end{itemize}     \end{block}     \begin{block}{item}         \begin{itemize}             \item item              \item item              \item item         \end{itemize}     \end{block}     \begin{block}{item}         \begin{itemize}             \item item              \item item              \item item         \end{itemize}     \end{block}   \end{frame}  \end{document} Display data from geoserver on click map with openlayers I tried to display data with popup from Geoserver or Postgres to my map, but it dosen't work at all. Can anyone please help me? I used this code:      function Info(e)          {                 if (e.features &amp;&amp; e.features.length)                  {                      //console.log(e.features[0].attributes);                     var popupId = e.xy.x + "," + e.xy.y;                     var popup = popups[popupId];                     if (!popup || !popup.map) {                     popup = new OpenLayers.Popup.FramedCloud(                     popupId,                      map.getLonLatFromPixel(e.xy),                     null,                     " ",                     null,                     true,                     function(e) {                         delete popups[this.id];                         this.hide();                         OpenLayers.Event.stop(e);                     }                 );                 popups[popupId] = popup;                 map.addPopup(popup, true);                 }                popup.setContentHTML(popup.contentHTML + "nom_user:" +e.features[0].attributes.nomservice +" region_user :"+ e.features[0].attributes.region);             popup.show();                                    }       selectcltr = {         click: new OpenLayers.Control.WMSGetFeatureInfo({             url: 'http://localhost/geoserver/wms',              title: 'Identify features by clicking',             layers: [ser],             queryVisible: true         }),         hover: new OpenLayers.Control.WMSGetFeatureInfo({             url: 'http://localhost/geoserver/wms',              title: 'Identify features by clicking',             layers: [ser],             hover: true,             // defining a custom format options here             formatOptions: {                 typeName: 'projet',                  featureNS: 'http://www.openplans.org/topp'             },             queryVisible: true         })     } What is the purpose of the EBP frame pointer register? How do you Xforward windows using ssh? Skip empty rasters in ArcGIS I'm trying to write some code to skip over empty raster sets when converting rasters to point (the empty sets stop the conversion, leaving me with a half complete model run).  I know that I can identify empty rasters using the ALLNODATA result from "Get Raster Properties" .  I know I can delete rasters using the Delete.management tool.  I know I should be able to create a branch using calculate value.  But I'm inexperienced with python and modelbuilder, and haven't been able to put this all together. Could someone help me figure out what I am doing wrong? My code and an image of the modelbuilder setup are below.  def fn(Property):  #Import modules      import arcpy  #ifelse statement if Property == "1":             return "true" elif Property == "0":            return "false" General Information about Eigenvalues for an 3x3 symmetric matrix What is the rigor behind U subsitution? $ \int f(g(x)) dx = \int \frac {f(u)}{u'} du$  requires that  $ \int f(x) dx = \int f(x) \cdot dx$  but dx just represents the variable that F(x) +c is a function of. So why is it legal for dx to be treated algebriacally?  I tried investigating this property by using riemann summation:  $\lim\limits_{n \to \infty}(\sum_{k=1}^n f(\frac{kx}{n} )\frac{x}{n}) = \int_0^x f(t)dt= (F(x)+c)-(F(0)+c)$  and so you can define   $ \lim\limits_{n \to \infty} (\sum_{k=1}^n f(\frac{kx}{n} )\frac{x}{n})+F(0)+c)  = \int f(x) dx$  you can write $\frac{x}{n} = dx$ and $k\frac{x}{n}=kdx= x_k$  then you have  $ \lim\limits_{n \to \infty} (\sum_{k=1}^n f(x_k)dx)+F(0)+c) = \int f(x) dx$  since  dx is being written to be multiplied by the series, then you can define dx in different terms to aquire an integral in terms of other variables.   but that exists only in abstraction. I can't quite sufficiently complete the task of doing so. Heisenberg famously derived his uncertainty principle by considering the disturbance that a measurement would have on a small enough system.  Of course in the mathematical formalism of Quantum Mechanics the relationship is derived from more basic principles.  How does String Theory account for it? Heuristically, and mathematically? How do I fix Google Chrome's latest new tab page? I've just updated to the latest Google Chrome Dev build, and they've changed from the annoying-but-ok new tab page to an ugly and less usable page.  On the old one, nine thumbnails was not enough, and now they've restricted it to eight.  There's an empty "Tips and Suggestions" box which I wouldn't want even if it had stuff in; they've added Recent Downloads - on the rare occasions when I need that, I'll hit Ctrl-J to open the Downloads tab; and the site-specific search box on the right has now gone!  Is there a way to get the old new tab page back, or a plugin to provide a superior one?   Update: I launch Chrome by pressing Win+R and entering "chrome" - I have a shortcut on the Path with a Target of "C:\Documents and Settings\&lt;user&gt;\Local Settings\Application Data\Google\Chrome\Application\chrome.exe" --old-new-tab-page and whilst double-clicking this shortcut brings up the previous new tab page, using the keyboard method results in the old one.  How do I get it working via the Run box? (Without using a batch file, which slows things down too much.)  Update 2: GRRR! Now they've gone and stopped the --old-new-tab-page parameter from working at all! I hate this new version of the page that they're forcing upon us - can anyone provide a method to get rid of it and get the decent page back? I've found a strange thing: some Stack Exchange sites have their own domain names instead of https://[site name].stackexchange.com. The websites which have their own domain names are:          Are there any parameters for allocating specific URLs to Stack Exchange sites? Python argparse: default value or specified value Proof that $A_n$ the only subgroup of $ S_n$ index $2$. I have what seems to me a very simple proof that $A_{n}$ is the only subgroup of $S_{n}$ of index 2.  Since I've seen other people prove it with what feel like really complicated methods (), I'm wondering if I've overlooked something.  Proof: Let $H$ be any subgroup of index 2 in $S_{n}$.  Then $H\cap A_{n}$ is a normal subgroup of $A_{n}$ and since $A_{n}$ is simple, $H$ is either the trivial or the improper subgroup of $A_{n}$.  If it's the trivial subgroup it doesn't have index 2 in $S_{n}$, and otherwise the theorem is proved.