I want to prove that  let S be the set on which group G operates. Let H ={ g∈G | g.s=s for all s∈S} prove that H is normal subgroup of G. This group action gives an homomorphism whose kernal is H. Then it follows directly from statement that H is normal.  How to prove that if the subgroup is the kernel of a homomorphism having the group as its domain then it is normal?  Any help clearing up the statement would be greatly appreciated. Thanks! Is there any other approch ?
My professor proved that the kernel is a normal subgroup under a homomorphism f by saying. Let $h\in H$  by applying f to it  $f(ghg^{-1})=f(g)f(h)f(g)^{-1}=e $  My question is how is this a proof that the kernel is normal. What is the reason we apply f to it ? By applying f dont we just prove that $f(ghg^{-1})$ is normal ?
Tracking down where disk space has gone on Linux? Transaction log reported full but is not You have two identical eggs. Standing in front of a 100 floor building, you wonder what is the maximum number of floors from which the egg can be dropped without breaking it. What is the minimum number of tries needed to find out the solution? What happens to Bruce Banner when Hulk loses a tooth? In Age Of Ultron we witnessed Hulk lose a tooth.   My question is: Has Bruce Banner lost a tooth as well? How to parse a date? If $a_k\geq0$ is a decreasing sequence converging to $0$, do we have $\lim_{n\to\infty}\frac{\sum_{k=1}^n a_k}n = 0$? Here's a pickle: reference style links in markdown are fantastic. Numbered footnotes, however, can be a pain. Named references are a beautiful solution, but the editor seems to choke on them in certain contexts, throwing an error:  your post appears to contain code that is not properly formatted as code...  I've looked at a  and tested it out quite a bit and I'm pretty sure line breaking isn't the problem, named references are valid , and I can get multi-word named references to work basically every time (in this post, even). Is there a reason for named references to be seen as improperly formatted code in certain contexts? At the risk of being verbose (but for the sake of completeness), here's the source of the specific question I was trying to post when I encountered the problem: I'm trying to build a [Google Apps Script] that integrates with [Trello], the idea being to use it to push information from spreadsheets and forms into the [Trello API] and create cards on a `pending` list on a certain board.  I found this [question] on here that pointed me in the right direction, and added in OAuth based on the [GAS OAuth Documentation]. The problem is I can't post the the board. I run the script, the OAuth prompt fires, and the script completes with no errors. I can also `GET` data from the private board, so I assume the authorization is working properly.  So, what am I doing wrong that prevents my script from `POST`ing to Trello?  Here's the code I'm working with:      var trelloKey = [Trello API key];     var trelloSecret = [Trello API key secret];     var trelloList = [the id of the list we're posting to];      var oauthConfig = UrlFetchApp.addOAuthService('trello');         oauthConfig.setAccessTokenUrl('https://trello.com/1/OAuthGetAccessToken');         oauthConfig.setRequestTokenUrl('https://trello.com/1/OAuthGetRequestToken');         oauthConfig.setAuthorizationUrl('https://trello.com/1/OAuthAuthorizeToken');         oauthConfig.setConsumerKey(trelloKey);         oauthConfig.setConsumerSecret(trelloSecret);      function createTrelloCard() {        //POST [/1/cards], Required permissions: write       var payload = {'name': 'apiUploadedCard',                       'desc': 'description',                       'pos': 'top',                       'due': '',                       'idList': trelloList};        var url = 'https://api.trello.com/1/cards'       var options = {'method' : 'post',                      'payload' : payload,                      'oAuthServiceName' : 'trello',                      'oAuthUseToken' : 'always'};        UrlFetchApp.fetch(url, options);     }  [Trello]: http://trello.com [Google Apps Script]: https://developers.google.com/apps-script/ [Trello API]: https://trello.com/docs/ [other question]: http://stackoverflow.com/questions/18710641/trello-api-how-to-post-a-card-from-google-apps-script-gas [GAS OAuth Documentation]: https://developers.google.com/apps-script/guides/services/external#making_requests_to_services_with_oauth  If I'm being a complete dunce, please let me know. It seems like one of those errors where it's either a hardcore backend bug or a really stupid user mistake and I'm ready to admit to the latter. how to scale math font only Change Gmail account to make in-app purchase? I know that the gravitational interaction of antimatter is expected to be the same as normal matter.  But my question is, has it ever been experimentally validated?  I think it would not be a trivial experiment, because electromagnetic effects have to be eliminated, so neutral particles would be needed. Maybe diamagnetically trapped antihidrogen atoms could be examined as to which direction they fall? Is globbing a feature of the shell? Is it OK to require users to have a third party login account? For a web app I'm developing I was dreading having to   Require users to create yet another account / password to keep in mind. Handle retrieval of lost passwords etc.   What many sites (like this one for instance) has started offering is to use a third party account to create an account.   What I wonder is whether it would be considered OK to require having third party account, i.e. Find a polynomial f(x) of degree 5 such that 2 properties hold. I have been trying to find a polynomial $f(x)$ such that these $2$ properties hold:   $f(x)-1$ is divisible by $(x-1)^3$ $f(x)$ is divisible by $x^3$   To start, I set $f(x) =ax^5 + bx^4 + cx^3 + dx^2 + ex + f$. This is divisible by $x^3$, so $d, e, f $  must be $0$. So polynomial $f(x) = ax^5 + bx^4 + cx^3$  This minus $1$ is divisible by $(x-1)^3$. So I used synthetic division and got that the remainder of $ax^5 + bx^4 + cx^3$ divided by $(x-1)$ is $a+b+c-1$. But it should divide out evenly, so $a+b+c-1 = 0$.  From here, I can't find any other equations involving the three variables. Could someone have any suggestions for how to continue? Thanks in advance. :) Rotate photo in moderncv I'm doing my CV with the template ModernCV Classic. I want to rotate my picture but the command used is \photo I don't know how to do that.  \documentclass[11pt,a4paper,sans]{moderncv}  \moderncvstyle{classic} \moderncvcolor{blue} \nopagenumbers{}  \usepackage[utf8]{inputenc}  \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage[francais]{babel}  \usepackage[scale=0.75]{geometry}  \name{John}{Doe} \title{...} \address{adress}{Town}{Country}  \extrainfo{age, nationality} \phone[mobile]{54466145745} \email{john.doe@mail.com} \photo[80pt][0.4pt]{Photo.jpg} \quote{Objectifs: ...} Named blocks to limit variable scope: good idea? I'm doing some web scraping and sites frequently use HTML entities to represent non ascii characters. Does Python have a utility that takes a string with HTML entities and returns a unicode type?  For example:  I get back:  &amp;#x01ce;   which represents an "ǎ" with a tone mark.  In binary, this is represented as the 16 bit 01ce.  I want to convert the html entity into the value  u'\u01ce' Drawing arrows on an HTML page to visualize semantic links between textual spans I have an HTML page with some textual spans marked up something like this:  ... &lt;span id="T2" class="Protein"&gt;p50&lt;/span&gt; ... &lt;span id="T3" class="Protein"&gt;p65&lt;/span&gt; ... &lt;span id="T34" ids="T2 T3" class="Positive_regulation"&gt;recruitment&lt;/span&gt; ...   I.e. each span has an ID and refers to zero or more spans via their IDs.  I would like to visualize these references as arrows.  Two questions:   How can I map an ID of a span to the screen coordinates of the rendering of the span? How do I draw arrows going from one rendering to another?   The solution should work in Firefox, working in other browsers is a plus but not really necessary. The solution could use jQuery, or some other lightweight JavaScript library. Prove Uncountable set minus a countable set is uncountable Let B be a countable set, and let A be an uncountable set. Prove that A-B is uncountable Socket Tester all indicators light up I used the above GFCI/RCD (up to 250v) socket tester on the 240v black and red live wire of the US split phase ac power system. When I connect the ground, all indicator lights (3 red) got on indicating "Live/Grd Reverse, Missing GRD" although it is connected properly.  What is inside the socket tester? Does it have a live and neutral leg such that it should only be used for ac system with polarity? How about if you use it on the red and black 240v phase to phase? Why would all the indicators light become on (3 red lights)?  The following is the specs of it:  Specifications: Brand Name: HABOTEST Color: Black Material: Flame Retardant ABS Plug Type: US Voltage: 48V-250V 45-65Hz RCD Test: >30mA (EU/UK Plug) RCD Working Voltage: 220V±20V GFCI Test: >5mA (US Plug) GFCI Working Voltage: 110V±20V Voltage Measurement: 48V~250V / 45~65Hz    Accuracy: ±(2.0%+2) Working Environment: 0°C ~ 40°C, 20% ~75%RH Storage Environment: -10°C ~ 50°C, 20% ~80%RH Safety Rating: EN61010-1,-2-030, EN61326-1,CAT II 300V Item Size: 62 * 65 * 55mm / 2.4 * 2.6 * 2.2in Item Weight: UK 64g/2.3oz, EU 59g/2.1oz, US 53g / 1.9oz Package Size: 7.5 * 7.4 * 7.3cm / 3.0 * 2.9 * 2.9in Package Weight: UK 90g/3.2oz, EU 86g/3.0oz, US 79g / 2.8oz Can a continuous function from the reals to the reals assume each value an even number of times?