There is a group $G$ and subgroup $H \subset G$. Their orders:  $$\left | H \right| = n,  \left |G \right| = 2n. $$  How can I prove that $H$ is a normal subgroup?
Please excuse the selfishness of the following question:     Let $G$ be a group and $H \le G$ such that $|G:H|=2$. Show that $H$ is normal.   Proof:  Because $|G:H|=2$, $G = H \cup aH$ for some $a \in G \setminus H$.   Let $x\in G$. Then $x \in H$ or $x \in aH$.  Suppose $x \in H$. Then $xhx^{-1} \in H$ because $H$ is a group.  Suppose $x \in aH$. Then $ahH(ah)^{-1} = ahHh^{-1}a^{-1} = a(hHh^{-1})a^{-1}$ ***1  ***1: Can I say now that $x \in H$, based on the that $hHh^{-1} \in H$ and that $aa^{-1} = e$ ?  Thank you for the time you spent reading my doubts.
Book series with college students finding a way to open portals to other planets I got the storyline of this book series (I believe it was a series) told to me by a colleague at a former workplace but I can't remember the name of the author or of any of the books in the series. The first book is about a couple of college students who find a way to relatively easily open portals to pretty much wherever they want to go. One day, NASA(?) finally succeeds in sending astronauts to Mars(?). When they have just left the spaceship, one of them suddenly gets a tap on the shoulder, and when he/she turns around, these college students stand there in their homemade space suits, chuckling. I believe these students later also start a company that helps other organizations establish themselves on other planets, or perhaps even &quot;conquers&quot; and sells planets to other organizations. A later book in the series also describes a system that lets people upload their minds to the cloud. Some people choose to live solely in the cloud as virtual creatures, only deciding to &quot;descend&quot; into a &quot;flesh&quot; body when they need to perform some task that requires them to be in physical form. Some people also choose to live in physical form but as several entities, or clones, and sync their mind between the clones through this system, basically allowing them to, for example, run a one-man company with several &quot;instances&quot; of the same human who all share their knowledge and all know exactly what all other clones know in the same moment, effectively creating a super-efficient team of workers. File access time not updating in ubuntu 12.04 Does there exist a continuous bijection from $(0,1)$ to $[0,1]$? Of course the map should not be a proper map. Find the area where dog can roam A dog is tied to circular pillar by a rope. Radius of this pillar is $1m$ and length of rope is $\pi m$. What is an area where dog can roam?   I tried to find the area of all semicircles and then to find its sum. It is easy to find an area at front side of the pillar. It is $\displaystyle\frac12\pi^2\pi=\frac{\pi^3}{2}$. Problem is how to find remaining area. I tried to write this area using compass and straightedge, but I couldn't. Then I wrote this in AutoCAD and it looks like this: Is it possible to find the exact value of this area? \clipbox of a \scalebox not working properly in xelatex Add an "I don't understand" option for review audits I wrote a Table Valued Function in Microsoft SQL Server 2008 to take a comma delimited column in a database to spit out separate rows for each value.  Ex: "one,two,three,four" would return a new table with only one column containing the following values:  one two three four   Does this code look error prone to you guys? When I test it with   SELECT * FROM utvf_Split('one,two,three,four',',')    it just runs forever and never returns anything. This is getting really disheartening especially since there are no built in split functions on MSSQL server (WHY WHY WHY?!) and all the similar functions I've found on the web are absolute trash or simply irrelevant to what I'm trying to do.  Here is the function:  USE *myDBname* GO SET ANSI_NULLS ON GO SET QUOTED_IDENTIFIER ON GO ALTER FUNCTION [dbo].[utvf_SPlit] (@String VARCHAR(MAX), @delimiter CHAR)  RETURNS @SplitValues TABLE (     Asset_ID VARCHAR(MAX) NOT NULL )  AS BEGIN             DECLARE @FoundIndex INT             DECLARE @ReturnValue VARCHAR(MAX)              SET @FoundIndex = CHARINDEX(@delimiter, @String)              WHILE (@FoundIndex &lt;&gt; 0)             BEGIN                   DECLARE @NextFoundIndex INT                   SET @NextFoundIndex = CHARINDEX(@delimiter, @String, @FoundIndex+1)                   SET @ReturnValue = SUBSTRING(@String, @FoundIndex,@NextFoundIndex-@FoundIndex)                   SET @FoundIndex = CHARINDEX(@delimiter, @String)                   INSERT @SplitValues (Asset_ID) VALUES (@ReturnValue)             END              RETURN END 14 months ago, an  supplement was published that discussed how the ranger described in the Player's Handbook was under powered. These rules reimagined the ranger's abilities, and suggested that an official version would be released later.     Finally, we come to implementation. If this iteration of the ranger, or a future revision of it, grades high enough, our plan is to present it as a revised ranger in a future D&amp;D sourcebook.   Does that Sourcebook exist yet? If not, have the devs given an update or told us their intentions? How can I open a cmd window in a specific location? How can I open a cmd window in a specific location without having to navigate all the way to the directory I want? tcolorbox manual tip box, how to clone? The difference between GPU and CPU It's become clear that last night's (Tues Jan 8th 2019) patches to Windows 7 broke SMB2 networking in Lanman Server.  One or both of KB 4480970 and KB 4480960 results in any OS that uses SMB2 (such as Windows XP, some Windows 7, and same generation server OS' and Mac OS X machines (certainly 10.10, 10.11 not sure about others yet)) being unable to connect to Windows 7 Pro shares.  Client machines repeatedly prompt for login despite credentials being correct. Why is area of a surface of revolution integral $2\pi y~ds$? not '$dx$'? Complex polynomial of degree $n$ I use the following command to clear a directory, of files and directories over 30 days old, and move them to an archive directory which I can delete after a few weeks if nobody asks for their files back. The target directory has subdirectories by user name, so will the archive directory.  This is the command I use:  find /path/to/directory/username/ -mtime +30 -exec mv "{}" /path/to/archive/username/ \;   I suggested a modified version of this to answer a question on ask ubuntu, another user edited the code to change the end of line \; for + as it's faster(and more correct?).   However, using + in this way works if the -exec command is ls -lh but not in the actual command that I use. If I try it with + I get an error message:  find: missing argument to '-exec'   I don't understand why it's behaving this way, or what the correct command would be. Please don't just post a command correction, I'd like to understand rather than just follow a suggestion blindly. Move all files of type to a new folder in windows 7 How long does mana from Burning-Tree Emissary stay around? Does mana added to your pool from  get destroyed when the card that added it is destroyed? JavaScript RegExp objects I try to write a simple Markdown parser in JavaScript. Therefore I want to check for the [link content][link id] syntax. I use the following code:  data = data.replace( /\[(.*?)\][ ]*\[([0-9]+)\]/g, '&lt;a href="$2"&gt;$1&lt;/a&gt;' );   It works well, but now I want to do this with a RegExp object. So I set up the following bit of code:  var r = new RegExp( '\[(.*?)\][ ]*\[([0-9]+)\]', 'g' ); data = data.replace( r, '&lt;a href="$2"&gt;$1&lt;/a&gt;' );   But it doesn't work. It even says that my regular expression (which works since the first example does a good job) is invalid:     unmatched ) in regular expression   I think it must have to do with some RegExp-object peculiarities I am not aware of. What am I doing wrong and how can the problem be solved? "Images with street signs" CAPTCHA impossible to solve Nexus 7 2013 error upgrading to 5.0.2 OTA I updated to 5.0.1 via OTA sometime ago, and it worked fine. Now I got a notification that 5.0.2 is ready but when Android shuts down to install update it returns error (with no other message. My Nexus 7 is rooted but has the stock recovery.  Is there an easy fix that doesn't require factory reset or installing recovery?