Let $H$ a subgroup of $G$ and the index of $G\backslash H$ is $2$. Prove that   $H$ is a normal subgroup of $G$.   I have already know that $1_G *H $ is an element of $G\backslash H$. I miss the little flash to continue the problem. Will someone be able to give me advice?
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.
"Nikki's and Alice's X" vs. "Nikki and Alice's X" If I did  \setcounter{chapter}{-1} \chapter{Introduction}   then I get Chapter 0 Introduction. I just want the word Introduction, and I want it also in the contents when I use   \tableofcontents   How can I do that? Beam size and material for 15 feet long swing I need to create a 15 feet span beam for kid swing (300 lbs load / 150 lbs and 2 swings assumption). One end will bolt to the tree (just for convenience).  What "treated lumber" lumber size and material I should use? Or is it better / cheaper to use metal pipe (from the plumbing department)?  I guess 4x8x16 or two (2) 2x8x16 would do it according to this website . But I do like a clean look of metal. My wife would concern about big piece of lumber running on top of the kid (that looks heavy visually). Start Mathematica without the menu bar? XML attribute vs XML element At work we are being asked to create XML files to pass data to another offline application that will then create a second XML file to pass back in order to update some of our data. During the process we have been discussing with the team of the other application about the structure of the XML file.   The sample I came up with is essentially something like:  &lt;INVENTORY&gt;    &lt;ITEM serialNumber="something" location="something" barcode="something"&gt;       &lt;TYPE modelNumber="something" vendor="something"/&gt;     &lt;/ITEM&gt; &lt;/INVENTORY&gt;   The other team said that this was not industry standard and that attributes should only be used for meta data. They suggested:  &lt;INVENTORY&gt;    &lt;ITEM&gt;       &lt;SERIALNUMBER&gt;something&lt;/SERIALNUMBER&gt;       &lt;LOCATION&gt;something&lt;/LOCATION&gt;       &lt;BARCODE&gt;something&lt;/BARCODE&gt;       &lt;TYPE&gt;          &lt;MODELNUMBER&gt;something&lt;/MODELNUMBER&gt;          &lt;VENDOR&gt;something&lt;/VENDOR&gt;       &lt;/TYPE&gt;    &lt;/ITEM&gt; &lt;/INVENTORY&gt;   The reason I suggested the first is that the size of the file created is much smaller. There will be roughly 80000 items that will be in the file during transfer. Their suggestion in reality turns out to be three times larger than the one I suggested. I searched for the mysterious "Industry Standard" that was mentioned, but the closest I could find was that XML attributes should only be used for meta data, but said the debate was about what was actually meta data.  After the long winded explanation (sorry) how do you determine what is meta data, and when designing the structure of an XML document how should you decide when to use an attribute or an element? Mac Mini (Mid-2011), 16GB SSD, 16GB RAM, El Capitan (v10.11.6)  The Mac ran fine until I upgraded to Sierra. The latest Mac OS conflicted with most of the specialized apps I use every day, So I immediately returned to El Capitan.  Activity monitor shows that two "Installer/Installer.app" items are always running and are they major users of all of my available memory. Consequently, all my others apps become unresponsive.  I run Disk Utility, and the Installer.app show up there as .dmg files; but I cannot erase, delete, or unmount them. None of my 3rd party app can remove the installers.  My Mac Mini, which should be running super-fast (as it used to), now runs dreadfully and disgustingly slow. I a truly upset with Apple for this kind of garbage. To get their help: $$$. What happens to waves when they hit smaller apertures than their wavelenghts? I was wondering this for quite a long time now. Let's say you have a water wave (like ripples, not the ones you see during tsunamis) with wavelength 10 m. Imagine you put a boundary with an opening of 1 m. Will diffraction be observed? According to my research, no. But, then, what would be seen? What LaTeX markup would one have to write to produce a result similar to the image below?    Please excuse my Paint-and-mouse drawing skills. :-) Poincaré's theorem about groups Links being changed to cppreference.com Someone just edited a link in one of my answers to point to a 'better reference'.    I've no problem with links being changed in general, but this smells like part of a concerted attempt to redirect traffic.   Is this kind of activity acceptable on SO? Find which date is missing in the results What is the difference between a field and a property? What's the difference between "established" and "founded" In this sentence: "CARE was ...... in 1945 to help people after World War II."    A. founded  B. built C. established D. constructed    The answer is C. But why don't we use "founded" in this case, I think they have the same meaning. Can you help me to give a reason for that? How do I match any character across multiple lines in a regular expression? For example, this regex  (.*)&lt;FooBar&gt;   will match:  abcde&lt;FooBar&gt;   But how do I get it to match across multiple lines?  abcde fghij&lt;FooBar&gt; How do I set up my LaTeX document to enable me to cite from my BibTeX bibliography? This may be an overly simplistic question but please bear with me...  I have a BibTeX bibliography containing all of my references, set up using BibDesk. I am now trying to cite these references in my LaTeX documents. Following the advice on the LaTeX wikibooks page, I have included the following lines into my LaTeX document...  \bibliographystyle{plain} \bibliography{/Users/Graeme/Documents/References/BIBliography.bib}   ... where BIBliography is my .bib file, created in BibDesk.  Unfortunately, when I cite something in the document, using \cite{citekey} and typeset the document, the log tells me that there were undefined references. In the .pdf that is produced, there is a [?] where the references should be.  Am I doing something wrong? Do I need to store my .bib for in a particular location? I know there can be some errors that mean you need to typeset the document several times, but I have tried countless times to no avail.  I would be very grateful if someone could help out. Problems updating Python from the repositories today (Feb 25, 2021) NOTE: This is a Master Canonical Post for the issues observed on February 25, 2020.  All issues on Feb 25 relating to Python updates not found should be dupe-linked here.  Errors have been seen such as this today, February 25, 2021: Failed to fetch http://security.ubuntu.com/ubuntu/pool/main/p/python2.7/libpython2.7-minimal_2.7.17-1~18.04ubuntu1.3_amd64.deb 404 Not Found  Is there something up with the repositories? Is it possible to produce a context free grammar for Tikz path? node? Even further, I would be happy to see a definition of the tikz language in the usual manner programming languages such as Java and Pascal are defined.   The reason I ask is that I keep getting mysterious error messages for the tiniest typos. Essentially, incorrect use of the underlying languages leads to a "crash" of the compiler. This crash sometimes gives useful error messages, but sometimes they are hard to decipher. What's a (big) word or phrase to describe a book that is so good that I cannot put down when I start to read it? Neatest proof that set of finite subsets is countable? Construction of a Borel set with positive but not full measure in each interval