Suppose that $G$ is a group in which $a^3 =1$ for all $a \in G$. Is $G$ abelian?   I tried to play around with algebraic expressions. I found e.g. $(ab)^2 = b^2 a^2$ for all $a,b$. I don't know what to do.  I also can't find a counterexample.  Thanks for help
I know that any group satisfying $x^2=1$ for all $x$ is abelian. Is the same true if $x^3=1$? I don't think it is, but I can't find a basic counterexample.
I have a quick question about updating my references from Bibtex.  I am using TexStudio as my Latex editor.  I have to make changes to some of the references.  I put the changes in the "references.bib" file.  I then compile the Bibtex and the main document I'm working on.  Sometimes the changes show up, and sometimes they don't, and I haven't figured out why. I have even gone as far as trying to delete the references by changing the filename, but the original references still appear.  Here's what I put in my main document:   \newpage %Puts in References  \bibliographystyle{unsrt}  \bibliography{references}   My references.bib file shows all the changes I have made.  Any suggestions are appreciated! MVC Model require true Showing 2 Distributions are the Same Let $X_1, X_2, \dots$ be i.i.d. exponentially distributed RVs. For $n = 1,2,\dots$ consider: $Y_n := \max(X_1, X_2, \dots, X_n)$ $U_n := \sum_{i=1}^{n}\frac{X_i}{i}$ Show that $Y_n$ and $U_n$ have the same distribution   What I've tried: $P(Y_n&lt;y)= P(X_i&lt;y)^n = (1- e^{\lambda y})^n =&gt; P(Y_n=y) = n(1- e^{\lambda y})^{n-1}$ But I get stuck with $U_n$. I tried an MGF, $U_n$ evaluates nicely but then $Y_n$ gets messy. Any thoughts? I zoomed in / out of one of my UI panels and I can't figure out how to zoom back out/ in to get it normal size.   How can I reset the zoom level of UI panels? Are there any more Asgardians? I've used following codes and no luck  1  //Hide admin footer from admin function change_footer_admin () {     return ' '; } add_filter('admin_footer_text', 'change_footer_admin', 9999);  function change_footer_version() {     return ' '; } add_filter( 'update_footer', 'change_footer_version', 9999);   2  function wpbeginner_remove_version() { return ''; } add_filter('the_generator', 'wpbeginner_remove_version');   3  function my_footer_shh() {     remove_filter( 'update_footer', 'core_update_footer' );  } add_action( 'admin_menu', 'my_footer_shh' );   Still I see those texts and version on my footer.   So what is the accurate way to remove these? I've added these code snippets into my child theme function.php Why disable Fast Boot on Windows 8 when having dual booting? Doom-style 2.5D Movement Animations I've started working on a demo for my 2.5D game. For a basic scene I figured I would just use the good old fashioned doom sprites textured to a double sided plane. Simple enough. But I've been scratching my head as to how to display the proper sprite animation relative to the player viewing the sprite...if that makes since.  For example suppose I have three players in the scene Players A, B, and C as such:    Player C should see player A as: Here's an example of what I think player A should look like from C's perspective:    While player B would see player A as follows:    My current idea is that for any given player, I will need to know the current direction of each other player, and where they are located/moving to relativly. From this, I could calculate which sprite image is required.  Is this correct? What improvements/optimizations could be made?  ** EDIT ** Total number of directions will be 8 as depicted below:    For each direction there are two sprites (to show walking movement, sprites for the right hand directions are mirrors of the left):    ** EDIT 2 ** Got this working and have a simple javascript/threejs example for anyone interested: solution :-  $21= 3 \times 7$  there is only one Sylow $3$ and Sylow $7$ subgroup  so, Sylow $3$ and Sylow $7$ subgroup are normal in group $G$  so $G$ is cyclic group of order $21$.  Am I right ?  somebody told me that group of order $21$ is not cyclic.  he gave me this .  If group of order 21 is not cyclic, then can we understand it by Sylow method ? I've tried to install Ubuntu on a VMWare Virtual Machine, but it never gets past the  bootloader :  ISOLINUX 4.01 debian-20100714 ETCD Copyright (C) 1994-2010 H. Peter Anvin et al No DEFAULT or UI configuration directive found! boot: _   I've also tried to burn this .iso image to a CD and go from therem but I got the same problem.  Some more details :   VMWare Server 2.0.2 Build 203138 on Windows 2008 R2 ubuntu-10.10-desktop-i386.iso image used (downloaded yesterday evening) Is there a way to recover files from a storage device partially overwritten with dd? I'm an idiot, I was writing an image to a USB drive using dd when instead of typing /dev/sdb I typed /dev/sdd and wrote the 135mb image to my 750gb External Hardrive full of files. Is there ANY possibility of recovering the files or did dd completely zero out the drive? Please help, as this is one of the lowest points of my life EVER.  dd finished writing the 135mb image in 5 seconds, so I didn't have a chance to stop it. By the time I knew what happened to was too late. The command was as follows:   sudo dd if=generic.img of=/dev/sdd bs=1M   EDIT2: The External Harddrive's filesystem was FAT32(I've had the drive for awhile and FAT32 was the most Universal FS at the time). I also have a 2TB drive I just got which I was planning on moving most of the files to from the 750gb, If needed I can use it to transfer the Recovered files to. Extruding surface while keeping the shape here is the picture of what I would like to get(example nr. 2). I need the edge frame to be same in width and length. Also the extrusion is on surface (without depth) I tried extruding region, but it did not give me the result I wanted.   I know I could just adjust the scale and do it manually, but that would be eyeballing without achievign accuracy. I would like to know if it is possible to extrude or inset while keeping the shape (even edge space) Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ if $m &lt; 0$. This statement came up in an algebraic geometry text with no explanation provided, and I'm trying to understand why it's true.   Thoughts so far:   Since $H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m))$ is the vector space of regular sections of $\mathcal{O}_{\mathbb{P}^n}(m)$ over $\mathbb{P}^n$, we can try to find a basis. Since the regular sections are the degree $m$ polynomials in $x_0, \dots, x_n$, it seems like the collection of degree $m$ monomials in $x_0, \dots, x_n$ should work. But I don't see how there are ${n + m \choose n}$ of these. Is it just a basic counting argument that I'm struggling with?  On a side note, I think I understand what global sections of $\mathcal{O}_{\mathbb{P}^n}(m)$ look like, but how can we describe the regular sections over a distinguished open set $D(f)$? Is it something like degree $m$ polynomials with denominator $f$? Cron jobs in Drupal 7 I have a PHP file that does some task in consistent intervals.   I have placed that PHP file inside a custom module folder, but I found that Drupal functions like drupal_set_message() are not available inside the PHP file.   How do I include Drupal functions inside this PHP file? How do I set up this php file to be run as a cron job? Now that I have Firefox 49 or higher, how do I watch Netflix? Consider a partial function $f$ that is defined only for a few values of its domain (my exact use case is $\delta$ transition functions in automata). One can 'complete' it by saying $$g(x)=0\iff f(x) \text{ is not defined.}$$  Is there a symbol to mean "undefined"? Would it be correct, or accurate, to write $\nexists f(x)$? I'm trying to set an RSA key as an environment variable which, as a text file, contains newline characters.  Whenever I attempt to read from the file and pass it into an environment variable, it will just stop at the newline character on the first line. How can I prevent this? Prove that any two nontrivial subgroups of $\mathbb{Q}$ have nontrivial intersection I need to prove that any two nontrivial subgroups of $\mathbb{Q}$ have a nontrivial intersection as part of a larger proof that $\mathbb{Q}$ cannot be represented as a nontrivial direct product.    (Yes, I realize that there are other ways to prove that $\mathbb{Q}$ cannot be represented as a nontrivial direct product, but I am not interested in those, and will not accept answers showing me how to do that).  Based on the density of the rationals, I guess this idea kind of makes sense, but I am doing this proof for an abstract algebra class, and I don't think real analysis proofs would be accepted.   Everywhere I've looked online has said this is "trivial" or "easy to see" without bothering to actually prove it for those of us mere mortals who don't see it as such. As for myself, I have literally no idea where to even get started proving this. Could somebody help me figure this out (without giving me cutesy examples that aren't really all that related with the expectation that I should be able to figure it out from looking at those - that's not really how my brain works)?  Thank you. Create a .desktop file that opens and execute a command in a terminal What to do before root and flash my phone?