Similar to how one can find a primitive root element of $\mathbb{Z}/p\mathbb{Z}$, I was wondering if, for any group of size $N$ that can be written as $(k\backslash\{0\},\cdot)$ where $k$ is a field, one can find an element of the group such that its order is $N$.   The sketch of my attempted reasoning is as follows:   Lemma 1: Legendre's Theorem says the order of every element of a group divides the size of the group, $N$.   Lemma 2: The order of every element divides the largest order of an element ().   Lemma 3: The maximum number of solutions to $x^n=1$ in a group given by $(k\backslash\{0\},\cdot)$, where $k$ is a field, is $n$ (, ).   From Lemma 2 and 3, if the largest order is $n$, then the maximum number of solutions is $n$, but $N$ elements must solve it, so $n\geq N$. From Lemma 1, $n|N$, so $n=N$.   Does this mean that every group that can be defined as $(k\backslash\{0\},\cdot)$, where $k$ is a field, is cyclic??? Or am I missing something? Is there an existing proof of this that has a name?
The book "A First Course in Abstract Algebra" by Fraleigh says      If $G$ is a finite subgroup of the multiplicative group $\langle F^*,\cdot\rangle$ of a field $F$, then $G$ is cyclic. In particular, the multiplicative group of all nonzero elements of a finite field is cyclic.    I wonder why that is. Take $a,b\in F$ such that $a^2=b^2=1$. Then $\{1,a,b,ab\}$ is a non-cyclic subgroup of $F$. Where am I going wrong?  Thanks in advance!
Where can I find 32-bit version of Ubuntu? How to scan an entire network using Nmap? Is there a way to scan an entire network using ?   What I want to do is scan my network for all the devices that are currently connected to it. Is there any noun in English which changes the first letter in the plural? Plenty of nouns change the second letter to become plural (man->men, goose->geese) but does anything change its first letter.  I've hunted high and low over the internet, and spent ages browsing the questions at  but I can't find anything. Is there any way to kill a Thread? Is it possible to terminate a running thread without setting/checking any flags/semaphores/etc.? How to evaluate environment variables into a string in Python? I have a string representing a path.  Because this application is used on Windows, OSX and Linux, we've defined environment variables to properly map volumes from the different file systems.  The result is:  "$C/test/testing"   What I want to do is evaluate the environment variables in the string so that they're replaced by their respective volume names.  Is there a specific command I'm missing, or do I have to take os.environ.keys() and manually replace the strings? Biometrika journal style file not getting along with TeXLive-2020 I cannot compile styleguide.tex in biometrika.zip (see ). Here is a MWE which leads to the same error message, though not sure yet that I've not tossed baby out with bath water. \documentclass{article} \RequirePackage[thmmarks]{ntheorem}  \makeatletter \def\arabic#1{{\rm\expandafter\@arabic\csname c@#1\endcsname}} \theoremnumbering{arabic} \newtheorem{definition}{Definition} \makeatother  \begin{document}  \begin{definition} This is a definition. \end{definition}  \end{document}  Here is the output of pdflatex: % pdflatex styleguide This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020) (preloaded format=pdflatex)  restricted \write18 enabled. entering extended mode (./styleguide.tex LaTeX2e &lt;2020-02-02&gt; patch level 5 L3 programming layer &lt;2020-03-06&gt; (/usr/local/texlive/2020/texmf-dist/tex/latex/base/article.cls Document Class: article 2019/12/20 v1.4l Standard LaTeX document class (/usr/local/texlive/2020/texmf-dist/tex/latex/base/size10.clo)) (./ntheorem.sty  Style `ntheorem', Version 1.24 &lt;2004/09/20&gt; ) (/usr/local/texlive/2020/texmf-dist/tex/latex/l3backend/l3backend-pdfmode.def ) (./styleguide.aux) ! Use of \@item doesn't match its definition. \update@series@target@value #1-&gt;\def \reserved@a {                                                   #1}\ifx \target@meta@famil... l.13 T       his is a definition. ? q OK, entering \batchmode How to navigate to C drive in bash on WSL-Ubuntu? sed one-liner to delete everything between a pair of brackets? I am working with some text that is full of stuff between brackets [] that I don't want. Since I can delete the brackets myself, I don't need the one-liner to do that for me, but I do need a one-liner that will remove everything between them.  What about parentheses () instead of brackets? Solving $\sqrt{x+5} = x - 1$ Show that every large integer has a large prime-power factor  That is, if $P(n)$ designates the largest number $p^a$ which divides $n$, then $\lim_{n\to\infty}P(n)=\infty.$ I'm using this code to get standard output from an external program:  &gt;&gt;&gt; from subprocess import * &gt;&gt;&gt; command_stdout = Popen(['ls', '-l'], stdout=PIPE).communicate()[0]   The communicate() method returns an array of bytes:  &gt;&gt;&gt; command_stdout b'total 0\n-rw-rw-r-- 1 thomas thomas 0 Mar  3 07:03 file1\n-rw-rw-r-- 1 thomas thomas 0 Mar  3 07:03 file2\n'   However, I'd like to work with the output as a normal Python string. So that I could print it like this:  &gt;&gt;&gt; print(command_stdout) -rw-rw-r-- 1 thomas thomas 0 Mar  3 07:03 file1 -rw-rw-r-- 1 thomas thomas 0 Mar  3 07:03 file2   I thought that's what the  method is for, but when I tried it, I got the same byte array again:  &gt;&gt;&gt; binascii.b2a_qp(command_stdout) b'total 0\n-rw-rw-r-- 1 thomas thomas 0 Mar  3 07:03 file1\n-rw-rw-r-- 1 thomas thomas 0 Mar  3 07:03 file2\n'   How do I convert the bytes value back to string? I mean, using the "batteries" instead of doing it manually. And I'd like it to be OK with Python&nbsp;3. I have the following code:  \documentclass[a4paper,12pt]{article} \usepackage[margin=2cm]{geometry} %showframe \begin{document} \tableofcontents \newpage \section*{Abstract} \section{Introduction} \newpage \bibliographystyle{apacite} \bibliography{references} \end{document}   And when I create the table of contents it adds the "References" in the TOC, how can I remove this to not be included in TOC? So far I have tried something like   \bibliography*{references} This inequality $a+b^2+c^3+d^4\ge \frac{1}{a}+\frac{1}{b^2}+\frac{1}{c^3}+\frac{1}{d^4}$ let $0&lt;a\le b\le c\le d$, and such $abcd=1$,show that   $$a+b^2+c^3+d^4\ge \dfrac{1}{a}+\dfrac{1}{b^2}+\dfrac{1}{c^3}+\dfrac{1}{d^4}$$   it seems harder than Is there a goto or any equivalent in Python to be able to jump to a specific line of code? Example of a finitely generated module with submodules that are not finitely generated Regulatory bodies and authoritative dictionaries for English How to shrink Windows 7 boot partition with unmovable files This question has bugged me for a while. I want to write a nested function, f(g(x)). If I naively enter it:  \begin{align}     f(g(x))=y \end{align}     The problem is that each and every parenthesis is the same size as all of the others. Shouldn't the outer parentheses be larger? But if I do this:  \begin{align}     f\left(g(x)\right)=y \end{align}     Now it looks even less like the functions are nested. What's the proper way to do this? I have heard that one ought to turn the pedals at 90rpm. However, I do so much slower, especially while standing, and would like to understand the argument behind this number (or any other number). So:  What is the optimal cadence of a cyclist. Why? Does it depend on the terrain? Does it depend if the rider is sitting or standing upright? What are some possible consequences of deviating too much from the proposed number?  Possible duplicate of , but I didn't find there the "whys" except for "it will grind your knees." However, I find riding upright with slow cadance no more demanding than walking up a hill with a backpack. How to add CC to a webform email? I'm using Drupal 7.  Is it possible to add a CC to a webform email?  I was using   function theme_webform_mail_headers($variables) {   $headers = array(     'X-Mailer' =&gt; 'Drupal Webform (PHP/' . phpversion() . ')',   );   return $headers; }   changing it with      function MYTHEME_webform_mail_headers($variables) {       $headers = array( 'CC' =&gt; 'ccmeail@domain.com', 'X-Mailer' =&gt; 'Drupal Webform (PHP/'. phpversion() .')',  );       return $headers;     }   It does not work, I've cleared all the chaches and it does not add the CC. How could I add the CC?  Thanks in advance.