$(G,*)$ is an abelian group and $a,b$ are elements of G. Let $n=ord(a)$ and $m=ord(b)$ in $G$. Show that the order of $a*b$ is equal to $nm$ if $gcd(n,m)=1$.  I have already proved that $ord(a*b)|nm$. But the other part is a problem.  Thanks
We want to show that if $a,b\in G$ where $G$ is a finite Abelian group, we have  $\operatorname{LCM}(|a|,|b|) = |ab|$ given that $ab \neq e$.   How I approached this question was by saying let $\operatorname{LCM}(|a|,|b|) = L$. Then if we show that $L$ divides $|ab|$ and $|ab|$ divides $L$ then $|ab| = L$. Showing that $|ab|$ divides $L$ was fine. But then I am having trouble with the second part that is showing $L$ divides $|ab|$. For this, so far I have:  Consider $(ab)^{|ab|} = e = a^{|ab|}b^{|ab|} $  since we know $a \neq b^{-1}$ by assumption then we can conclude that $|a|$ divides $|ab|$ and $|b|$ divides $|ab|$. We know that $L = \frac{|a||b|}{\operatorname{gcd}(|a|,|b|)}$ from here can we conclude that $L$ divides $|ab|$?
How do I find my follower if and when they leave me? I had parked my follower at an inn with a bunch of valuable items that I didn't want to carry. I guess I have been out a bit too long as this message appears:      Your follower tires of waiting and leaves.    There have been other situations where my follower has mysteriously disappeared as well.  How can I find my follower when they get up and leave or disappear into thin air? Can I take a sex-toy on plane through UAE? I will be traveling from Australia to Austria, transferring in Abu Dhabi. Is it alright for me to bring a vibrator in the check-in luggage, or are there laws against that?  It is approximately 17cm long. Are there any precautions to take? Elantech Touchpad not working on Ubuntu 16.04 and Arch Linux So I have and Asus R558UF notebook with an Elantech Touchpad. After installing Ubuntu 16.04, the touchpad worked once but after restarting, it failed to work and has not been working ever since.  This is output of xinput:   ↳ Elan Touchpad                            id=12   [slave  pointer  (2)]   In Mouse and Touchpad settings, the touchpad is enabled. What should I do to fix this? When a user on our site loses his password and heads off to the Lost Password page we need to give him a new temporary password. I don't really mind how random this is, or if it matches all the "needed" strong password rules, all I want to do is give them a password that they can change later.  The application is a Web application written in C#. so I was thinking of being mean and going for the easy route of using part of a Guid. i.e.  Guid.NewGuid().ToString("d").Substring(1,8)   Suggesstions? thoughts? I've connected a USB to Serial bridge dongle, and running dmesg | grep tty outputted the following:  [    0.000000] console [tty0] enabled [603199.380677] usb 2-2: cp210x converter now attached to ttyUSB0   So now I'm trying to write to it as per  by running the following:  cat hello.txt &gt; /dev/ttyUSB0   and   sudo cat hello.txt &gt; /dev/ttyUSB0   But both result in the following error:  bash: /dev/ttyUSB0: Permission denied   What am I doing wrong?  Thanks How to detect the Internet connection is offline in JavaScript? Controversy over verb choice in "neither you nor I {is/am/are} in control" Evaluating $\int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$ I need to evaluate $I = \int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$ where $a$ is some integer.   I get: $\dfrac{2a(a^2-7)\sin(\pi a)}{a^4 - 10a^2 + 9}$. However $\sin(\pi a)$ is $0$ for all $a$ so $I=0$. But as noted by an answer, there are answers for $a = 1,3,-1,-3$. Moderncv adding date of birth to personal information I'm doing my cv with the moderncv package and I'm using the classic style. I would like to add a date of birth line directly above the adress. Does anyone know how to do this? I've tried various solutions but they all seem to be broken. How to model effectively using exact measurements? I always struggle when trying to create a floor plan or furniture. How do I model using exact measurements. Why are home networks prefixed with 192.168? Why are home networks prefixed with 192.168 ?  Why those numbers?  For some historical reasons? How would I give a simple example of a function $f$ differentiable in a deleted neighborhood of $x_0$ such that $\lim_{x\to x_0}f^\prime(x)$ does not exist? I cannot seem to think of an example.  A delete neighborhood is an open interval about $x_0$ which does not contain $x_0$. So, $(x_0-\delta,x_0+\delta)-\{x_0\}$ for some $\delta&gt;0$.  How would something be differentiable in a deleted neighborhood if at the point of the derivative, the limit does not exist. Presumably, the derivative ends up looking something like $\lim_{x\to x_0} \dfrac{1}{x}$, if it does not exist. Are any of the alien languages in Star Wars real Earth languages? Are any of the alien languages (other than English and binary) in Star Wars movies or TV shows real Earth languages? Is there a possibility to change the font-style in the table of contents? Everything I wrote down in my document was written with Times. However, the ToC does not have the same font family. Is there a way to change the font of the ToC to Times New Roman? Thank you : )    \documentclass[a4paper, fontsize=11pt]{scrbook} \usepackage{geometry} \geometry{a4paper, top=25mm, left=25mm, right=25mm, bottom=20mm, headsep=10mm, footskip=12mm}  \usepackage[ngerman]{babel}  \usepackage[T1]{fontenc}        \usepackage[latin1]{inputenc} \usepackage{acronym}      \usepackage{anyfontsize}  \begin{document}  \tableofcontents  \chapter{Test} Is there a possibility to change the font-style in the table of contents? Everthing I wrote down in my document was written with Times. However, the ToC does not have the same font family. Is there a way to change the font of the ToC to Times New Roman?  \end{document} On the  blog post, nearly all the images are broken. I assume this is because they are all hosted on stackoverflow.blog/wp-content/uploads/ rather than zgab33vy595fw5zq-zippykid.netdna-ssl.com/wp-content/uploads/, and they got destroyed in the removal of Blog Overflow. I looked around at a few other blog posts, and this post seems to be the only one having an issue. Theorem numbering I am trying to number my theorems, corollaries, Lemmas, Propositions, Definitions, Remarks etc. I put the following code:  \theoremstyle{theorem}  \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[theorem] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition}  \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[definition]  \theoremstyle{remark}     \newtheorem{remark}{Remark}[section]   I want to produce the following:  theorem 1.1  proposition 1.2  definition 1.3  example 1.4  remark 1.5  lemma 1.6  but the code above gives me  theorem 1.1  definition 1.1  remark 1.1  lemma 1.2  Can anyone tell me if there is a code that numbers all these respectively? I just did a clean install of 20.04 and I noticed that no key shortcuts for files manipulation work when my focus in on Desktop (not in Nautilus).   When I have copied a file to clipboard, I cannot Ctrl+V it to Desktop, when I click on a file on Desktop, I cannot Ctrl+C it, or I cannot Delete it with keys. Only ways to do it is either navigate to the Desktop folder in Nautilus, or right-click the file on the Desktop and copy/move to trash/paste...  Sorry if I'm missing something obvious, all the key shortcuts worked normally for me in 18.04. Excluding empty(Null) fields when using EntityFieldQuery query condition Can you be tripped by an AoO while standing up? transpose columns to rows