Prove that $\mathbb{Z}_m\times\mathbb{Z}_n$ is isomorphic to  $\mathbb{Z}_{mn}$ whenever $m$ and $n$ are coprime.   $\mathbb{Z}_m$ represents a cyclic group of order $m$.  I have no knowledge of rings and fields. I only know group theory.  In general, I would like to know how to prove that $2$ groups are isomorphic to each other? Is the only way to solve it is to show a bijective homomorphism?
Let $m,n$ be positive integers bigger than $1$. Show that $\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$.    I have no idea on how to start. Anyone hints are much helpful.
Extract Raster Value into Polygon Attribute I have Raster Data for a State which has Elevation Data in its pixels. I also have Vector layer which has the 300 divisions of State. what I need is, I want to get the average elevation value of particular division from the raster data using ArcGIS. I tried extract by mask in ArcGIS to extract individual polygon to clip the raster into 300 raster then from individual raster i have collected average elevation but it has taken days together to complete the work. Is there any other easy way to get it? I have a device on which I would like to install Ubuntu. How do I find out if Ubuntu will work on it? Why do Ruby setters need "self." qualification within the class? Is it possible to change the line style in a tikz figure in the same \draw command?  I am looking for something like:  \documentclass[tikz]{standalone} \begin{document}     \begin{tikzpicture}         \draw[style1] (0,0) -- ++(1,0)  **CHANGE STYLE** -- ++(1,0);     \end{tikzpicture} \end{document}   The main motivation is that I don't want to loose the relative coordinate as I go on with a long \draw command. Is there a way to monitor an SSD state and predict its failure in case of active read-write usage? As far as I know, the only con against switching to an SSD except its price is the tendency to fail soon (relatively to conventional HDDs) in case used in active read-write operation. Is there an efficient way to predict the failure? What are the general rules regarding when can you refreeze food that was frozen and then thawed in the fridge? and what are the reasons behind them? (Hygiene? Food quality?). Let's assume for the sake of the argument that the items were in a non-frozen state only a few hours.  To be more specific, I'm interested in the application of these rules for  - Ginger garlic paste - Dough - Indian Curries - Green Chilly paste - Frozen green peas How do I install drivers for a Canon LBP-2900 printer? I seen many times how people describe how to's about LBP-2900, and 10-12 steps from start to end of how to download, unpack, build driver, fix driver, make it work... Could be there some .deb package with ready-to-go driver for Ubuntu? This printer is very popular... How can I pad a String in Java? Is there some easy way to pad Strings in Java?  Seems like something that should be in some StringUtil-like API, but I can't find anything that does this. How to manage and restore a modern iPhone or iDevice on ubuntu with iTunes This question—how to install iTunes on ubuntu—has been asked many times before on this site, but the answers are all outdated and involve Wine or some other tool that no longer works for the most recent version of iTunes. Put simply, I need to regularly back up and occasionally restore the firmware on an iPhone 6.   I cannot:   dual boot Windows—for some reason installing one OS makes the other unbootable &amp; after days of headache &amp; dozens of reinstalls I’ve decided to just leave Ubuntu be &amp; not try to preserve Windows. Plus “just use Windows” is not really a solution on an Ubuntu forum. use libimobiledevice-dev. libimobiledevice-dev doesn’t let you manage iOS updates or sync music anymore. Buy another phone. This one should be obvious but for some reason some asshat always comments “too bad lmao shouldntve bought an iphone ;P”. How do you iterate backwards through an STL list? What is the closest general-relativistic equivalent of a "time slice"? In a newtonian universe, one can talk of a "time slice", that is, the state of the universe at a given point in (global) time. In a "typical" classical universe, a time slice would contain enough information to fully compute the state of the universe at any other point in time, backwards or forwards.  So, disregarding what we know of quantum effects and talking purely of a general-relativistic universe, which concept is closest to a time slice? How to evaluate potential advisers on grounds other than their research/publications? Notify us when one of our posts is edited Your summary page shows "N Edits to your posts" for the selected time period but you really have to be looking for it.  Could we add edits to the notification feed? You could put the reason for the edit (entered--or not--when the editor changes it) and just a link to the post. I am using MiKTeX 2.9 on Windows 7.   I'm getting the error: "can't write on pdf file."  The pdf doesn't reflect what I wrote.  What did I do wrong?  My input:  \documentclass[12pt]{article} \usepackage{epsf} \usepackage{amssymb} \usepackage{graphicx} \usepackage{setspace} \usepackage{amsmath} \usepackage{rotate} \usepackage{endnotes} %\usepackage{harvard} \usepackage{vmargin} \usepackage[nolists,noheads,tablesfirst,nomarkers]{endfloat} \usepackage{natbib} \bibliographystyle{apsr}  \oddsidemargin=1.0in\evensidemargin=0.in \textheight=8.5in\textwidth=6.6 in \baselineskip=18pt \parskip=6pt  \begin{document}  % added, was missing  TEXT  \newpage \baselineskip=12pt \doublespacing  \bibliographystyle{apsa} \bibliography{Dissertation_Proposal}   \end{document} pgfplots fillbetween with multiple curves I have a TikZ picture like this:  \begin{tikzpicture}   \begin{axis}[     axis lines=middle,     xmin=-0.5,xmax=3.5,ymin=-0.5,ymax=3.5   ]   \addplot[very thick,blue, samples=300, domain=0:3, name path=A] {sqrt(x)};    \addplot[very thick,yellow!70!red, samples=300, domain=0:3, name path=B] {9-3*x};    \addplot[very thick,red!90!teal, samples=300, domain=0:3, name path=C] {3};    \addplot[very thick,green!70!magenta, samples=300, domain=0:3, name path=D] coordinates{(1/9,-6) (1/9,6)};   \end{axis} \end{tikzpicture}     I’d like to fill the area enclosed by these four curves.  Similar questions have been answered using pgfplots’s fillbetween package, but it only seems to support filling between two curves. What are my options? How do you make a script or app to run or stop running automatically when the system is booted, a user logs on, at a specific time, when a disk is attached, etc.? In C#/VB.NET/.NET, which loop runs faster, for or foreach?  Ever since I read that a for loop works faster than a foreach loop a  I assumed it stood true for all collections, generic collections, all arrays, etc.  I scoured Google and found a few articles, but most of them are inconclusive (read comments on the articles) and open ended.  What would be ideal is to have each scenario listed and the best solution for the same.  For example (just an example of how it should be):   for iterating an array of 1000+ strings - for is better than foreach for iterating over IList (non generic) strings - foreach is better than for   A few references found on the web for the same:               [Edit]  Apart from the readability aspect of it, I am really interested in facts and figures. There are applications where the last mile of performance optimization squeezed do matter. Is the differentiation operator an open mapping? Consider the vector spaces $C([0, 1])$ and $C^1([0, 1])$ with norm, \begin{align*} \displaystyle \|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|, \end{align*} and let $T:C^1([0, 1])\to C([0, 1])$ the operator given by, \begin{align*} \displaystyle Tf=f'. \end{align*} Is $T$ defined this way an open mapping? If it's, how to prove it? Why is 3D stress tensor acting only on three surfaces? I'm trying to learn about the stress tensor (in 3D)    The tensors are said to have directions (the first subindex $i$ in $\sigma_{ij}$) and specify the surface upon which they act (the second subindex $j$ in $\sigma_{ij}$).  What confuses me is why is it defined only to act on three surfaces, even when the cube has six surfaces? When Frodo put on the One Ring, both the Nazgul and Sauron himself found Frodo. However when Bilbo had the ring and possibly used it many times, Sauron sought Gollum to get a hint on where to look for the ring and send the Nazgul after it.   Sauron sought Gollum before Bilbo wore the ring at his birthday party, slightly after that time, he was very effective at locating Frodo when Frodo put on the ring. So I guess it's not a matter of Sauron gaining any extra powers with time.  Also Bilbo seemed to use the ring without it bothering him at any point, unlike Frodo who seemed to be very troubled when using the ring. How is that possible?