Take an Abelian group G, and let $g,h \in G$ with $o(g) = x$ and $o(h) = y$, prove that if $x,y$ are coprime, then $o(gh) = xy$  (o(g) denotes the smallest integer x s.t. $g^x = e$)  My (bad) attempt:  since $o(g) = x$ then $o(g) | x$, similar $o(h) | y$  Now let $o(gh) = k$ for some integer k,  then since $x,y$ are coprime $o(g) | o(h)o(gh)$ implies $o(g) | o(gh)$, similarly $o(h) | o(gh)$ i.e. $k = z_1 x$ and $k = z_2 y$, so the only way for this to be possible is if $z_1 = y$ and $z_2 = x$ and we're done  is this the right approach? Could someone correct me somewhere? Also, why does G have to be Abelian?
Prove if $\operatorname{ord}(a)=m$, $\operatorname{ord}(b)=n$, and $\operatorname{gcd}(m,n)=1$, then $\operatorname{ord}(ab)=mn$.   I was reading this and was thinking how this proof would look like. I tried to do it and am not sure if this is correct. Here is what I did:  If $a$ and $b$ commute then $(ab)^{mn} = a^{mn} * b^{mn} = (a^m)^n * (b^n)^m = 1 * 1 = 1.$ So $ord(ab) | mn$.  Now, take $k = \operatorname{ord}(ab) = m^\prime * n^\prime$ where m' is relatively prime with $n$ and $n'$ is relatively prime with $m$. By the result above $m' | m$ and $n' | n$. Now we have $((ab)^k)^{m/m'} = 1$ since $(ab)^k=1$. But on the other hand, by the commutativity:  $$((ab)^k)^(m/m') =$$  $$((ab)^(m' * n'))^(m/m') =$$  $$ a^{n'm} * b^{n'm} = $$  $$(a^m)^{n'} * b^{n'm'}=$$  $$b^{n'm'} = 1$$  This implies that $n' * m'$ is divisible by $n$. But $m'$ is relatively prime with $n$, so we must have $n' = n.$ By symmetry, $m' = m$. So $ord(ab) = mn$.  Just to say this again, I want to prove if $a$ and $b$ commute and $m$ and $n$ are relatively prime then   $$ord(ab)=mn.$$   The comments are vague. I guess this must be the answer then and there probably is not another way to do this.
I was going to ask , but I came to the conclusion that I'm not going to do it (asking and building).  Now my question is: why is it (usually, if we leave the silly high-end prices alone) more expensive to build a bike from scratch? How do I replace the *first instance* of a string in .NET? I want to replace the first occurrence in a given string.   How can I accomplish this in .NET? Problem printing LaTeX from lua in lualatex I am having difficulty printing LaTeX from a lua function in a LaTeX document processed with lualatex (version 0.70.2). I have a sample that works (if I remove all newlines from the code) and a case that fails. I don't know why these get different results. Here is the sample that fails:  \documentclass{report} \usepackage{luacode} \begin{document}  \begin{luacode*} tex.sprint([[ \begin{tabular}{ | l | l | p{4.in} | }  \hline  {\bf File} &amp; {\bf Cards} &amp; {\bf Summary} \\ \hline  \end{tabular} ]]) \end{luacode*}   \end{document}   I get an error saying:  ! Misplaced \noalign. \hline -&gt;\noalign               {\ifnum 0=`}\fi \hrule \@height \arrayrulewidth \futurelet... l.22 \end{luacode*}   When I put all of the LaTeX code printed from lua on one line the code works fine:  \documentclass{report} \usepackage{luacode} \begin{document}  \begin{luacode*} tex.sprint([[ \begin{tabular}{ | l | l | p{4.in} | } \hline {\bf File} &amp;{\bf Cards} &amp; {\bf Summary} \\ \hline \end{tabular} ]]) \end{luacode*}   \end{document}   So I have a few question.  First, why does the first example shown above fail? It seems like something I am not aware of is being inserted into the generated LaTeX.  Also is there a way for me to see the actual LaTeX that is generated and processed so that I can debug issues like this better?  Is there a way to tell the lualatex to log the actual LaTeX it tries to process?  Any other suggestions for how to debug a problem like this? Restrictive vs Non-restrictive Relative Clause Can you tell me the difference in the meaning of the two sentences below?  As a defining relative clause.     The location which was called Central Park was a park in New York.    As a non-defining relative clause.     The location, which was called Central Park, was a park in New York. Why is there no 32 bit ISO for the new Ubuntu 18.04? Why is there no 32 bit ISO for the new Ubuntu 18.04? There are some .exe and .dll files belonging to some programs on my laptop which communicate with the internet without my permission (for self-update generally). I want to prevent those files one by one from accessing the internet. I don't want to install any external firewalls. I'm using Windows 7 Home Pro and if this can be done by Windows Firewall, it's OK. I took a look at Windows Firewall but couldn't find a setting to disable file access. Comments are useful. Many, many posts need some small amount of meta discussion, and comments fulfill this need.   Comments are also noisy and distracting. Occasionally, someone will post an insightful and informative comment that complements the post its attached to... But more often, they're tangential, quickly irrelevant and in the way: stale comments show up first, making newer comments less visible.  The guidance here for a long time has been that comments are transient, . And yet, some do hang around for months or years because .   So let's fix that.   I propose that all comments older than 7 days be hidden by default unless the comment has been up-voted in the past 30 days. Hidden comments should be indicated by a note below the post ("add / show 3 more comments") and made visible to anyone who cares to click it.   This doesn't destroy anything, but it would get it out of the way of most readers - and hopefully encourage folks to edit it into the actual answers as a result.  Note: I previously suggested this . Other  have been . However, I'm not suggesting that comments be automatically deleted, and they should be hidden based purely on the age / activity of the comment itself, not of the post to which they're attached. Yesterday I was copying a single, 8 GB file to a USB with a slow, 7 MB/s write speed, while my RAM is 3 GB. While copying the system froze, to the point where I couldn't even move the cursor.  I managed to log into the text console, and ran iotop, it showed that a process named kswapd0 was taking 99.99% of IO.  Are there workarounds so copying a large file doesn't make my system unusable? If $f$ is a concave function and $f(0) \geq 0$ then $f(a+b) \leq f(a) + f(b)$ for all $a,b >0$ I'm planning to use ZFS on my system (HP ML370 G5, Smart Array P400, 8 SAS disk). I want ZFS to manage all disks individually, so it can utilize better scheduling (i.e. I want to use software RAID feature in ZFS).  The problem is, I can't find a way to disable RAID feature on the RAID controller. Right now, the controller aggregates all of the disks into one big RAID-5 volume. So ZFS can't see individual disk.  Is there any way to acomplish this setup? Can't install wine on Ubuntu 14.04 I cannot install Wine on Ubuntu 14.04.  When I try to install it on the Ubuntu Software Center I always get a message that says I have unmet dependencies. I tried installing Wine through the terminal and I get this:  Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  wine : Depends: wine1.6 but it is not going to be installed E: Unable to correct problems, you have held broken packages. Is it possible to use a primitive type (int) in as a generic type in Java? Specifically, with a SortedMap&lt;Vector&lt;String&gt;, int&gt; I get "dimensions expected after this (int) token." Help! I am using pgfplot to draw graphs in latex. Now this is the problem I am facing:  \begin{tikzpicture}[scale=1] \begin{axis}[ybar,grid=major,bar width=2pt,width=8cm, height=8cm,xmin=1,xmax=540,ymin=1,ymax=240] \addplot [draw=black,fill=green]  table [x=size, y=cluster, col sep=comma] {sim-0.4-0.6.csv}; \end{axis} \end{tikzpicture}   It generates the following graph:    Since, I have values mainly between 0-50 on x-axis, I want to zoom there. After that, values comes around 500. S0, there are no graph points between 50 - 450. How should I do that? If I have a $2\times 2$ or $3\times 3$ matrix, how should I go about computing the eigenvalues and eigenvectors of the matrix?  NB: I am making this question to provide a unified answer to questions about eigenvalues of small matrices so that all of the specific examples that come up can be marked as duplicates of this post. See . If RSA is limited to 117-200 bytes or so, is that a very limited use case? Am I missing something, or is RSA very very limiting when it comes to ecrypting data when it comes to the actual message size?  I have read that you can only encrypt a message of around 117 to 200 bytes in size (depending on your key size: 1024 or 2048 bits). Is this correct?  If so, I think I have read people suggesting on using it for larger files. I guess they are just repeating the process many times over and over in blocks of 117 bytes? How to create a progress bar? How can I separate a separable function Black screen when creating movie I've rendered about 300 frames for a short clip. The frames look fine, but when I compile into a movie all I get is a black screen. I haven't been able to find anything to solve this problem. The individual frames are well lit, capture what I want them to, look great. Just creating the AVI or MPEG results in a clip that has nothing but black. A possible incorrect application of Law of Large numbers Using image sequence of medical scans as volume data in cycles I know it's possible to do this in BI, but I need your help to do this in cycles. For now, I have a setup to use an image sequence as volume but it only shows 1 slice/frame. I'm using generated texture coordinates for easy mapping (0-1 on each axis). The images are mapped on XY and I would like to use normalized frame# as Z coordinate to map the sequence as a volume.       Images sequence can be found  (I'm using 16-bit TIF files of MRBrain in this exemple).    Any solution is welcome but try to keep it simple and user friendly :).  Final Edit: I had a hard time choosing between Lemon's and PGmath's answers. Lemon's is more user friendly and flexible, and uses the image sequence without any change inside blender. PGmath's uses a 3rd party converter to make the sequence 1 single image, but allows GPU rendering. Well, ImageMagick's command line is : "convert +append *.tif anim.tif" from the folder where the sequence is unzipped. Not so hard. That's why I accepted PGmath's answer.      Now, i played with it, and here's the result.  The "volume" object contains the volume shader (playing with the color ramp gives funny things) and the boolean modifiers (turn them off to see the complete volume, but reduce the final density multiplier for better view). The "Slice Control" arrow empty controls the slice position/rotation. The "Texture Mapping" object controls...the mapping, if it need some tunning (position, rotation, scale, and then tiling if necessary?).