If $x^m=y^n=e$ and $(m,n)=1$, then $o(xy)=mn$ If $G$ is a group with $xy=yx$ for $x,y\in G$, and  $x^m=y^n=e$, then: (1) $o(xy)|mn$, where $o(xy)$ is the order of $xy$, and (2) if $(m,n)=1$ then $o(xy)=mn$.   Well, I've already proved (1), but I'm having trouble with (2), I was thinking that maybe, if we already know (1), $o(xy)|mn$ then we just have to show that $mn|o(xy)$, that way we get equality. So we get: $mn|o(xy)$ iff $o(xy)=mn*l$ for some $l\in\Bbb Z$. On the other hand, we know that $(xy)^{mn}=e$ then it's obvious that for any $l\in\Bbb Z$, $(xy)^{mn*l}=e^l=e$, we get what we wanted, but if there's an infinite number of $l's$, what does this means? I feel that what I'm doing is wrong since I didn't used the principal hypothesis: $(m,n)=1$, but this is the only way so far I've been able to do something.
Relationship between $\operatorname{ord}(ab), \operatorname{ord}(a)$, and $\operatorname{ord}(b)$ Another homework problem from my Group Theory class.  Let $a,b$ be elements of a group, $G$.  Let $\operatorname{ord}(a)=m$ and $\operatorname{ord}(b)=n$. Let $a$ and $b$ commute. Prove: If $m$ and $n$ are relatively prime, then the $\operatorname{ord}(ab)=mn$.  So I am having trouble starting off on this problem.  Here is what I believe I know:  $a^m=e$ and $b^n=e$ [as defined by the order of the elements] which leads $a^m=b^n$  $\operatorname{ord}(ab)=mn=\operatorname{ord}(ba)=nm$ as defined by commuting.  $\gcd(m,n)=1$ as defined by $m,n$ being relatively prime.    I don't know how to start this proof. Do I want to to go the route of $a^mb^n=e$?  Then $(ab)^{mn}=e$?   Any tips to start this would be greatly appreciated!!!
I know Newton's third law of motion might be the answer for this but still I am wondering how the rockets could thrust in the empty space and move in the opposite direction. I guess an astronaut wouldn't be able to push in the empty space with his hands or legs to move himself, but with a rocket engine it's possible. How? What might be the explanation for this in General Relativity? Draft mode for pgfplots The output from the following script \documentclass{article} \usepackage{tikz} \usepackage{graphicx} \begin{document} \begin{tikzpicture}   \node(title)[text width = 0.25\textwidth, text height = 3 cm, draw]{title};   \end{tikzpicture} \end{document}  is  How can I vertically align the text in the node? Specifying the text depth as in \node(title)[text width = 0.25\textwidth, text height = 1cm, text depth = 1 cm, draw]{     title line 1\\     title line 2\\     title line 3};  Does not result in the text being vertically centred. As seen in this video: (Skip to 5:53)  When a player is slain, they drop an Enchanting Table on top of their inventory.  How was this achieved and how can I build a system (using command blocks) that does that? Come to think of it, what happens to a player's location when he/she dies? Does it reset to the spawnpoint? Stay the same (where they died) until they respawn? Or becomes a null value until they respawn? Arrays in Java and how they are stored in memory Remove all packages installed from a PPA? I have installed LibreOffice 4 using PPA. I was using LibreOffice 3, then  I had upgraded to LibreOffice 4, because I was somewhat eager to try a new version.   Now since I found that LibreOffice 4 has some issues, including handling my native language, I want to move back to LibreOffice 3.  This question  has already some answers which has some similarity with my above mentioned issue, but it does not solve my problem. This question talks about downgrading from a specific version of LibreOffice, namely from 4.0 to 3.6. The solutions mentioned are not the ones I am looking for. They will work, but the  solutions suggest either downloading .deb files for LibreOffice 3.6 or adding PPA for them. Furthermore, some of the answers put out-of-proportion~(applicable for the solution, however) stress on use of , not general command-line-solution.     I want a general solution without using PPA or downloading .deb files. All I want is to move back from the (higher) PPA version to the (lower) official version.  Anyway, in order to accomplish this, I removed the LibreOffice config directory from my home and then purged LibreOffice from my machine.  sudo apt-get purge libreoffice-*   Then I removed the relevant PPA's using the sudo apt-add-repository --remove command. And then ran sudo apt-get update.  Now, when I try to install LibreOffice using the command  sudo apt-get install libreoffice   I get an avalanche of output about unmet dependencies, something like,   The following packages have unmet dependencies:  libreoffice : Depends: libreoffice-core (= 1:3.5.7-0ubuntu4) but it is not going to be installed (snipped)    If I dig the issue further, by using the command,  sudo apt-get install libreoffice-core   I get    The following packages have unmet dependencies:  libreoffice-core : Depends: libreoffice-common (&gt; 1:3.5.7) but it is not going to be installed                 Depends: libexttextcat0 (&gt;= 2.2-8) but it is not going to be installed                 Depends: ure (&gt;= 3.5.7~) but it is not going to be installed E: Unable to correct problems, you have held broken packages.    Could you please tell me how do I install LibreOffice 3 in my machine?  I am using Ubuntu 12.04 LTS.    EDIT  Before receiving , I was wondering, at this very moment, if I take a fresh computer, and install Ubuntu 12.04, LibreOffice installation will work without a hitch. Then why I can not install LibreOffice in my 12.04 machine today from simple command line?  Then the accepted answer clarified everything. I need to use ppa-purge so that this resets all packages from a PPA to the standard versions released for my distribution.  Basically it is like a way to restore my system back to the way it was before my installed packages from a PPA.  further elaborates the idea.    worked perfectly for me. Actually, this was an education for me since it taught me how do downgrade a package that was added via PPA. And I realized that the answer actually teaches us to "move from PPA installation  of a package to official lower version package." Problem: Let $f$ be defined for all real $x$, and suppose that   $$|f(x)-f(y)|\le (x-y)^2$$  for all real $x$ and $y$. Prove $f$ is constant.  Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exercise 1. Is there a case where including the same header twice is actually helpful? Presentation of SL$(n,\mathbb{Z}_p)$ I know a presentation of the special linear group SL$(2,3)$(). My question is that-     Can we give a presentation for SL$(n,\mathbb{Z}_p)$ in general or in more general can we  give a presentation for SL$(n,\mathbb{F})$,where $\mathbb{F}$ is field.   As J.P. Serre had given presentation for SL$(2,\mathbb{Z})$   $$\mathrm{SL}_2(\mathbb{Z}) = \langle \,S, T \mid S^4 = 1, (ST)^3 = S^2 \,\rangle$$ where, \begin{align}     S &amp;=     \begin{pmatrix}     \phantom{-}0&amp; 1 \\ -1 &amp; 0     \end{pmatrix},     &amp;     T &amp;=      \begin{pmatrix}     1 &amp; 0 \\ 1 &amp; 1     \end{pmatrix}. \end{align} I wants it to be more general. I know how to rename files in Unix:  $ mv ~/folder/subfolder/file.txt ~/folder/subfolder/file.sh       ^-------this part------^    ^------this part-------^   It takes too long time to repeat ~/folder/subfolder/file twice.  Is there a quicker way? I use the term "100% crop" to mean a portion of a picture that has been cropped but not resized, but I have also seen some confusion around that term.   Is there an accepted definition for "100% crop?" What is its opposite called?   Here's what I think it means:   If I'm right, what is the proper name for "something else?" If I'm wrong, please include illustrations.  Either way, what evidence supports the definitions? sci-fi book where a meteorite gives 3 different kinds of telepathic powers to people who then vie for control 3 groups of telepaths each with different powers.  They received their powers from a meteorite.  One group is the strongest and are kind of in control.  Another group wants to gain control.  One group, I believe the strongest, have telepathic battles to determine the strength of one another conjuring up swords, walls, etc. in their minds. It was written by a lawyer.  Could have been written in the ‘80s, or earlier. Display Animated GIF How to find chromatic number of the $n$-dimensional hypercube $Q_n$? Upgrade from MaaS 2.6.2 to MaaS 2.7 I was using my MacBook Pro (with 16GB of RAM) as usual and out of blue I've this popup that:  Your system has run out of application memory. To avoid problems with your computer, quit any applications you are not using.   I've obviously closed few which I could, but it didn't help. After checking on memory, it seems kernel task ate 7GB and 22.36GB swap memory was used of total 23GB (which obviously was the case). However I've still 20GB of space free on my SDD.  Activity Monitor didn't help much whilst my OS X was heading to the destruction.    My top stats before my kernel freeze: Processes: 344 total, 2 running, 5 stuck, 337 sleeping, 2580 threads                                                                                                                         19:23:56 Load Avg: 1.56, 1.62, 2.09  CPU usage: 3.51% user, 8.47% sys, 88.1% idle   SharedLibs: 46M resident, 0B data, 6572K linkedit. MemRegions: 757970 total, 2139M resident, 56M private, 907M shared. PhysMem: 9410M used (6198M wired), 556M unused. VM: 1155G vsize, 1311M framework vsize, 112872658(320) swapins, 122668389(0) swapouts. Networks: packets: 299419263/363G in, 142126838/14G out. Disks: 58970173/1079G read, 20012389/1120G written.  At the end my OS X frozen and I had to do hard reset, repair my SDD in recovery mode and fixing afterwards (recovering lost work, fixing application conflicts, checking my lost+found folder, Chrome/Terminal tabs gone, headache, etc.). My question is, how do I check on high memory usage of kernel task or how to properly deal with that kind of situation? I've tried to take Sample with Activity Monitor, but it's greyed out.  My MacBook Pro details: 2.3GHz Intel Core i7 (late 2013) with 16GB RAM. OS X: 10.9.5 Taking the * of these two large shapefiles   locally using ArcMap takes appr. 10 min.   Over the last few weeks, I've tried to obtain the same results using other, open-source approaches without success. I am curious why only ArcMap seems to generate results in a reasonable amount of time without errors.   The approaches I've tried so far:  QGIS  The most straight-forward alternative. There are two "union" functions available in QGIS. One default (part of vector overlay), one based on SAGA. Both functions create the desired results on smaller samples but fail to run (or take forever) on my shapefiles. I tried running union for 30 minutes after which the progress bar was at appr. 4%    PostGIS  I've ingested the shapefiles on an AWS RDS instance running postGreSQL with PostGIS enabled. The approach is not so straight forward and requires a hideous long SQL query to obtain the same result. Most importantly, the query did not create a meaningful result on the full datasets. It did run properly on a sample though.   See separate question   CREATE TABLE test.sqlislelijk AS  -- input data with polys1 AS (   SELECT      pfaf_id as df1,     geom as g   FROM hybas06_v04 ), polys2 AS (   SELECT      aqid as df2,     geom as g   FROM y2018m11d14_rh_whymap_to_rds_v01_v01 ), -- intersections intersections AS (   SELECT df1, df2, ST_INTERSECTION(a.g, b.g) i, a.g AS g1, b.g AS g2    FROM polys1 a, polys2 b WHERE ST_INTERSECTS(a.g, b.g) ), -- per-row union of intersections with this row diff1 AS (   SELECT df1, ST_UNION(i) i FROM intersections GROUP BY df1 ), diff2 AS (   SELECT df2, ST_UNION(i) i FROM intersections GROUP BY df2 ), -- various combinations of intersections pairs AS (   SELECT df1, df2, i AS g FROM intersections   UNION ALL   SELECT      p.df1,     NULL,     CASE       WHEN i IS NULL THEN g        ELSE ST_DIFFERENCE(g, i)     END   FROM polys1 p LEFT JOIN diff1 d ON p.df1 = d.df1   UNION ALL   SELECT     NULL,     p.df2,     CASE       WHEN i IS NULL THEN g       ELSE ST_DIFFERENCE(g, i)     END   FROM polys2 p LEFT JOIN diff2 d ON p.df2 = d.df2   ) SELECT    df1 as pfaf_id,   df2 as aqid,   g as geom FROM pairs WHERE NOT ST_IsEmpty(g);   Google BigQuery  Google's GIS extension for BQ looks promising but I experienced weird behavior and internal errors, probably due to mixing of GEOMETRY and GEOGRAPHY objects. Also, the syntax is similar to PostGIS and a command of 1 line in geopandas takes >20 lines in SQL.   GeoPandas  Probably my favorite approach. The out-of the box approach does't give any results after running it for several hours. My latest approach includes tiling the shapefiles, use multiprocessing on each tile and merge them. Not very straight forward and one 10x10 degree tile till takes 40+ minutes to calculate.   See   It is the first time in my career that I am impressed by the performance of ArcMap. It seems that ArcMap handles this use-case much, much better than anything else. The issue seems persistent over all solutions I've tried so far. Is there something wrong in the underlying libraries of the open-source/alternative solutions?    Note that Union in ArcMap means something different than in the Shapely / PostGIS world. Is it ever revealed what Crookshanks is? When Hermione first got her pet thing Crookshanks, there was a mention that it looked kinda like a cat, but wasn't.  Was it ever revealed what Crookshanks really was? I always thought the book was Manifest Destiny but its not. The story that sticks in my head was a human in a cage over a pit? Aliens walk in and discuss how dangerous humans were. He eventually escapes. Man I miss this book. wish I wouldve written down every book I ever read. Ugh Do I need to apply for a visa to enter South Korea as a tourist on an Australian passport? Do I need to apply for a visa to enter South Korea as a tourist for a few days when travelling on an Australian passport? Or will I be able to get one on arrival?