I'd like to prove the following theorem :     Let $G$, $H$, $G'$, $H'$ four groups. If they are finite then   $$ \left\lbrace \begin{array}{ll} G\times H \simeq G'\times H' \\ G\simeq G' \end{array}\right. \quad \Rightarrow\quad H\simeq H'.$$   This not stands with infinite groups. Consider for example $G=\mathbb{Z}[X]$, $G'=\bigoplus_{n\geq 0}\mathbb{Z}$ and $H=\mathbb{Z}$, $H'=\lbrace \mathrm{Id} \rbrace$.  Our teacher proved the theorem using the following method : we first show that for all finite groupe $L$ the number of morphism beetwen $L$ and $H$ is the same that beetween $L$ and $H'$. Then we proved that the number of one to one morphism beetwen $H$ and $H$ is the same that beetwen $H$ and $H'$. We concluded by cardinality.  I'm okay with this proof but I find it a bit artificial. I've tried to find this theorem in books but after hours of rechearch I wasn't able to find any book which prove this theorem.  If you see another proof, or if you know where I can find it, please let me know. Thank you for your help.
Let $G\times K$ be a finite group. Suppoose $G\times K\cong H\times K$. Is this sufficient to imply $G\cong H$?
I'm having some problems with listings and UTF-8 in my document. Maybe someone can help me? Some characters work, like é and ó, but á and others appear at the beginning of words...  \documentclass[12pt,a4paper]{scrbook} \KOMAoptions{twoside=false,open=any,chapterprefix=on,parskip=full,fontsize=14pt}  \usepackage[portuguese]{babel} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{listingsutf8}  \usepackage{inconsolata} \lstset{     language=bash, %% Troque para PHP, C, Java, etc... bash é o padrão     basicstyle=\ttfamily\small,     numberstyle=\footnotesize,     numbers=left,     backgroundcolor=\color{gray!10},     frame=single,     tabsize=2,     rulecolor=\color{black!30},     title=\lstname,     escapeinside={\%*}{*)},     breaklines=true,     breakatwhitespace=true,     framextopmargin=2pt,     framexbottommargin=2pt,     extendedchars=false,     inputencoding=utf8 }  \begin{document} \begin{lstlisting} &lt;?php  echo 'Olá mundo!'; print 'Olá mundo!'; \end{lstlisting}  \end{document} \end{lstlisting} Proof of $Y=F_X(X)$ being uniformly distributed on $[0,1]$ for arbitrary continuous $F_X$ Word for killing off or attempting to kill off an entire bloodline? Why does $\lim_{n\to\infty} z_n=A$ imply $\lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A$? I'm self-studying a bit of complex analysis, and I'm attempting to figure out the following.   Suppose $\lim_{n\to\infty}z_n=A$. How can I show that $$ \lim_{n\to\infty}\frac{1}{n}(z_1+\cdots+z_n)=A. $$  Is there a clever way to write the limit to make it more approachable? Thank you. Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective Let $f: A \rightarrow B$ be a function. How can we show that for all subsets $S$ of $A$, $S \subseteq f^{-1}(f(S))$? I think this is a pretty simple problem but I'm new to this so I'm confused.  Also, how can we show that $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective? Is it possible to set a CNAME record at the top of a domain? (i.e. @ CNAME www, @ CNAME foobar.com., etc.)  My ISP says that it's only possible to use CNAME's for subdomains but I've read somewhere else that is should be possible even if not recommended. Canned reasons for up/down vote (w/ example of how it would work) I'm annoyed by down voters that don't explain why they down voted a question or an answer. Although, at the same time, I understand that they . I think there is a middle ground, though, where the voter can remain anonymous, while the poster can get aggregated feedback on ways to improve the post.  Rather than having to imagine what it might be like, the current stackoverflow engine can emulate this rather effectively with the use of comments and the up-voting of those comments (essentially saying "I agree"). So I thought I would give it a try. The only difference is that in this testing post, you will always see the list of "canned responses" where as in the complete version, it will most likely be hidden in a popup (e.g. when you hover over the score of an answer/question). To try it out, simply up or down vote this question (or one of its answers), and press the up arrow next to the comment which best describes why are voting that way. For example, if you hate this idea, give this a down vote, and then press the up arrow near -1: I don't like this idea.  In addition to retaining anonymity, this will also save time for the voter so that they don't need to write out the comment. This wouldn't be required, obviously, and there could be a default option selected don't leave a reason. In addition, there could be a custom option which allows you type exactly why you are down-voting this way. Depending on the final implementation, this could either simply be converted to a comment automatically prefixed with -1:  or +1:; or, to retain anonymity, it would be added to this same list that shows the canned responses and how many users have voted that way.  Here are some ideas for the canned responses:  Question:   +1: Well formatted and thought out question. -1: It doesn't seem like enough time was spent trying to solve this issue on your own. -1: I don't like this idea. (Could be especially useful on meta.stackoverflow.com feature-request.) -1: No prior research/work. (Idea from .) -1: Poor question. Please see . -1: Doesn't contain code. (This would be useful on stackoverflow.com. Idea from )   Answer:   +1: Concise. +1: Insightful. +1: Worked for me. -1: It's wrong. -1: Confusing. -1: Encouraging bad behavior (e.g. answering a question that shouldn't have been asked; see ).   I made this a community-wiki since there will probably be testing of up/down votes. Several of these canned responses have been taken from . This same idea was also . we are currently experiencing some issues with the ambiguity feature of apa for biblatex/biber (we use biber) in TexLive 2012.  What would be correct with the ambiguity feature:  Doe J., Peters M., Kramer N., Conan B. (2012) Doe J., Mayer M., Smith N., Walker J. (2012)  Citations in text (subsequent ones) - AS IS AND CORRECT Doe, Peters et al (2012) Doe, Mayer et al (2012)   But in case of different years, it shouldn't actually do this ...  Doe J., Peters M., Kramer N., Conan B. (2012) Doe J., Mayer M., Smith N., Walker J. (2009)  Citations in text (subsequent ones) - AS IS: Doe, Peters et al (2012) Doe, Mayer et al (2009)  Citations in text (subsequent ones) - HOW IT SHOULD BE: Doe et al (2012) Doe et al (2009)   Can anyone help here?  Thank you! Chat related privileges are not site specific Files disappearing on linux server I've got 4 specific files that seem to keep disappearing from a user's home directory. As far as we know, there are no cronjobs or other automated tasks that would be removing them.  I've setup auditd on them but the logs aren't really showing anything of interest.  I can see our backup utility accessing them every night until the point they aren't there anymore, but nothing else.  Is there anything that would be causing those files to be removed that would get around auditd?  The files in question are these:  /home/username/.bashrc /home/username/.bash_profile   as well as a couple of files in that user's .ssh directory.  Copies of these files placed into a subfolder called "keepers" get deleted at the same time as well.  Changing the permissions on them to 000 and having them owned by root hasn't helped.   I've currently got inotifywait setup to log create,delete,move on that subfolder, so hopefully that will turn up something, although it doesn't log much aside from when it happened, not what caused it. How to make ttf-mscorefonts-installer package download fonts, after it says it installed? When I install "Installer for Microsoft TrueType core fonts" (ttf-mscorefonts-installer) from the Software Center on 14.04, it gives a positive response--but after this I get a download error. It seems that the files actually providing the fonts are never retrieved. Is there any way to resolve this? Is it possible to use the fact that $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$ to compute $\sum_{n=1}^\infty \frac{(-1)^n}{n^4}$? Short sci-fi story of plan devised by marooned spaceship pilot to save paradise world from development Spaceship crashes on beautiful, remote planet. The pilot survives and lives quite well on the beautiful "island paradise" world amongst friendly, good-natured people... but, one day (while relaxing in a hammock near a beach) he "realized he was dying" (actually, that he would eventually die, then being unable to help the people he had come to love). He fears future "discovery" of his adopted world and the inevitable plundering of the resources and people by "developers"... so he devises... "a plan" "convinced island elders to support his plan" "his students taught others" "achieved 80% literacy... a remarkable feat" "established world government" "more generous than developer expected" "successfully argued for independent status" "established 100% taxation rate" "finished resort buildings converted to school" "developer hired to build resort in isolated area"  ...'not having luck finding it... my recollection of the story is good, but apparently not EXACT/searchable phrases How do I update Ubuntu manually? I wanted to update Ubuntu manually, because I used CDMA USB modem and connected using wvdial, but it won't able to update Ubuntu via Update Manager, because my Ubuntu won't showing "Connected" message via network-manager.  Can I update Ubuntu manually? Am I should download separate files, or use terminal instead? Would this look natural:     StackExchange is a helluva site! I found out about this site at Facebook.    Or should we use the preposition on:     StackExchange is a helluva site! I found out about this site on Facebook.    Or are both okay?   (Inspired by a ) When there is less than 5% battery left on a mac, the battery indicator goes red and system performance significantly drops - presumably in an attempt to make that remaining 5% last a little longer.   Is there a way to manually trigger this, so that the system performance is low (and battery life longer), even when there is plentiful battery?   This would theoretically be a way of extending the number of hours on a battery beyond the rated amount. I'm looking for a javascript that can limit the number of lines (by line I mean some text ended by user pressing enter on the keyboard) the user is able to enter in textarea. I've found some solutions but they simply don't work or behave really weird. The best solution would be a jquery plugin that can do the work - something like , but it should be able to limit text line count not character count. Show there are infinitely many primes that are equivalent to 1 mod 8. Show there are infinitely many primes that are equivalent to $1 \pmod{8}$.  Hello there! I have been trying to do this problem for a pretty long time with no avail.    I noticed that this is really similar to Euclid's proof that there are infinitely many primes.  However, I can't find a way to use that here.  Can anyone help me? What is the difference between "source x", ". x" and "./x" in Bash? Any chance of MathJax in chat? It would be really nice to be able to use MathJaX (math formulas) in chat.  Not specially for formulae, but for things like $V_{CC}$ in electronics chat would be very useful.