Group of order $p^2$ is abelian. Yes, I know that there are tons of solutions of this up here, but I, essentially, wanted to try it a different way and ah, well.  Let $|G| = p^2$ for some prime $p$.  Consider $x \in G$. So, $|x| = p, $ or $ p^2$. If it is the latter, then, we are done.  So, if $|x|= p$, then, let, $H = [h|  h=x^i, i \in Z] \Rightarrow |H|= p$.  As $p$ is the smallest prime dividing $|G|$, $H$ is a normal subgroup.  Now, consider some $y \in G$, but not in $H \Rightarrow y^p= e$. Then, its easy to show that $G = \langle x,y \rangle$.  Now, consider a homomorphism $f:G \rightarrow \operatorname{Aut}(H), g \to c_g $, wherein $c_g$ represents conjugacy by $g$. So, for example, $c_g(x) = gx g^{-1}$.  Then, it is clear that, $H \subset \operatorname{Ker}(f) $, as it is cyclic and therefore abelian. So, we need only look at $G/H$.  Basically, I am asking for a hint as to why $c_y(x) = yxy^{-1} = x = c_e(x)$, where $x$ and $y$ are the generators of $H$ and $G/H$, respectively?  Because, if I establish it for the generators, then it follows that it applies for for all elements, i.e, that $\operatorname{Im}(f)$ is trivial and therefore that $G$ is abelian.  Thank You!
Showing group with $p^2$ elements is Abelian I have a group $G$ with $p^2$ elements, where $p$ is a prime number. Some (potentially) useful preliminary information I have is that there are exactly $p+1$ subgroups with $p$ elements, and with that I was able to show $G$ has a normal subgroup $N$ with $p$ elements.   My problem is showing that $G$ is abelian, and I would be glad if someone could show me how.   I had two potential approaches in mind and I would prefer if one of these were used (especially the second one).   First: The center $Z(G)$ is a normal subgroup of $G$ so by Langrange's theorem, if $Z(G)$ has anything other than the identity, it's size is either $p$ or $p^2$. If $p^2$ then $Z(G)=G$ and we are done. If $Z(G)=p$ then the quotient group of $G$ factored out by $Z(G)$ has $p$ elements, so it is cylic and I can prove from there that this implies $G$ is abelian. So can we show theres something other than the identity in the center of $G$?  Second: I list out the elements of some other subgroup $H$ with $p$ elements such that the intersection of $H$ and $N$ is only the identity (if any more, due to prime order the intersected elements would generate the entire subgroups). Let $N$ be generated by $a$ and $H$ be generated by $b$. We can show $NK= G$, i.e every element in G can be wrriten like $a^k b^l $. So for this method, we just need to show $ab=ba$ (remember, these are not general elements in the set, but the generators of $N$ and $H$).   Do any of these methods seem viable? I understand one can give very strong theorems using Sylow theorems and related facts, but I am looking for an elementary solution (no Sylow theorems, facts about p-groups, centrailzers) but definitions of centres and normalizers is fine.
I have been stuck in a problem, that has thrown my brain out of the coding. This problem is at very high priority and I need the solution as early as possible. Problem is as :  There are exactly N advertising boards on the highway. Now a company want to advertise on some of these advertising boards(each advertising board costs some money) .  Company strategy is that, they want at least 'K' advertisement should be there among M consecutive advertising boards. But at the same time Company want to pay minimum for its advertisement .  Now, what is the total number of ways, different ways Company can advertise meeting its minimum cost strategy.  As for Example: N= 3, M = 2, K=1 ==> there is only one way for minimum cost, ie. 0C0 , where '0' denotes No company advertisement, and 'C' denotes company advertisement board.  Similarly, for N =4, M =2, K =1 ==> there is 3 possible ways, ie. C0C0, 0C0C, 0CC0.  Hope to get a reply from someone soon. Thanks. I have a problem when I try to start a Workflow 2013 in SharePoint Server 2013. I believe the error is related with permissions, but I cannot be sure.  Here is the error message in the log:  RequestorId: ef9edb7c-cc3d-5e10-3681-eada907596a4.  Details: System.ApplicationException: HTTP 401 {"Transfer-Encoding":["chunked"],"X-SharePointHealthScore":["0"],"SPRequestGuid":["ef9edb7c-cc3d-5e10-3681-eada907596a4"],"request-id":["ef9edb7c-cc3d-5e10-3681-eada907596a4"],"X-FRAME-OPTIONS":["SAMEORIGIN"],"MicrosoftSharePointTeamServices":["15.0.0.4420"],"X-Content-Type-Options":["nosniff"],"X-MS-InvokeApp":["1; RequireReadOnly"],"Cache-Control":["max-age=0, private"],"Date":["Thu, 31 Jan 2013 16:36:44 GMT"],"Server":["Microsoft-IIS\/8.0"],"WWW-Authenticate":["NTLM"],"X-AspNet-Version":["4.0.30319"],"X-Powered-By":["ASP.NET"]}  The HTTP response content could not be read.  'Error while copying content to a stream.'. at Microsoft.Activities.Hosting.Runtime.Subroutine.SubroutineChild.Execute(CodeActivityContext context) at System.Activities.CodeActivity.InternalExecute(ActivityInstance instance, ActivityExecutor executor, BookmarkManager bookmarkManager) at System.Activities.Runtime.ActivityExecutor.ExecuteActivityWorkItem   The question is: What permissions do I need to configure in SharePoint for the WF publisher and the WF manager service account?  I’ve set a WF Manager service account named “wfservice” (this account is a local admin and has the sysadmin permissions in SQL Server), and I published the workflow and tested it using the account “wfuser”. Both accounts have full control permissions in the web app and are site owners, for testing WF 2013 purposes.  I’ve already done the DNS flush, as mentioned in Andrew Connell’s article, and I’m still getting an Unauthorized exception in the Item Workflow Status.  Anyone knows what’s going on? Which permissions do I need to configure? Let $R=\frac{\mathbb{C}[x,y]}{\langle x^2-y^2-1 \rangle}$. Prove that (1) $R$ is an integral domain. (2) Any nonzero prime ideal of $R$ is maximal.  My idea: (1) The first part is easy. Since $x^2-y^2-1\in \mathbb{C}[y][x]$ is Eisenstein with respect to the prime $y+i$, it is irreducible. Hence the ideal generated by it is a prime ideal. (2) I am struggling with this part. I have one approach (not sure if correct). Approach : Let $I=\langle x^2-y^2-1 \rangle$. Then $R=\mathbb{C}[x,y]/I$.  Any ideal of $R$ is of the form $J/I$ where $J\subseteq \mathbb{C}[x,y]$ is an ideal containing $I$. If we can show that any prime ideal $J$ containing $I$ is maximal, then we are done. How to prove this? Is there any other way to prove the second part? Edit: I am trying to avoid Hilbert Nullstellensatz and the notion of dimension of a ring, as these concepts are not there in the syllabus of the exam from which I found this question. Before asking my question, I'll use an example to clarify the idea.  Say that a company used to sell horses and then they turn into a sheeps business. While some of the old pages can be mapped to the new pages using a 301 redirect, because they're really related in the content and the company don't want to lose users and link juice, there are a lot of pages strictly related to horses that they want to shutdown.  What's the best practice? Is there a way to say to users and Search Engines the content you were looking for won't be available any more?  Basically a 301 redirect of the closed pages to new ones seems not so good, because you'll show a content totally unrelated to the searched one.  P.S: I'm aware that probably the best fallback is to simply map those pages to the homepage but I hope in a smarter solution. How do I boot my PC from GRUB? If $\sum_{n=1}^\infty \frac{1}{a_n}$ converges, must $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ converge? Suppose $\sum_{n=1}^\infty \frac{1}{a_n} = A$ is summable, with $a_n &gt; 0,$ $n = 1,2,3,\cdots.$ How can we prove that $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ is also summable?  This question came from a problem-solving seminar, but I'm quite stuck without a push in the right direction. I tried a few things, including Cauchy-Schwarz (which says $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n} &lt; \sum_{n=1}^\infty \frac{A}{n}$) and also the idea of assuming the latter series diverges and attempting to deduce the divergence of the former series from that, using facts such as $\sum a_n = \infty \implies \sum \frac{a_n}{a_1 + \cdots + a_n} = \infty$. Nothing has worked so far. "no public key available" on apt-get update Mysql crashed, dumped DB's in recovery mode. What now? Entering Unicode characters in LaTeX Antisymmetric Relations apt-get install vs pip install I am a bit confused about the cases in which the above commands must be used when downloading python packages. I was trying to download a package named pyudev in accordance with an answer with this . I executed this command :  sudo pip install python-pyudev   but received the following message :  Downloading/unpacking python-pyudev    Could not find any downloads that satisfy the requirement python-pyudev Cleaning up... No distributions at all found for python-pyudev Storing complete log in /home/vineet/.pip/pip.log   However the following worked fine :  sudo apt-get install python-pyudev   When is apt-get supposed to be used to install packages and when is python-pip used? I remember reading a sci-fi story where astronauts visit a planet, which is covered in stationary aliens with mile long necks that filter feed from the atmosphere.  It was published between 1960 and 2000.    It was in English and involved a pair of astronauts that went from planet to planet on a tour or something, though I don't know details of any other planets they visited. I'm trying to install Wifite from , but I can't install cowpatty.  See below:  root@vitor-hp:~# apt-get install cowpatty Reading package lists ... ready Building dependency tree Reading state information ... ready E: Could not find package cowpatty root@vitor-hp:~#    Translated from Portuguese  I use Ubuntu 13.10 64-bit. Black artifacts when looking through many transparent objects? Number formation from given digits I was asked this question in a recent interview. You are given two digits $0$ and $9$. Using only these two digits you must form the least positive number that is a multiple of a given '$n$' . The value of $n$ will be provided before hand. For example if we are given $n$ as $7$ then the answer will be $9009$ .  I tried using some basic divisibility principles but was not able to come up with a correct solution and code it up. Probably I need some help in number theory Difference in number of communities on main site Gnome Panel becomes fully transparent when window is not maximised! Including Bitmap (.bmp) I can't include bitmaps with pdflatex. Here is my mwe:  \documentclass[12pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[turkish]{babel} \addto\extrasturkish{\uccode`i=\string"9D \lccode`I=\string"19 } \addto\noextrasturkish{\uccode`i=`I \lccode`I=`i }   \usepackage{graphicx}  \begin{document}  \shorthandoff{=}  \includegraphics[width=5cm,height=5cm]{000.bmp} \end{document}   Gives the error Unknown graphics extension: .bmp. "To your left" versus "on your left" There is a security camera on the building to your left      There is a security camera on the building on your left   Is there any real difference in meaning here? Which sounds more natural to native speakers? I've seen many boads with weird zig-zag shaped tracks just like the one below. Why are they designed like that?