An automorphism of a finite group which sends more than three quarters of elements to their inverses Question is :  Let $G$ be a finite group and suppose the automorphism $T$ sends more than three quarters of elements of $G$ onto their inverses.  Prove that $T(x)=x^{-1}$ for all $x\in G$ and that $G$ is abelian.  What I could see is...  Once I prove $T(x)=x^{-1}$ for all $x\in G$ then I would have that $G$ is abelian.  Because, for any $a,b\in G$ we have   $b^{-1}a^{-1}=(ab)^{-1}=T(ab)=T(a)T(b)=a^{-1}b^{-1}$  Thus, $ab=ba$ for all $a,b\in G$ Thus $G$ is abelian.  So, Only problem is to prove that $T(x)=x^{-1}$ for all $x\in G$  I am unable to use the fact that      "$T$ sends more than three quarters of elements of $G$ onto their inverses"   May be I should take  : $A=\{x\in G : T(x)=x^{-1}\}$ and see if this is a subgroup (It is Not) or something like that.  Please help me to clear this.  I am very excited about this question So please do not spoil my excitement by posting full answer at once (This is a request cum order).  Please help me to do this by myself.   Thank you so much :)
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group. Assume that $\pi$ is an automorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|&gt;\frac{3|G|}{4}$, then $G$ is an abelian group.    Anyone has any idea on how to solve this ? I have no idea how to start.
Where does Minecraft for Windows 10 store its data? I have searched in %appdata%, pulling up the list of files there, but there is no file labelled .minecraft. Searching %appdata%\.minecraft returns "no results found".   I also opened Minecraft to click on the resources button and could not locate that either. This may be a very basic question.      Let $R$ be a commutative ring with unity (may be assumed to be an integral domain, if necessary) and $I$ be an ideal of $R$. Assume that $R/I$ is isomorphic to $R$. Does this mean $I=(0)$ ? I am trying to prove that if I have $f:\mathbb{R} \to \mathbb{R}$ satisfying $\forall x,y\in\mathbb{R},f(x+y) = f(x) + f(y)$. 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I could not figure out which one to download but downloaded one on a windows PC and pasted the zip file on my desktop then unzipped it from the terminal using unzip &lt;filename&gt;.zip. The page says you have to go to control panel system hardware and all that on the windows system. I did not do it. Windows computer not being mine, I am afraid I should not install any additional driver on it. After unzipping it has one .inx file and one .bin file but no .sys file. I don't know whether I am completely right till here or have I missed something? And if I am right then how to proceed from here? I also made a new directory and copied the unzipped .inx and .bin files in it. Then in that directory I ran the ndiswrapper -i &lt;filename&gt;.inf but it says ndiswrapper not installed and when I do sudo apt-get install ndiswrapper-common, I get the reply that ndiswrapper-common is already the latest version. What to do? here is my dmesg | grep ath10k result  [   7.882758] ath10k_pci 0000:03:00.0: pci irq msi-x interrupts 8 irq_mode 0 reset_mode 0 [    8.190016] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/cal-pci-0000:03:00.0.bin failed with error -2 [    8.334587] ath10k_pci 0000:03:00.0: Direct firmware load for ath10k/QCA9377/hw1.0/board-2.bin failed with error -2 [   10.140677] ath10k_pci 0000:03:00.0: qca9377 hw1.0 (0x05020000, 0x003820ff sub 105b:e09a) fw WLAN.TF.1.0-00267-1 fwapi 5 bdapi 1 htt-ver 3.1 wmi-op 4 htt-op 3 cal otp max-sta 32 raw 0 hwcrypto 1 features ignore-otp [   10.140682] ath10k_pci 0000:03:00.0: debug 0 debugfs 0 tracing 0 dfs 0 testmode 0 Here is the output of lspci -nnk | grep -iA2 net  02:00.0 Ethernet controller [0200]: Realtek Semiconductor Co., Ltd. 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Like so:  \newcounter{howmany} There are \thehowmany items in this list: \begin{enumerate}   \item one   \item two   \item three \setcounter{howmany}{\theenumi} \end{enumerate}   And it should say "There are 3 items in this list". But that doesn't work. Well, sure. I need to use \refstepcounter to get that save-functionality. But \refstepcounter only increments it by one, and there doesn't seem to be a \refsetcounter ;) Any suggestions? A loop after the enumerate over \refstepcounters? Redefining \item to include \refstepcounter? There gotta be a simpler way, no?  Thanks! I have an external USB 3 disk drive (2TB capacity) that is most likely going to be moved from machine to machine. The disk has a GUID partition table and an ext4 partition. I am unable to write to the disk unless I elevate the process (sudo).  As of now I am thinking of trying one or both of the following and would like to know the cons of each -   chmod 777 /mnt/externalDrive chown nobody:nogroup /mnt/externalDrive   If I give 777 permission and user1 (UID:1005) writes to it and I later move the disk to another computer where user7 is UID:1005, what happens? Does user7 become the owner of the file on that computer? It seems to me that I will periodically have to run chown -R nobody:nogroup /mnt/externalDrive on the disk.  Is any of what I am considering an obvious bad practice? The disk will most likely contain videos, music and pictures none of it need to be protected like some financial data. Do I choose the target or the ally for the Help action? Uniform continuity of $\sin(x\sin x )$ for all $x\in (0,\infty)$ solving third-order nonlinear ordinary differential equation Vertical spacing for two consecutive figure environments I would like to download the alternate installation ISO for the current release, Ubuntu 13.10 Saucy Salamander, but I can't find it on the releases.ubuntu.com website. Aren't alternate ISOs provided anymore?  I need the alternate ISO because Xorg crashes on recent versions of Ubuntu when using old SIS on board graphics.  In fact, I just patched Ubuntu sources and generated a custom package for the xserver-xorg-video-sis driver on my PPA to try fix this problem, after installing with the minimal ISO. How can I create a new extensible symbol? Given an arbitrary thin vertical symbol, can I create a new math delimiter from it (and if so, how)?   Edit: Ideally, I would like to be able to create an extensible symbol, either built out an existing normal symbol, or even made from scratch.  Edit 2: I am pretty sure that what I am after is an explanation of how to create a .tfm file for an extensible symbol (which Werner's answer describes the general structure of, but not how to make one).    For example, let's say I wanted to use the \dagger () or \wr () symbol as a delimiter, and write expressions like    \[\left\ldagger \sum_{n=1}^\infty a_n \right\rdagger\]   where the symbol scaled appropriately. Would there be any distinction between declaring a symmetric delimiter (like \vert), as opposed to one that has inherently different "left" and "right" versions like [ and ]?  Now, even being unaware of the inner workings of TeX, I imagine that there has to be lots of information in the fonts for how a delimiter is to be scaled in different situations, and if I try to make my own delimiter, this information won't be present. Would I need to get deep into FontForge or equivalent to achieve my custom delimiters, or is there an easier way? 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Software/App for electronic research notebooks I want to keep a research notebook for my computational "experiments". Basically, I should at least be able to write text and attach images like plots. Other nice features to have would be   linking to past experiments/pages latex equations uploading papers, or other urls being able to transfer data dating and version control open-source tool   I am already familiar with trello and tiddlywiki. Trello is good for attaching stuff and organising tasks, but I need a notebook where I attach a plot, write stuff around it, attach another below it. It doesn't have a paper or canvas. Tiddlywiki is a little painful for attaching pics and equations, the file bloats, it seems I am keeping a blog.  What is a good, free electronic notebook tool for research/lab work? How can I automatically forward Facebook messages to my email or phone? I can create email addresses that forward to other email addresses.  Likewise, can I automatically forward Facebook messages to either my email or phone?