Let B be a set, and let * be a binary operation in B. Suppose * satisfies the associative law. Let  $$P=\{b \in B : b * w = w * b \quad\forall\, w \in B\}$$  Prove that P is closed under *.
Here is the problem:  Suppose that $*$ is an associative binary operation on a set $S$. Let  $$H:= \{a \in  S\mid a * x = x * a \mbox{ for all }x\in S\}.$$  In other words, $H$ is consisting of all the elements of $S$ that commute with every element in $S$.  Show that $H$ is closed under $*$.
Four complex numbers $z_1,z_2,z_3,z_4$ lie on a generalized circle if and only if they have a real cross ratio $[z_1,z_2,z_3,z_4]\in\mathbb{R}$ Let $[z_1,z_2,z_3,z_4]$ denote the  of the complex numbers $z_1,z_2,z_3,z_4\in \mathbb{C}$. Show that the distinct points $z_1,z_2,z_3,z_4\in\widehat{\mathbb{C}}$ lie on a generalized circle if and only if $[z_1,z_2,z_3,z_4]\in\mathbb{R}$.    I saw this statement claimed in , and wondered how one would go about showing it. I know of some facts we could work with:   First, the definition: A  in $\mathbb{C}$ is either a circle or a line in $\mathbb{C}$. I know that a Mobius transformation maps generalized circles to generalized circles. I know that if $f$ is a Mobius transformation and $z_j'=f(z_j)$, $j=1,2,3,4$, then $[z_1',z_2',z_3',z_4']=[z_1,z_2,z_3,z_4]$.   I have posted a potential proof as an answer below. Please let me know if you think it's complete. Thank you! How to print file content only if the first line matches a certain pattern? What's the 'correct' way of registering/installing an Assembly to the GAC? Nvidia graphics card no longer works in Windows 10 (error: Code 43) but works in Ubuntu Currently, when I boot into Windows, one monitor stays black (undetected), and the other can only display 800x600 resolution.  When I look at Device Manager &gt; Display Adapters &gt; NVIDIA GeForce GTX 970 &gt; General &gt; Device Status, it says "Windows has stopped this device because it has reported problems. (Code 43)".  Details &gt; Problem Code says "00002b". Details &gt; Status says:  01802400 DN_HAS_PROBLEM DN_DISABLEABLE DN_NT_ENUMERATOR DN_NT_DRIVER   When I then click into the Events tab, it says "Device PCI\VEN_10DE&amp;DEV_13C2&amp;SUBSYS_29763842&amp;REV_A1\4&amp;25438c51&amp;0&amp;0008 requires further installation."  I have a EVGA Nvidia GeForce GTX 970 graphics card with the latest drivers (376.19 from 12/1/2016).  My Windows 10 Pro has the latest updates and is at Version 1607, OS Build 14393.576.  My ASRock Z170 Pro4 motherboard BIOS is also updated to the latest version (7.00 from 10/4/2016).  I have dual Acer monitors with 1920x1080 resolution.  I downloaded Ubuntu 16.04.1 LTS onto a bootable USB drive and booted into the Ubuntu operating system instead of Windows. Both monitors immediately worked at full 1920x1080 resolution. So the hardware seems functional! My remaining challenge is to figure out why my video card has stopped working within Windows.  What I Have Tried   I've tried using Display Driver Uninstaller from Guru3d in Safe Mode to delete all drivers and try to install fresh.   I've tried many different versions of the driver dating back more than 12 months. I've tried reseating the card. I've tried moving the card to a different slot. EVGA's phone support told me to reinstall windows (Windows &gt; Reset this PC &gt; Keep my files). Unfortunately, I followed their advice (and now don't have any of my programs), and I still get Code 43 with the EVGA Nvidia card.   What I Have NOT Tried   I would love to try installing this card onto a different Windows 10 computer to see what happens, but I don't have access to any other computers. If you live northeast of Atlanta, let me know. ;-)   I'd appreciate any other ideas you have!. Limit of ${a_n}^{1/n}$ is equal to $\lim_{n\to\infty} a_{n+1}/a_n$ How can we get rid of misspelled and unused (or "zombie") tags? During the re-tagging of questions, tags sometimes become orphaned from existing questions.  Are these zombie tags ever removed from the ?  What if a tag is misspelled and needs to be removed? How do we get rid of it? Odds made simple How does commands like ls know what its stdout is?  It seems ls is operating different depending on what the target stdout is. For example if I do:  ls /home/matt/tmp    the result is:  a.txt b.txt c.txt   However if I do  ls /home/matt/tmp | cat   the result is (i.e. new line per result):  a.txt b.txt c.txt   The process is passed a file descriptor 1 for stdout right? How does it determine how to format the result? Does the file descriptor reveal information? show that $x$ is in $\ell^1$ if $\sum x_ny_n$ is convergent for all $y=(y_n)\in{c_0}$ If $x=(x_n)$ is a sequence of complex numbers such that the series $\sum  x_ny_n$ is convergent for all $y=(y_n)\in{c_0}$. Then prove that $x\in{\ell^1}$. Can anyone tell me what is the meaning of $c_0$  in this question and how to proceed on proving it. How can I donate to Ubuntu without PayPal? When I wanted to download Ubuntu 12.10 from ubuntu.com, I was asked to make a donation. However, there is only a PayPal option to make the payment. But I don't use PayPal. Is there any other option to make the payment? I once saw watch a sci-fi movie or an episode of a tv series (not sure) in which:   Every police officer was a judge, and hence could act by themselves on the streets. Each "judge" had a gun, which appended their DNA to the bullet. This was used later in investigations about a crime scene, and I think was confidential before the trial was held. Only a judge could use the gun, try to use it and you've been electrified. Also the guns had a self-destruction feature. The main villain was an ex-judge who purchased and activated a very old "killer robot" which was a residue of a long-ended war. When a judge was exiled, they were given a copy of the book of law, and sent outside the city to enlighten the people outside cities.   Can you help me find it?  PS: It was difficult to select a title mathtools: \dddot cause the term to rise Why doesn't a bike/bicycle fall if going with a high speed? Why does a bike/bicycle fall when its speed is very low or close to zero and is balanced when going with a high speed? Is there a difference in ++i and i++ in a for loop? Is it simply a syntax thing? A problem from Isaacs' FGT This is problem 3B.12. from the afore mentioned book. It asks to show that, if $G$ is a solvable Frattini-free group (that is, $Φ(G)=1$) and $H$ is a subgroup of some maximal subgroup $M$ of $G$, then $G$ contains a subgroup of index $|M:H|$. It is easy to see that, if   $N \unlhd G$ and $M\cap N=1$, then $G=MN$, $M\cong G/N $ and $H\cong K/N $ for some $ K \leqslant G $, where $K$ contains $N$. Then $|G:K|=|G/N:K/N|=|M:H| $ and we are done. However, the hypothesis will not necessarily hold. But, if $N$ is minimal among normal subgroups of $G$ (in the sense of containment), then $N$ is elementary abelian, hence nilpotent, which implies that $M \cap N \lneq N_G(  M \cap N ) \cap N $. Thus, $M \lneq N_G(  M \cap N )$ and by the maximality of $M$ we deduce that $M \cap N \unlhd G$. So, for this particular $M$, if it is not true that there exists some normal subgroup of $G$ with which it intersects trivially, then $M \cap N = N$ by the minimality of $N$. Hence $M$ contains every minimal subgroup of $G$.     How should I proceed from here? Libraries module. Where to extract? Check and fix SD card errors within Android itself? How can I correct this code to get an answer? My usb mouse sleeps after 5 seconds of inactivity when running on battery power. It wakes uo again when i press any button on the mouse. This is really annoying.  I have made lsusb, and I noticed my mouse is   Bus 001 Device 012: ID 04d9:2083 Holtek Semiconductor, Inc.  I have checked my /sys/bus/usb/devices/usb1/power/autosuspend but it is already set to 0, and the directory usb1 has many subdirectories which also have autosuspend files, and i dont know how to edit them.  Also, I would like a permanent solution, which would work for any mouse, on any of my 3 usb ports. Any ideas?  Using: Ubuntu 12.04 64bit Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$ Prove the identity:  $$n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$$  I tried using the binomial coefficients identity $2^n = \sum_{k=1}^n {n \choose k}$  but got stuck along the way.