I have to prove that there is no group $G$ such that $G/Z(G)$ is isomorphic to $Q_8$.  Anytone can give me an idea to begin? thanks
Let $Q_8$ be the quaternion group. Show that $G/Z$ can never be isomorphic to $Q_8$, where $Z$ is the center of $G$. Hint: if $G/Z\cong Q_8$, show that $G$ has two abelian subgroups of index $2$.  I'm trying to prove the hint by looking at the canonical homomorphism $\pi:G\rightarrow G/Z$. Well, $Q_8$ has some abelian subgroups of index $2$, such as $H=\{1,-1,i,-i\}$, but that doesn't seem to map to abelian subgroups of index $2$ of $G$. (We know that $\pi(ab)=\pi(a)\pi(b)=\pi(b)\pi(a)=\pi(ba)$ for $a,b$ in $\pi^{-1}(H)$, but that only yields $aba^{-1}b^{-1}\in Z$, not $aba^{-1}b^{-1}=1$.)
Trichromelibrary was suddenly downloading from Google Play Store From nowhere, my friend got a notification that com.google.android.trichromelibrary was downloading together with a progress bar.  Why? how many semantically different boolean functions are there for n boolean variables? In short, this is an assignment question for a course I am taking - the exact wording is this:  "Given n Boolean variables, how many 'semantically' different Boolean functions can you construct?"  Now, I had a crack at this myself - and got pretty stuck. The question doesnt state how many boolean operators there are (and, or, xor, nand, nor, iff, implies, not) nor does it state whether brackets should be used, i.e. a ^ (b v c) is different from (a ^ b) v c.  So, my question for you is - is this question possible given the limited information available?   Is it going to be something like ${n^x}$ where x is the number of boolean operators.  Any direction here would be greatly appreciated. How good is Hard AI? Is there a way to spellcheck the word under the cursor with a keyboard shortcut? Why does correlation matrix need to be positive semi-definite and what does it mean to be or not to be positive semi-definite? I have been researching the meaning of positive semi-definite property of correlation or covariance matrices.   I am looking for any information on   Definition of positive semi-definiteness; Its important properties, practical implications; The consequence of having negative determinant, impact on multivariate analysis or simulation results etc. Let $f_n, f \colon [a,b] \to \mathbb{R}.$ Show that, if $f_n \to f$ uniformly, then the Lebesgue integrals are equal, i.e. $\int f = \lim \int f_n$.  This is clearly true for continuous functions, but how do I handle the case of non-continuous functions? What is the right ffmpeg output format for a v4l2-loopback device? Following the guide in  I did:  $ sudo modprobe v4l2loopback $ ffmpeg -f x11grab -r 15 -s 1280x720 -i :0.0+0,0 -vcodec rawvideo -pix_fmt yuv420p -threads 0 -f v4l2 /dev/video0   The device /dev/video0 shows up as expected but the ffmpeg command fails with following output:  $ ffmpeg -f x11grab -r 15 -s 1280x720 -i :0.0+0,0 -vcodec rawvideo -pix_fmt yuv420p -threads 0 -f v4l2 /dev/video0 ffmpeg version 4.0 Copyright (c) 2000-2018 the FFmpeg developers ... [x11grab @ 0x559767794cc0] Stream #0: not enough frames to estimate rate; consider increasing probesize .... [v4l2 @ 0x55976779e6c0] ioctl(VIDIOC_G_FMT): Invalid argument Could not write header for output file #0 (incorrect codec parameters ?): Invalid argument Error initializing output stream 0:0 --  Conversion failed!   I already tried to follow the suggestions below the answer to no avail. I noticed that v4l2 is listed as demux only in ffmpeg -formats. Does that mean I can't use it for output to /dev/video0? If so, what should I use instead? I tried h264 but that crashes after ~1000 frames with following error message:  $ ffmpeg -f x11grab -r 15 -s 1280x720 -i :0.0+0,0 -vcodec rawvideo -pix_fmt yuv420p -threads 0 -f h264 /dev/video0 ffmpeg version 4.0 Copyright (c) 2000-2018 the FFmpeg developers ... [x11grab @ 0x55da61e29cc0] Stream #0: not enough frames to estimate rate; consider increasing probesize ... File '/dev/video0' already exists. Overwrite ? [y/N] y .... Error writing trailer of /dev/video0: Invalid argument .... Conversion failed! How do I execute a program X times in the terminal?  I read that I have to do a bin/bash txt, but I don't know how to execute this X times. In propositional logic, the statement "If pigs can fly, then elephants can lay eggs." validates to true because the antecedent validates to false.  In other words, given $a \rightarrow b$, if a is false, the entire statement is true. Why?  Just because the antecendent is false doesn't mean that another fact depends on it, right? I am compiling LaTeX files with Texmaker on Ubuntu 12.04 system.   When I run pdflatex and then view the pdf file (internal), it does not show me the exact place where I am editing on Texmaker i.e. the generated pdf file does not show me the position of the cursor.   But some days before it was working fine. I don't know what causes this problem.  I already tried uninstalling and then reinstalling, I did update and upgrade too. But the problem still persists. Suppose f is a function such that $f'(x) = \frac{1}{x}$ for all $x &gt; 0$. Prove that if $f(1) = 0, then f(xy) = f(x) + f(y)$ for all $x,y &gt; 0$  As stated above, how do i go about doing this question?   Hi all, i would like to apologize if the question is missing context or other details.   This question was part of my calculus tutorial. I was on the topic of application of differentiation when this question was given to me.  Thus far i have learnt about intermediate value theorem, mean value theorem, rolle's theroem, Fermat's thereom.  As of this point in my learning, i have not touched upon integration. The only reason why i accepted the answer with integration in it was because i understood it.  Thus i believe i am suppose to use one the few theorems that i've stated to solve this question. But i have no idea where to start.  At first i differentiated the equation $f(xy)=f(x)+f(y)$ with respect to x  i believed the answer to be $f'(xy)\frac{dy}{dx}=f'(x)+f'(y)\frac{dy}{dx}$  which would be $\frac{1}{xy}\frac{1}{x}=\frac{1}{x}+\frac{1}{y}\frac{1}{x}$  At this point a few questions came to my mind  1) am i suppose to differentiate the eqn in the first place 2) what has $f(1)=0$ got to do with anything 3) what theorem am i suppose to use to solve this question  This was my thought process throughout the question and i hope someone would point out my mistakes. Thank you How can I add the session name to my screen status display? This is my current .screenrc files which is mostly copied from the link in the code  #First iteration 2014-09-06, not quite sure of stuff to add # got a fancy hardstatus line noted below hardstatus on hardstatus alwayslastline # hardstatus string "%w"  # blagged this hardstatus like from https://bbs.archlinux.org/viewtopic.php?id=55618, not quite sure what it does hardstatus string '%{= kG}%{C}Screen: %{c}%{= kG}[%= %{= kw}%?%-Lw%?%{r}(%{W}%n*%f%t%?(%u)%?%{r})%{w}%?%+Lw%?%?%= %{g}][%{C} %d-%m %{W}%c %{g}]'   I now want the session name to appear in it as well, and the one I have settled on is from this answer -   The solution runs along these line:  hardstatus string '%H:%`' backtick 0 30 30 sh -c 'screen -ls | grep --color=no -o "$PPID[^[:space:]]*"'   It is an area I am not familiar with and I wonder how the backtick command fits in it, ie whether it is a screen command at all. Printing the value of a variable in SQL Developer Is there a better name for this than "emphatic and"? In semi-poetic writing, I have had occasion to deviate from the standard prose list practice (interpose commas or semicolons between items, with an "and" preceding the final item), by utilizing a second or third and to separate three or four items.  That is,       Bob likes beer, wine and liquor.   The variant would be:     Bob likes beer and wine and liquor.   Now, the usage is obviously nonstandard and should be avoided in formal writing, but for those instances where poetic variation may be justified.    What is it called?  "Emphatic And" is the best I can coin on my own, but I wonder if there is a more established name for this variation. What did Enel do after being defeated? As shown in the Manga and the Anime, Enel survived along with the Ark.  He then Embarks to Fairy Verth      The Moon.   In the Manga it shows his adventures on the Moon.  But what happened?  I didn't understand the situation that happened.  What exactly happened on the Moon? Formula across objects Is it acceptable to take photos of slides at a conferences for private use? Is it accepted/appropriate to take pictures of presentation slides at conferences for private use?   I am not talking about putting those pictures out in the public afterwards, I know this is illegal without consent of the presenter.    Should it matter, I am talking about math conferences.  And I am asking because I don't see anyone taking pictures. How to calculate $5^{3^{1000}}\bmod 101$? Why does the Community user have votes cast? If Community is a background process and not a real person then why does its profile show that it has votes cast, and how does it cast them?  How can it decide which answers to upvote or downvote?  See the massive number of upvotes and downvotes cast in its profile: Why do sunbeams diverge even though the sun is much more than a few kilometers away? Consider this picture of sun beams streaming onto the valley through the clouds.    Given that the valley is only (at a guess) 3km wide, with simple trigonometry and the angles of the beams, this gives the result that the position of the light source is being a few tens of km away at most.   What is wrong with the analysis?