Automorphisms of Quaternion Group $Q$. Prove that $\operatorname{Aut}(Q)$ contains no element of order $6$, and so $\operatorname{Aut}(Q)\cong S_4 $ Let  $Q$ be the quaternion group where $|\operatorname{Aut}(Q)|=24$ and consists of elements  $\phi$ with $\phi(i)\in \left \{ \pm i, \pm j,\pm k \right \}$ and  $\phi(j)\in \left \{ \pm i, \pm j,\pm k \right \}\setminus \left \{ \pm\phi(i)\right \}$.  Prove that $\operatorname{Aut}(Q)$ contains no element of order $6$, and so $\operatorname{Aut}(Q)\simeq S_{4}$.  Any help is appreciated.  I have seen the other posts, but I am not sure how to approach this proof showing $\operatorname{Aut}(Q)$ contains no element of order $6$, then to conclude that it is isomorphic to $S_4$. There is a link in another post, that refers to no element of order $6$, but unable to be accessed.
Automorphism group of the quaternion group Let $Q_8$ be the quaternion group. How do we determine the automorphism group $Aut(Q_8)$ of $Q_8$ algebraically? I searched for this problem on internet. I found some geometric proofs that $Aut(Q_8)$ is isomorphic to the rotation group of a cube, hence it is isomorphic to the symmetric group $S_4$. I would like to know an algebraic proof that $Aut(Q_8)$ is isomorphic to $S_4$.
What does ;; do in sh? How to get all possible combinations of a list’s elements? I have a list with 15 numbers in, and I need to write some code that produces all 32,768 combinations of those numbers.   I've found  (by Googling) that apparently does what I'm looking for, but I found the code fairly opaque and am wary of using it. Plus I have a feeling there must be a more elegant solution.  The only thing that occurs to me would be to just loop through the decimal integers 1–32768 and convert those to binary, and use the binary representation as a filter to pick out the appropriate numbers.   Does anyone know of a better way? Using map(), maybe? Can we insert or update FieldSets using apex code? Although we can create or update fieldsets using salesforce UI but Is there any way to insert or update FieldSets using apex code (using metadata or API calls) ? Finding an isomorphic subring of matrices I'm struggling a fair amount with this exercise:     Find a subring of $M(2,\mathbb{Q})$ which is isomorphic to a) $\mathbb{Q}$ x $ \mathbb{Q}$ b) $\mathbb{Q}$ c) $\mathbb{Q}[x]$/$x^2$   Now I know a subring must be a subgroup, must contain the elements $0,1$ and must be closed under multiplication. As we are looking for isomorphisms then they must also be ring homomorphisms and must be bijective.  I tried to come up with a random subring and attempt to prove it is an isomorphism. E.g $\left\{\left.\begin{matrix}  a &amp; b \\ c &amp; 0  \end{matrix}\right|a,b,c \in \mathbb{Q}\right\}$ is a subgroup of $M(2,\mathbb{Q})$ and then try and show it's an isomorphism. However I'm stuck here as I don't actually know how to go about finding these specific isomorphisms and I completely lose what I'm doing. Any help would be great. How do I capture the milky way? Can the milky way actually be photographed ? I know the image is manipulated, and is probably a composite, but how do you capture such contrast in the milky way? Is an equitorially mounted scope required? Any advice or pointers are welcome, thanks.  I'm heading out towards much less light polluted areas middle of next week and I want to try some nighttime sky captures with my 7D.  Additional info: I just found  on wikipedia that says: 54s, tripod mounted 5D, 16mm lens, f/2.8, ISO800. What else do I need to be aware of? My 7D is only an APS-C and the widest lens I have is 19mm. I'm thinking that might not be wide enough to capture an interesting shot? How to import a module given the full path? How to do selective-color in Photoshop? How to make the single color in a photography to be color and other parts changed to B&amp;W?.  Don't know exactly the name of the effect, to be more clear in the following photo I want the green part to be green and other parts( leg and sandal) in B&amp;W..    How to achieve that effect in Photoshop? I have just updated MiKTeX and have found that pstricks now appears to cause a problem. The error message I'm getting is     ("C:\Program Files\MiKTeX 2.9\tex\generic\pstricks\pstricks.tex"      ! Undefined control sequence.      l.31 \if@check@engine   Not sure what's happening here. If I stop pstricks from being loaded things work ok.  Is there a known problem with the latest pstricks? Best way to parse RSS/Atom feeds with PHP I'm currently using  but it sometimes falls over when the RSS or Atom feed isn't well formed. Are there any other options for parsing RSS and Atom feeds with PHP? What novel had an isolationist Earth and the rest of humanity in a war of attrition against aliens? I was practicing with modeling a leaf using a image reference. Everything went fine and I finished modeling my leaf, but I wanted to make it more realistic (I'll add textures later), I saw lots of people using the Subdivision Surface modifier but it doesn't work for me.  My model:   My model when I add the Subdivision Surface modifier Scenario: I have a custom button on a standard layout, which will call an apex class to do some validation on a record. If not validated, it needs to throw out some errors on that page. What are the best UX options to achieve this?  I am not too sure how to achieve this. Within a trigger, you can use the addError method to add the custom errors.  Can my Apex class reference the fields of a standard object? The ideal approach would be to do the following:  Contact.Email.addError('Please enter the right format here');   How can I achieve this via an apex class that the custom button will call?  What I have thought: Having a VF Page popup on the screen so that user can see all the errors, but that's not the best approach as UX is a big deal for this application  Thanks. When I went to the review overview on Meta on 08.01.2015, 19:16, it listed two open items in the suggested-edits queue, however when I clicked on the queue, I was told that the queue had been cleared. When I went back to the review overview, however, the edits were there again.  Since the icons for recent reviews had not moved, I suppose that these were still the same two items waiting for review, and since phenomenon lasted for half an hour now, I would rule out  or  to be the reason for this.  After a while, the phenomenon ceased, with one new reviewer popping up in the icon list of recent reviewers.  I had not performed or skipped any suggested-edit review from the queue at that point.    A similar phenomenon occurred for several hours up to 11.04.2015, 11:05: The review overview showed two suggested edits and when I went to the suggested-edits queue, it was cleared. Sometimes, edits that I could review appeared, but two phantom edits remained. How is (p ∧ q ∧ ¬r) ∨ (p ∧ ¬q ∧ ¬r) ≡ (p ∧ ¬r) ∨ (¬q ∧ q)? Is it really distributive property? Is there a particular reason why vote times are omitted in the data dump?  IMO this could be a very useful data-set. Is this one of the steps that was done to anonymize the data, or does it simply not exist? This would be a very good addition to future dumps to help build time-based relationships, which are currently very rigid and full of assumptions. How to upgrade to MySQL 8.0? I have a Drupal site 8.6.10 on an Ubuntu server 18.04 with MySQL 5.7.25  I want to update MySQL to version 8.0  Here are the steps I followed :   I backed up my database. I uninstalled MySQL from my server with the commands :  $ sudo systemctl stop mysql $ sudo apt remove mysql-* $ sudo apt purge mysql-* $ sudo apt autoremove $ sudo dpkg -l | grep mysql | grep ii  I added the repository with the command :  $ wget https://repo.mysql.com//mysql-apt-config_0.8.12-1_all.deb  I installed the package with the command :  $ sudo dpkg -i mysql-apt-config_0.8.12-1_all.deb            My problem :   I install MySQL with the command :  $ sudo apt update $ sudo apt install mysql-server    When I do step 5, it installs MySQL 5.7 why does not it install version 8 ? How to Check If A Plugin Is Enabled Through API? In biblatex, treat periods in journal as abbreviation dots I would like to consistently have a comma between the journal and volume number for article references using biblatex with biber as the backend. Unfortunately, the punctuation between the journal and volume is not consistent. Journals whose last word is abbreviated have no comma between the journal and the volume, and the word "Vol." is capitalized. Journals whose last word is not abbreviated have a comma and the word "vol." is not capitalized. I can fix the capitalization by un-commenting the commented lines in the below MWE, but the comma problem is quite perplexing. Apologies if this fix is in an answer somewhere, but I cannot find it (or at least, I don't know what to search for!).  Current output: [1] First Author. “Blah Blah 2”. In: Am. J. Phys. Vol. 72, no. 3 (Mar. 2004), pp. 367–375. [2] Second Author. “Blah blah”. In: Energy Fuel, vol. 27, no. 12 (Dec. 2013), pp. 7778–7789.  Expected output: [1] First Author. “Blah Blah 2”. In: Am. J. Phys., vol. 72, no. 3 (Mar. 2004), pp. 367–375. [2] Second Author. “Blah blah”. In: Energy Fuel, vol. 27, no. 12 (Dec. 2013), pp. 7778–7789.  MWE:  \documentclass{article} \usepackage{filecontents} \usepackage[     backend=biber,     citestyle=numeric-comp, ]{biblatex} \addbibresource{\jobname} % \DefineBibliographyStrings{english}{%     % volume = {\lowercase{v}ol\adddot}, % avoid capitalization after dots. % } \DeclareFieldFormat[article]{volume}{\bibstring{volume}\addspace #1} \DeclareFieldFormat[article]{number}{\bibstring{number}\addspace #1} \renewbibmacro*{volume+number+eid}{%   \printfield{volume}%   \setunit{\addcomma\space}%&lt;---- was \setunit*{\adddot}%   \printfield{number}%   \setunit{\addcomma\space}%   \printfield{eid}} \renewbibmacro*{journal+issuetitle}{%   \usebibmacro{journal}%   \setunit*{\addcomma\addspace}%&lt;---- was \setunit*{\addspace}%   \iffieldundef{series}     {}     {\newunit      \printfield{series}%      \setunit{\addspace}}%   \usebibmacro{volume+number+eid}%   \setunit{\addspace}%   \usebibmacro{issue+date}%   \setunit{\addcolon\space}%   \usebibmacro{issue}%   \newunit}  \begin{filecontents*}{\jobname.bib} @article{York2004, author = {Author, First}, journal = {Am. J. Phys.}, month = mar, number = {3}, pages = {367--375}, title = {{Blah Blah 2}}, volume = {72}, year = {2004} } @article{Baumgardner2013a, author = {Author, Second}, journal = {Energy Fuel}, month = dec, number = {12}, pages = {7778--7789}, title = {{Blah blah}}, volume = {27}, year = {2013} } \end{filecontents*}  \begin{document} \nocite{*} \printbibliography \end{document} Google Docs thumbnails do not update, can I force an update? Learn Bayesian inference applied to astronomy / astrophysics?