I am trying to understand the best way to find the conjugacy classes of $S_3$. What I am doing to find these is first consider what elements would be in the center of the group $Z(S_3)$. Then, after that, I try finding the conjugacy classes for each element of $S_3$ that is not in the group. Namely, I try finding the conjugacy class of 2 cycles, so for $(12)$ and after that I try finding the conjugacy class for 3 cycles like $(123)$. Doing all the work of computing the conjugacies, I found taht the conjugacy class of $(12)$ is $\{(12),(13),(23)\}$ and the conjugacy class of $(123)$ is $\{(123),(132)\}$. Now my question is, is there a more efficient way to find these conjugacy classes, without computing all the conjugacies? For instance for the group $A4$, I think this approach is not efficient and definitely harder to apply. Thanks!
Is there a systematic way of finding the conjugacy class and centralizer of an element? Could the task be simplified if we are working with "special groups" such as $S_n$ or $A_n$? Are there any intuitive approaches? Thanks.
C state-machine design After enabling hibernate grub menu appears on start-up with 30s timeout My system is not dual boot just single OS UBUNTU 16.04. Prior to making the following changes, the Grub screen would not appear at start-up (OK).  After following these steps from  to enable hibernate in the system menu:   Open terminal and enter;   sudo nano /etc/polkit-1/localauthority/50-local.d/com.ubuntu.enable-hibernate.pkla  add the lines into the empty file;  [Re-enable hibernate by default in upower] Identity=unix-user:* Action=org.freedesktop.upower.hibernate ResultActive=yes  [Re-enable hibernate by default in logind] Identity=unix-user:* Action=org.freedesktop.login1.hibernate;org.freedesktop.login1.handle-hibernate-key;org.freedesktop.login1;org.freedesktop.login1.hibernate-multiple-sessions;org.freedesktop.login1.hibernate-ignore-inhibit ResultActive=yes  ^X then Y to save (a filename was suggested and I accepted without taking note).  Reboot for the changes to take effect.   hibernate now works and appears in the menu (OK) but whenever I reboot I now get a Grub menu at start up with a full 30 second timeout (NOK). I tried playing around with the Grub menu timeout settings but still get the full 30 sec timeout.  How do I stop the Grub menu from appearing and reduce my boot time to what it was before I made these changes.   Current Grub settings;   GRUB_DEFAULT="0" GRUB_HIDDEN_TIMEOUT="0" GRUB_HIDDEN_TIMEOUT_QUIET="true" GRUB_TIMEOUT="1" GRUB_DISTRIBUTOR="`lsb_release -i -s 2&gt; /dev/null || echo Debian`" GRUB_CMDLINE_LINUX_DEFAULT="quiet splash" GRUB_CMDLINE_LINUX="" Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime. Is it possible to get the information for a device tree using /sys of a running kernel? Commonly for arm systems, device trees supply hardware information to the kernel (Linux). These device trees exist as dts (device tree source) files that are compiled and loaded to the kernel. Problem is that I do not have access to such a dts file, not even to a dtb file.   I have access to /sys and /proc on the machine and I wanted to ask if that would allow me to "guess the correct values" to be used in a dts?  Also potential answer could highlight additionally the aspect if the answer to this question also depends on whether the device tree interface was used in the first place (i.e. a dtb was created and provided to the kernel) instead of some more hacking "we simply divert from vanilla and patch the kernel so as to solve the device information problem for our kernel only"-solution? Non-trivial cyclic group with unique generator Let $G$ be a non-trivial cyclic group with an unique generator. Show that $G$ has exactly two elements.   Let $g$ be a generator of $G$. My first idea to approach the problem was to show that if $|G|=n &gt; 2$ then there is an integer $k$ between 1 and $n$ such that $\langle g^k \rangle = G$ or, equivalently, $g \in \langle g^k \rangle$. My second idea was to analyse the problem in 2 separate cases. For the first one, if $n$ is prime, let $k$ be a positive integer then, by Lagrange's Theorem, $|g^k| \mid n$ and, therefore, $\langle g^k \rangle = n$. Hence, we must conclude that, if $n &gt; 2$ the generator cannot be unique. For the second case, where $n$ is composite, I was hoping to get to a contradiction, but could not advance any further. Is there a way to develop either of these ideas or is it necessary a new approach? New global preferences: keyboard shortcuts and sticky topbar Following a global preference for , as of now, the preferences for enabling keyboard shortcuts and disabling the fixed topbar are network-wide* as well.  If you had either checked off on any site in the network, that setting has been preserved and applied to the whole network for your account already. Future changes will also apply to all* Q&amp;A sites.  Find them in their usual spot as previously on the  pages.    * With the exception of , which will support the fixed topbar after the . I am having trouble with (a) and (b):  I evaluated (a) and I got that sun of the first three terms does not equal $\ln(5/8)$. I got that it is equal to $\ln(3/4) + \ln(2/3) + \ln(15/16)$ which is not equal to $\ln(5/8)$.  I am also having trouble with part (b). I cannot figure out how to calculate the math to cancel out the $(k+1)^2$ and turn it into $2(n  +1)$.  Any help would be amazing. How to loop through a HashMap in JSP? Show the series $a_n/(1+a_n)$ converges absolutely Fantasy book series with non-human races and labyrinth Creating a large amount of convex polygons using existing coordinates Is it possible to have two background images?  For instance, I'd like to have one image repeat across the top (repeat-x), and another repeat across the entire page (repeat), where the one across the entire page is behind the one which repeats across the top.  I've found that I can achieve the desired effect for two background images by setting the background of html and body:  html {     background: url(images/bg.png); }  body {     background: url(images/bgtop.png) repeat-x; }   Is this "good" CSS?  Is there a better method?  And what if I wanted three or more background images? What is a word that means to say in a positive sense? For instance, when writers say that a person was sexually assaulted instead of saying raped.  Or people use let go instead of fired.  What is that called? independent variables in Random forest model Shell command to sum integers, one per line? I am looking for a command that will accept (as input) multiple lines of text, each line containing a single integer, and output the sum of these integers.  As a bit of background, I have a log file which includes timing measurements. Through grepping for the relevant lines and a bit of sed reformatting I can list all of the timings in that file.  I would like to work out the total. I can pipe this intermediate output to any command in order to do the final sum.  I have always used expr in the past, but unless it runs in RPN mode I do not think it is going to cope with this (and even then it would be tricky).  How can I get the summation of integers? Delaunay triangulation to generate abstract networks and nodes Is it possible to use Delaunay triangulation to generate an image like the below from a raster or vector of an object (in this case, a human heart)? What software and techniques would be required to make something like this? My intention is to make and use something like this in Inkscape without having to draw all the nodes/lines/effects manually. Best compiler warning level for C/C++ compilers? Number after list name I'm sure this is a really simple setting I'm having trouble finding. If I delete a web part then add it again, there is a number in brackets after the title that goes up by 1 each time. How do I get rid of this? Windows 8.1 missing font files after restart An elementary proof of $\int_{0}^{1}\frac{\arctan x}{\sqrt{x(1-x^2)}}\,dx = \frac{1}{32}\sqrt{2\pi}\,\Gamma\left(\tfrac{1}{4}\right)^2$