We were asked to prove the following today:     Let $G$ be a group. Prove that if $G$ has order $2n$ for some odd integer $n$ greater than $1$, then $G$ contains a proper non-trivial  normal subgroup and cannot be a simple group.   We were told that Cayley's Theorem would be very beneficial to the proof but all I managed to get was to say that with a group $G$, we have a map $\varphi$ such that $$\varphi (g) = T_g $$ where $T_g$ is simply a function $G \to G$ where $T_g(x) = gx \ \ \ \forall x \in G$  Next I am told we need to show that $G$ definitely has an element of some order. Naturally I wanted to used LaGrange for this but I was told this isn't the correct approach. Searching for help after this I see that Cauchy's Theorem is used but we haven't covered that in class so we can't use that. Is there a way that I can continue from here using only Cayley's Theorem? As for a bit of reference, the exact proof of Cayley's Theorem we went over is "Every group is isomorphic to a group of permutations"  After proving we were supposed to use it to show that a group of order $216$ cannot be simple. Assuming the theorem is true, I still don't know how this can be, seeing that $216 = 2*108$. $108$ is an even number so does the theorem we are supposed to prove even play a role in this case?  This has been bugging me all morning and I'm not getting any further along looking at it by myself. So any help would greatly be appreciated.
Problem  Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. Show that $H$ is a proper normal subgroup of $G$.  So basically the result of this problem is that every group of order $2n$, with $n$ odd, is not simple.   The morphism $\phi$ can be seen as the action of $G$ on itself by left multiplication. Since $A_{2n} \lhd S_{2n}$, then $H=\phi^{-1}(A_{2n}) \lhd G$. I am stuck at showing that $H \neq \{1\},G$. All I could think of is that $G$ has an element $g$ of order $2$, then $\phi(g)$ is of order $1$ or $2$, but it is easy to see that $\phi(g)=1$ if and only if $g=1$, so $ord(\phi(g))=2$. I don't know which are the elements of order two in $S_{2n}$. I would appreciate some help to solve the problem, thanks in advance.
How to toggle Scroll Lock in Excel 2010 without a Scroll Lock key in the keyboard I have a notebook with no Scroll Lock key (or any shortcut using the 'Fn' key).  Somehow Excel 2010 got the Scroll Lock on. I see it left down in the status bar and now I can't navigate through cells using the arrow keys, which is pretty annoying. It wasn't turned on some hours ago, so I think I must have hit some Scroll Lock toggle shortcut in Excel 2010 by mistake.  Anyone knows what hot to turn off Scroll Lock in Excel (shortcut or menu option)? Travel in the Schengen area after my residence card was lost or stolen I have already booked my flight from Italy to the Netherlands through Ryanair. I'm staying in Italy legally and have a carta di soggiorno. However, after booking my flight, my wallet was stolen, which contains all my IDs except my passport. I filed a police report, and I'm trying to get a new residence card. However, I need to go to the Netherlands ASAP. Will Ryanair accept the police report and receipt of getting a new residence card from the post office?  UPDATE (17.07.2014)  Thanks for helping me out. BTW it's OK now. I went to our local CISL here in Florence, Italy, where they process your envelope for renewing your permesso di soggornio or carta di soggiorno (residence permit for foreigners) to be submitted to the post office. That's where you pay and get your postal receipt, and they make an appointment for you at the questura. Anyway, long story short, they processed a application for IMPROVISED residence permit which is good for 30 days to travel within the Schengen area, so all I have to do is go to the questura, bring my travel booking, the police report and copy of my passport. Now I can travel, and it's my first time to fly with Ryanair because I always use Transavia to go to Holland. =) Hope this helps for future persons who will have the same issue with lost or stolen residence permits...Just wish me luck that the questura will process my improvised residence permit immediately. How can I see flags I cast from all sites? I remember coming across this fact a while ago in a pdf somewhere, but I haven't been able to find it again. Can someone show me how to prove it? I would appreciate easier proofs.  EDIT: I'm very sorry, I forgot to actually state it.  Prove that $-1$ is not a quadratic residue modulo primes of the form $4k+3$.  Thanks! Probability expected value football problem The number of injuries at a college football game has the following probability distribution:      $$\begin{array}{|l:l|} \hline \textrm{Number of injuries} &amp; 0 &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\ \hdashline \textrm{Probability} &amp; 0.14 &amp; 0.33 &amp; 0.23 &amp; 0.17 &amp; 0.09 &amp; 0.04 \\ \hline \end{array}$$      The coach must file a report whenever there is more than one injury in any game.      a. Calculate the expected number of reports filed in the next 20 games.      b. Calculate the standard deviation of the number of reports filed in the next 20 games.      c. Calculate the expected number of football games to be played before the first game in which there are at least four injuries. How to configure logback to skip logging messages from org.package.* with all levels below WARN? Prevent iTunes from starting when iPhone/iPod/iPad is plugged in on Windows Is there any way to prevent iTunes from starting under XP when an iPhone is plugged in?  Using iTunes v 9.  Usually I just want to charge my phone and don't want iTunes to start.  Note: Exact  of this question - except this is for v9 on Windows.  The answer in the related question doesn't work for me. I have many tags marked as "interesting" and others as "ignored". (I use Java at work, I'm a Cocoa guy by night, any anything dealing with Windows/.NET/C# simply doesn't apply to me.) When tags are added to the list in the right navbar, they appear in the order in which they were added. There are several problems with this:   It can be difficult to visually locate a tag quickly, unless you have memorized its location. The tags appear as the same gray boxes like everywhere on SO, but with a small "x" afterward, and they wrap based on size. This tends to leave a ragged right margin and wasted space. Ordering based on when the tag was added is not always the most intuitive default. Various users might prefer to view them alphabetically, by popularity, recent activity, etc.   It would be nice to introduce a tabbed interface to switch between various orderings, and remember the value in the cookie, profile, etc. Coupled with this, perhaps users could specify a custom ordering that works best for them? (Currently the workaround is to copy all the tag names, remove them all, and re-add them in the desired order, but this is obviously annoying.)  Thoughts? Why does the logo not show up when signing up for a site and confirming the account? Was Voldemort allowed to visit Hogsmeade while in school? We know that in order for third-years and up to be allowed to visit the village of Hogsmeade, they must present a form duly signed by their parent or guardian. When Harry’s guardians, the Dursleys, miss their opportunity (!) to sign his, he is not allowed to go to Hogsmeade. Both Cornelius Fudge (as Minister for Magic) and Professor McGonagall (as head of Gryffindor House) refuse to sign the form on his guardians’ behalf—probably at least in part because they think there’s a mad killer on the loose looking for Harry, but also, as McGonagall specifically states, simply because:  “You heard what I said. No form, no visiting the village. That’s the rule. […] The form clearly states that the parent or guardian must give permission.” Prisoner of Azkaban, Bloomsbury/UK paperback, p. 113  Obviously, Tom Riddle did not have any parents or guardians ready to sign his form for him: his mother was dead, his father likely did not even wish to know of his existence, and if he did have a legal guardian at all (I don’t know how orphanages worked in Britain in the 1930s), it was likely to be  (warning: Wikia link) as the matron of Wool’s Orphanage where Riddle grew up. As Mrs. Cole was perfectly Muggle in every way and was never told by Dumbledore—nor, we must assume, by the boy himself—that Riddle was a wizard attending a wizardry school, it seems unlikely in the extreme that she ever signed Riddle’s form to visit Hogsmeade.1 I don’t recall any incidents in the books where Voldemort speaks specifically of him visiting Hogsmeade as a school boy (then again, why would he?); but being content to remain at the castle while his classmates have fun in the village seems quite out of character for him, too. Are there any (canonical) indications whether special provisions were made for orphans with no legal guardians to sign, so that they were allowed to visit Hogsmeade after all? Or, oppositely, whether such cases were played strictly by the book and unguardianed orphans, and thereby Riddle, were not allowed to visit Hogsmeade?    1 One might of course easily envision that Riddle simply Confounded Mrs. Cole to sign the form; but that would lead to detection of underage magic (assuming this was even a thing in 1940). Riddle may of course have forged Mrs. Cole’s signature, but that would be rather likely to be discovered as a forgery, since at least Dumbledore was aware that Mrs. Cole did not know of Riddle’s magical abilities. How can I set conditional formatting on a particular cell that depends on another cell's value? Why do I get "Required key not available" when install 3rd party kernel modules or after a kernel upgrade? This issue happens only on UEFI systems with enabled Secure Boot.  When I try to install DKMS modules like VirtualBox, Nvidia, or Broadcom drivers, they do not install and I get Required key not available when I try to modprobe them.  VirtualBox complains that vboxdrv is not loaded.  Broadcom wl driver is shown in lspci -k as a kernel module but is not in use. sudo modprobe wl throws Required key not available.  Also this issue may happen when I install some kernel modules from git sources.  This issue may appear after a kernel update as disabled wireless adapter, black screen after a reboot, etc.   How can I fix it? Bibtex true unsort Why is my profile messed up on a single site and how can I revert it? I was just asking a question on Gardening SE and suddenly my user name there is a25bedc5-3d09-41b8-82fb-ea6c353d75aand my picture is replaced by a Gravatar. My location is also left blank.  Here's a   Curiously, my profiles on all other sites are fine.  I don't have a separate login for SE, I log in using my Google account. Nobody other than me has access to my Google account. I use two step authentication and I can see on "recent devices" in Google that no one but me has accessed my Google account. I only access SE from home, and nobody other than me has used my devices.   I tried to edit my Gardening SE profile back to the way I want it to be and I got the following message:     Oops! There was a problem updating your profile:   Display name may only be changed once every 30 days; you may change again on Jun 20 at 10:49   I'm assuming this is because I changed the "about me" on all SE sites two days ago.  I never edit my profiles individually. I always apply changes to all sites, using the same profile on all sites.  I have certainly not made such strange (or any) edits to my profile on Gardening SE.  Why did this happen, and how can I get my proper user name and picture back on Gardening SE? Do I really have to wait for a whole month? I want to use the site, and I want them to recognize me as the user they know. I don't want to wait a whole month.  Update: The Gardening SE moderator Niall C. helped me out, and he offered the following potentially relevant information:     I changed your display name on Gardening &amp; Landscaping back to "Fiksdal". Your profile history shows that your profile image (from your headshot on imgur.com to a gravatar), email, about me also changed at the same time as your display name on May 21. It also shows that I changed your profile image to a different gravatar when I changed your display name just now... weird! – Niall C. 1 hour ago Can a negative charge ionize air? Adding to this question, I have studied that positive charge ionizes air but never studied that negative charge ionize air. If possible, please explain how does it ionize air? How to display a block with menu child items programmatically? Difference between LiveCD, LiveUSB, full-install, and persistence? I am currently running Ubuntu as a guest vm on a Win 7 host via VirtualBox, but I would like to have a portable version of Ubuntu when I am not using my laptop.   I am unsure as to the difference between the following:  1)LiveCD with persistence; 2)LiveCD without persistence; 3)LiveUSB with persistence; 4)LiveUSB without persistence; 5)Full-install.   I have a USB flash drive (32GB), and I also have empty DVDs lying around. I know that persistence means that data can be saved, but I am unsure as to which option to choose. What is the difference between a LiveUSB and a full-install on a USB thumbdrive and why would I want to pick one over the other? Thanks for any help. Finding the contact point with SAT How to use GROUP BY to concatenate strings in SQL Server? How do I get:  id       Name       Value 1          A          4 1          B          8 2          C          9   to  id          Column 1          A:4, B:8 2          C:9 When on the road as a tourist, I often do not carry a reflector of flash. Now I am at a location with very strong sunlight and I want to take a photo of a companion in front of a nice scenery or building, the classic "we have been here" photo.  With very strong sunlight however, specially at a slight angle, I either get it into the person's eyes and they squint like crazy, or they are in the shade and their whole face is black.  Please note that I do not have aspirations to take a professional photo in such circumstances, but at least something acceptable.  Is there an easy trick to deal with that?