Let $G$ be a finite group, and let $H$ be a subgroup of $G$ such that the index $(G\colon H)$ of $H$ in $G$ is the smallest prime that divides the order of $G$. Can we say anything about whether or not $H$ is normal in $G$?   The answer is of course yes if $G$ is abelian.
Generalizing the case $p=2$ we would like to know if the statement below is true.  Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.
This is probably pretty basic, but to save me an hour or so of grief can anyone tell me how you can work out the number of bits required to represent a given positive integer in Java?  e.g. I get a decimal 11, (1011). I need to get the answer, 4.  I figured if I could work out how to set all the bits other than the most significant bit to 0, and then >>> it, I'd get my answer. But... I can't. There exists $C\neq0$ with $CA=BC$ iff $A$ and $B$ have a common eigenvalue Question:  Suppose $V$ and $W$ are finite dimensional vector spaces over $\mathbb{C}$. $A$ is a linear transformation on $V$,  $B$ is a linear transformation on $W$. Then there exists a non-zero linear map $C:V\to W$ s.t. $CA=BC$ iff $A$ and $B$ have a same eigenvalue.  ===========This is incorrect==========  Clearly, if $CA=BC$, suppose $Ax=\lambda x$, then $B(Cx)=CAx=C(\lambda x)=\lambda (Cx)$, so $A$ and $B$ have same eigenvalue $\lambda$. On the other hand, if $A$ and $B$ have a same eigenvalue $\lambda$, suppose $Ax=\lambda x, By=\lambda y$. Define $C:V\to W$ s.t. $Cx=y$, then $BCx=By=\lambda y=C\lambda x=CAx$. But I don't kow how to make $BC=CA$ for all $x\in V$.   ====================================== How does 3-wire RTD or resistance measurement works in bridge of wheatstone? I cannot understand how 3-wire measurement will improve the resistance measurement.    They say that IF lead resistance \$ R_L \$ is the same, it will compensate. How can this happen?  If bridge is balanced (without \$ R_L \$):  \$ \frac {R_1}{R_2} = \frac {R_g}{R_3} \$  If \$ R_L \$ is added: \$ \frac {R1}{R2}= \frac {R_g+R_L}{R_3+R_L} \$  \$ \frac {R_g}{R_3} \$ is not same as \$ \frac {R_g+R_L}{R_3+R_L}\$; is this true only if \$ R_G=R_3 \$ in the beginning? So this means that \$R_L\$ does not get compensated? I have both Windows 7 and Ubuntu installed on a shared machine. Because a lot of the non-developers use Windows, I'd like to change the boot order to make it easier for them.  Currently the boot order looks like the following:   Ubuntu 11.10 kernelgeneric *86 Ubuntu 11.10 kernelgeneric *86 (safe boot)  Memory test Memory test Windows 7 on /dev/sda6   How do I change the default order so that Windows 7 is at the top of the list?   Windows 7 on /dev/sda6 Ubuntu 11.10 kernelgeneric *86 Ubuntu 11.10 kernelgeneric *86 (safe boot)  Memory test Memory test ++ is dangerous for C programmers How can I count unique selections from multiple selections in a drop down list (Excel) I would like to count the number of unique selections from a column that contains multiple selections from a drop down list.  For example, column B3 contains,     Monday, Tuesday, Wednesday   The count function returns a value of 1 instead of 3 - is there any way to count the three days distinctly? Thank you!  Multiple selections from a drop down list was made possible using the VBA code from: How can I get Gnome3 to remember my session? I am running Gnome3, but I had the same problem with Unity, on Ubuntu 12.04.  I have a fairly elaborate setup I enjoy using day-to-day. A LibreOffice spreadsheet with everyone's extension numbers; a window with reference data I use regularly; two mulit-tabbed Konsole/Terminal windows with several remote systems sessions, firefox with a set of open tabs, and so on. But whenever I log out, when I return, I must set up everything from scratch. I remember the good 'ol days, when I rebooted, I'd re-log in, and all my windows came back up in the same places as they were when I logged out, and my Konsole/Term would come up with the same remote logins as before.... I'd have to re -setup  some of my programs, but that's better than having to move every win into place, and resize them, etc.  How do I get this functionality back? Where does YouTube's offline feature store video files? I'm using YouTube offline feature on my Android device, so that I can playback the offline videos within the YouTube app even there is no internet connection. When I searched for any of the video files, I can't find them on device/SD card memory. Are they encrypting the file so that it can be played only with YouTube app? Does anyone know the specific folder for offline video data? I love recursion. I think it simplifies things a lot. Another may disagree; I think it also makes the code a lot easier to read. However, I've noticed that recursion is not used as much in languages such C# as they are in LISP (which by the way is my favorite language because of the recursion).   Does anybody know if there is any good reasons not use recursion in the languages such as C#? Is it more expensive than iteration? System.Drawing in Windows or ASP.NET services According to , it is not a particularly good idea to use classes within the System.Drawing namespace in a Windows Service or ASP.NET Service. Now I am developing a class library which might need to access this particular namespace (for measuring fonts) but it cannot be guaranteed that the host process is not a service.  Now there is a less optimal method I can fall back to if System.Drawing is unavailable, but if it is possible, I would rather use classes in System.Drawing. So what I would like to do is to determine at runtume if System.Drawing is safe or not, and if it is, use it, otherwise fall back to the suboptimal option.  My problem is: How could I possibly detect if System.Drawing is safe to use?  I think I should either   Detect if the current process is a Windows Service or ASP.NET service Detect if GDI is available Or maybe there is a way to ask System.Drawing.dll itself if it is safe to use   Unfortunately, I cannot come up with a way to implement any of these approaches. Does anyone have any idea? "invalid EFI file path" while booting Windows The kernel and range of the powers of a self-adjoint operator Suppose $L: V \rightarrow V$ self-adjoint (on a finite-dimensional space $V$); $b$ is a real number. Show   The minimal polynomial has distinct roots; $\ker(L) = \ker(L^k)$ for $k\geq1$; $\operatorname{im}(L) = \operatorname{im}(L^k)$ for $k\ge 1$; $\ker(L-bI) = \ker((L-bI)^k)$ for $k\geq1$;     My thoughts:  1) I think I'm supposed to assume root is repeated and reach a contradiction but I'm not sure how to proceed mathematically with that.  2-4) pretty sure there's some huge theorem that I'm missing. I've just been searching around the net for version control that has some support for graphic/Photoshop files. We're currently using BitBucket which is free and unlimited when it comes to file size but doesn't have any design review tools.  Here's what I've found. Does anyone else have any recommendations?  Pixelapse – Free/Beta, works well. Now has layer comp support.   Shipment – Private Beta (waiting for an invite)   LayerVault (out of business) – Windows/Mac, has layer comp support   PixelNovel – SVN Version control for Photoshop   CAD only?! But looks really good.   FileTrek – Very enterprise. No price or demo.   Kaleidoscopeapp – GIT with image compare   Apps that copy files with an incremented version Are the hoses in toilets in Asia something I should know about? I have noticed random bits of hose without proper fittings before I think for example in India and just assumed they are for use by the cleaners or for filling the water jug in .  But Now I'm travelling in Malaysia and Singapore and there are more western toilets than squat toilets in the places I've been yet proper hoses with proper fittings are very common in toilets:  (Hose next to a Western style toilet in my hostel in Chinatown, Kuala Lumpur.)  At one place in Singapore there were even signs on each cubicle door stating whether it contained a western or squat toilet, but also whether there was a hose! (I didn't take a photo of the sign, unfortunately)  This makes me think there is a use for the hose for clients, not just cleaners. But for what purpose? Should I know how to use it as a tourist? Please tell me it's not a kind of alternative to a bidet!  Possibly related is the fact that many toilet floors are awash with water. This can be a big problem if you're not wearing waterproof shoes or have to lower your trousers and keep them dry. Other than this toilets are pretty clean and I've seen both the hoses and the wetness in what I would regard as somewhat "fancy" places. What's the nice 'trick' to showing that the following expression is irrational?  $2^{1/3} + 2^{2/3}$ 0 ram free on ICS I'm new to the ICS os, coming from Gingerbread from another phone. The phone's quite nice and fast when I first used it, but just this morning it started exhibiting weird actions. Camera360 doesn't open up automatically when I click it, the Camera closes for no reason, Pocket restarts when an article has an image, and TTPod resets the song to the previous song on my widget in the middle of a new song I'm playing.  I checked out the ram, and and was surprised to see 500+ mb used, 0 mb available. I looked at the list of programs and I see the usual culprits like Facebook, but the ram they're using doesn't add up to 500+ mb. Yeah, I know that the whole ram indicated isn't usable by the apps, but I don't think I've done anything to warrant the 0 mb. I close my apps. There are times when there's free 4 to 5 mb, but I think that too shouldn't be the case. I tried installing Advance Task Killer, it listed less than 10 apps, killed it, still the ram is what it is above.   I did a reboot, and now it reads 300+ mb usage, 200 something free.   What can I do? How do I ensure an app's closed on ICS? I installed GoLaucher because it has a handy task killer and I can see the ram freed in an instant, but I would like to know if there's a better way to handle the RAM. I am using the Service Console in Lightning Experience, and using the Chatter Feed component on the Case layout to display updates to the Case.  If I use the "Log a Call" Quick Action from the Chatter publisher, these calls are displayed in the Chatter feed.  This is the desired behaviour.  However, if I insert a Log a Call record related to the Case any other way - e.g. using the Developer Console, or my CTI system's custom components - the call does not display in the feed.  If I query the record, it looks exactly the same as the record that is created with the Quick Action.  It just seems that the Chatter post is never made in this situation.  I have checked my Feed Tracking settings:   Task has Enable Feed Tracking and Subject checked Case has All Related Objects checked   How can I enable calls created with a custom component (part of a CTI system) to display in the case feed? Rename profile folder on Windows 7/8 Why does adding red light with blue light give purple light? Now that Unity is gone, the usual way of setting a Compose Key (via Keyboard Preferences > Layout) no longer works (because in Ubuntu 18.04, Devices > Keyboard has no Layout tab).  How do we set a Compose Key in Ubuntu 18.04 Bionic Beaver?