This was a previously asked question that only had answers referring to Sylow's theorem, which is not something I can use here. Let $G$ be a group, $|G|=35.$ We know that $G$ must contain an element of order $5$ and an element of order $7$.  (1) Prove that $G$ must have a normal subgroup of order $5$ or of order $7$ or both.   (2) Deduce that $G$ is isomorphic to $C_{35}$.  (1) Attempt: It is shown earlier in the question that if $\Omega$ is the set of subgroups of $G$ isomorphic to a subgroup $H\leq G,$ and $G$ acts on $\Omega$ by conjugation, then $$\text{Stab}(H)=G \iff H\; \text{is a normal subgroup}.(*)$$ Assume for contradiction $G$ has no normal subgroups of order $5$ or $7$. Take $g_1,g_2$ as order $5$ and order $7$ respectively, then $\langle g_1\rangle$,$\langle g_2\rangle \leq G$ are not normal. Now by $(*)$ we can find an element in $G$ such that conjugating $\langle g_1\rangle$ by it gives a subgroup of $G$, isomorphic to but distinct from $\langle g_1\rangle$. By Lagrange's theorem then, and noting the intersection of subgroups is a subgroup, these two order $5$ subgroups must have trivial intersection Similarly we can find two order $7$ subgroups with trivial intersection. And it is clear too that all non-identity elements in all four of these subgroups, of which there are $20$, are distinct. This leaves $14$ non-identity elements outside of these subgroups, but I am stuck here. I think if $G$ has no order $35$ element then noting the only non-negative integer solution to $4p+6q=14$ is $p=2,q=1$ shows that we can distribute these elements into two new distinct order $5$ subgroups and one new order $7$ subgroup, but I can't see how this would help. I'm sure I can complete this as a counting argument but I'm not sure where to go.
If $|G| = pq$,  $p,q$ primes, $p \gt q, q \nmid p-1 $, then how do I prove $G$ is cyclic without using Sylow's theorems?
Sum and Product of Infinite Radicals Explanation behind Second Derivative of a Parametric Equation Formula How to add current date to cell if the rest of the cells (or some cells) in that row are edited, but only the first time? I try to get rounded corners on a UIImage, what I read so far, the easiest way is to use a mask images. For this I used code from TheElements iPhone Example and some image resize code I found. My problem is that resizedImage is always nil and I don't find the error...  - (UIImage *)imageByScalingProportionallyToSize:(CGSize)targetSize {     CGSize imageSize = [self size];     float width = imageSize.width;     float height = imageSize.height;      // scaleFactor will be the fraction that we'll     // use to adjust the size. For example, if we shrink     // an image by half, scaleFactor will be 0.5. the     // scaledWidth and scaledHeight will be the original,     // multiplied by the scaleFactor.     //     // IMPORTANT: the "targetHeight" is the size of the space     // we're drawing into. The "scaledHeight" is the height that     // the image actually is drawn at, once we take into     // account the ideal of maintaining proportions      float scaleFactor = 0.0;      float scaledWidth = targetSize.width;     float scaledHeight = targetSize.height;      CGPoint thumbnailPoint = CGPointMake(0,0);      // since not all images are square, we want to scale     // proportionately. To do this, we find the longest     // edge and use that as a guide.      if ( CGSizeEqualToSize(imageSize, targetSize) == NO )     {          // use the longeset edge as a guide. if the         // image is wider than tall, we'll figure out         // the scale factor by dividing it by the         // intended width. Otherwise, we'll use the         // height.          float widthFactor = targetSize.width / width;         float heightFactor = targetSize.height / height;          if ( widthFactor &lt; heightFactor )             scaleFactor = widthFactor;         else             scaleFactor = heightFactor;          // ex: 500 * 0.5 = 250 (newWidth)          scaledWidth = width * scaleFactor;         scaledHeight = height * scaleFactor;          // center the thumbnail in the frame. if         // wider than tall, we need to adjust the         // vertical drawing point (y axis)          if ( widthFactor &lt; heightFactor )             thumbnailPoint.y = (targetSize.height - scaledHeight) * 0.5;          else if ( widthFactor &gt; heightFactor )             thumbnailPoint.x = (targetSize.width - scaledWidth) * 0.5;     }       CGContextRef mainViewContentContext;     CGColorSpaceRef colorSpace;      colorSpace = CGColorSpaceCreateDeviceRGB();      // create a bitmap graphics context the size of the image     mainViewContentContext = CGBitmapContextCreate (NULL, targetSize.width, targetSize.height, 8, 0, colorSpace, kCGImageAlphaPremultipliedLast);      // free the rgb colorspace     CGColorSpaceRelease(colorSpace);          if (mainViewContentContext==NULL)         return NULL;      //CGContextSetFillColorWithColor(mainViewContentContext, [[UIColor whiteColor] CGColor]);     //CGContextFillRect(mainViewContentContext, CGRectMake(0, 0, targetSize.width, targetSize.height));      CGContextDrawImage(mainViewContentContext, CGRectMake(thumbnailPoint.x, thumbnailPoint.y, scaledWidth, scaledHeight), self.CGImage);      // Create CGImageRef of the main view bitmap content, and then     // release that bitmap context     CGImageRef mainViewContentBitmapContext = CGBitmapContextCreateImage(mainViewContentContext);     CGContextRelease(mainViewContentContext);      CGImageRef maskImage = [[UIImage imageNamed:@"Mask.png"] CGImage];      CGImageRef resizedImage = CGImageCreateWithMask(mainViewContentBitmapContext, maskImage);     CGImageRelease(mainViewContentBitmapContext);      // convert the finished resized image to a UIImage      UIImage *theImage = [UIImage imageWithCGImage:resizedImage];      // image is retained by the property setting above, so we can      // release the original     CGImageRelease(resizedImage);      // return the image     return theImage; } Meta SE account is shown 3 times in accounts Earlier today I asked my first question on this site. So Meta SE was added to my accounts. But this account is shown three times:        One of them also has a badge while the other two don't.   The network profile doesn't show MSE at all:        Also I have got this long wierd user name and  I am not able to change it. In the edit- display name this long name is not there.My user name on other sites is shown there.   After all this happened Meta SE questions are not getting loaded in my SE app.    Edit: the issue of name is solved now. I changed the profile picture and saved changes and it worked. Why is lock(this) {...} bad? The  says that public class SomeObject {   public void SomeOperation()   {     lock(this)     {       //Access instance variables     }   } }  is &quot;a problem if the instance can be accessed publicly&quot;. I'm wondering why? Is it because the lock will be held longer than necessary? Or is there some more insidious reason? Convert audio stream to WAV byte array in Java without temp file Given an InputStream called in which contains audio data in a compressed format (such as MP3 or OGG), I wish to create a byte array containing a WAV conversion of the input data. Unfortunately, if you try to do this, JavaSound hands you the following error:  java.io.IOException: stream length not specified   I managed to get it to work by writing the wav to a temporary file, then reading it back in, as shown below:  AudioInputStream source = AudioSystem.getAudioInputStream(new BufferedInputStream(in, 1024)); AudioInputStream pcm = AudioSystem.getAudioInputStream(AudioFormat.Encoding.PCM_SIGNED, source); AudioInputStream ulaw = AudioSystem.getAudioInputStream(AudioFormat.Encoding.ULAW, pcm); File tempFile = File.createTempFile("wav", "tmp"); AudioSystem.write(ulaw, AudioFileFormat.Type.WAVE, tempFile); // The fileToByteArray() method reads the file // into a byte array; omitted for brevity byte[] bytes = fileToByteArray(tempFile); tempFile.delete(); return bytes;   This is obviously less desirable. Is there a better way? polynomial converges uniformily to analytic function Force ssh to not print "Remote host identification has changed" warning Is there a way to avoid ssh printing warning messages like this?   @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ @    WARNING: REMOTE HOST IDENTIFICATION HAS CHANGED!     @ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@   Although the remote host identity has changed but I know it is fine and  just want to get rid of this warning. Permission Denied To Delete File First of all, I'm not looking for an answer on how to remove the files. I am looking for a reason why this doesn't work - and a suggestion on how to remove the files without sudo.  I have a directory in my Desktop that has a few inner directories (don't judge the naming - I just extract what I need from files I'm given) and after getting the files I try and remove the files with rm -rf Bengal\ Tiger\ Creative/, I am surprised to see the following warning for each image.     rm: cannot remove ‘Bengal Tiger Creative/Bengal Tiger Creative/images/customer-services-landscape.jpg’: Permission denied   Below is some code outputs.  tree  nathan@nathan-pc:~/Desktop$ tree . ├── Bengal Tiger Creative │   └── Bengal Tiger Creative │       └── images │           ├── awards.jpg │           ├── blank.gif │           ├── bullet1-light-green.gif │           ├── bullet2-light-green.gif │           ├── bullet3-light-green.gif │           ├── bullet4-light-green.gif │           ├── customer-services-landscape.jpg │           ├── desktop.ini │           ├── header-img1.jpg │           ├── header-img2.jpg │           ├── header-img4.jpg │           ├── header-img5.jpg │           ├── softhr.gif │           ├── Thumbs.db │           ├── tiger_main-img1.jpg │           ├── tiger_main-img2.jpg │           ├── tiger_main-img3.jpg │           ├── tiger_main-img4.jpg │           ├── Vanquis-benefits-icons_iphone-green.gif │           └── white-arrow-green.gif   ls -laR  nathan@nathan-pc:~/Desktop$ ls -laR Bengal* Bengal Tiger Creative: total 12 drwxrwxr-x 3 nathan nathan 4096 Aug 10 11:30 . drwxr-xr-x 6 nathan nathan 4096 Aug 10 11:34 .. drwxrwxr-x 3 nathan nathan 4096 Aug 10 11:34 Bengal Tiger Creative  Bengal Tiger Creative/Bengal Tiger Creative: total 12 drwxrwxr-x 3 nathan nathan 4096 Aug 10 11:34 . drwxrwxr-x 3 nathan nathan 4096 Aug 10 11:30 .. dr-xr-xr-x 2 nathan nathan 4096 Aug 10 11:36 images  Bengal Tiger Creative/Bengal Tiger Creative/images: total 356 dr-xr-xr-x 2 nathan nathan   4096 Aug 10 11:36 . drwxrwxr-x 3 nathan nathan   4096 Aug 10 11:34 .. -rwxrwxrwx 1 nathan nathan  14910 Mar 18 12:29 awards.jpg -rw-rw-r-- 1 nathan nathan   1093 Mar 21  2014 blank.gif -rw-rw-r-- 1 nathan nathan   1978 May  8  2014 bullet1-light-green.gif -rw-rw-r-- 1 nathan nathan   1969 May  8  2014 bullet2-light-green.gif -rw-rw-r-- 1 nathan nathan   2416 May  8  2014 bullet3-light-green.gif -rw-rw-r-- 1 nathan nathan   2449 May  8  2014 bullet4-light-green.gif -rw-rw-r-- 1 nathan nathan 113138 Apr 14  2014 customer-services-landscape.jpg -r--r--r-- 1 nathan nathan    142 Mar 25 11:58 desktop.ini -rw-rw-r-- 1 nathan nathan  14312 Mar 23 11:39 header-img1.jpg -rw-rw-r-- 1 nathan nathan   1480 Mar 23 11:41 header-img2.jpg -rw-rw-r-- 1 nathan nathan   1278 Mar 23 11:41 header-img4.jpg -rw-rw-r-- 1 nathan nathan   1168 Mar 23 11:41 header-img5.jpg -rw-rw-r-- 1 nathan nathan   1703 Mar 21  2014 softhr.gif -rw-rw-r-- 1 nathan nathan  88064 Aug  4 10:39 Thumbs.db -rw-rw-r-- 1 nathan nathan  23607 Mar 23 11:56 tiger_main-img1.jpg -rw-rw-r-- 1 nathan nathan  23400 Mar 23 11:56 tiger_main-img2.jpg -rw-rw-r-- 1 nathan nathan   6970 Mar 23 11:56 tiger_main-img3.jpg -rw-rw-r-- 1 nathan nathan   8996 Mar 23 11:57 tiger_main-img4.jpg -rw-rw-r-- 1 nathan nathan    275 Jul 31  2014 Vanquis-benefits-icons_iphone-green.gif -rw-rw-r-- 1 nathan nathan   1821 Jul  9  2014 white-arrow-green.gif   Doing chmod 777 and then delete does not work.  Doing rm -rf * inside the directory does not work.  Doing sudo rm file does work.   I can easily remove these files using sudo, however, I cannot see why they cannot be removed without sudo. As a relatively new Ubuntu user, I would like to learn why.  Thanks in advance, Nathan Does JavaScript have the interface type (such as Java's 'interface')? My brother owns Skyrim on steam. We use Steam Family Share, but we were wondering how both of us could play it at the same time? There is a sentence:     No sooner had he sat down than he fell asleep.   I just do not understand, is this an inversion? And if so, I still do not understand the sentence. Identify $P(2014)$ if $P(i)=1/i$ for every positive integer $1\le i\le2013$ Suppose $P(x)$ is a polynomial of degree $2012$ and $P(x) = 1/x$ when $x$ takes the integer values $1\cdots2013$ (inclusive). What is the value of $P(2014)$?   I get $1/1007$ but I'm not sure if it's right, and my method is very inelegant. Badge symbol in notification is of the site you're on, not where badge was earned Possessive form of "one of [a list]"? How to find continued fraction of pi I have always been amazed by the continued fractions for $\pi$. For example some continued fractions for pi are:  $\pi=[3:7,15,1,292,.....]$ and many others given .  Similarly some nice continued fractions for $e$ and it's derivatives are given .I have tried to prove that the are indeed the continued fractions but did not get very far.If any one can can help it would be great(especially if there is some easy way to get continued fraction of derivatives of e from the original continued fraction of e).Here derivatives of e I mean $\sqrt e,\frac{e-1}{e+1} $ etc. Classify all groups with order $pq$, $p$ is not equal to $q$ I am reading the following material:  . At page 5, there is a proof classifying all groups with order $pq$(they are all primes and $p$ is not equal to $q$). I am lost in the last two sentence, which claims the semidirect product are all isomorphic. Could someone help me to understand it? Thanks! Why was the universe in an extraordinarily low-entropy state right after the big bang? Let me start by saying that I have no scientific background whatsoever. I am very interested in science though and I'm currently enjoying Brian Greene's The Fabric of the Cosmos. I'm at chapter 7 and so far I understand most of general ideas he has talked about. Or at least, I think I understand them :-)  There is however one part, at the end of chapter 6 that I can't grasp.  It is about entropy and the state of the universe a few minutes after the big bang.  On page 171 he says:     Our most refined theories of the origin of the universe -our most refined cosmological theories- tell us that by the time the universe was a couple of minutes old, it was filled with a nearly uniform hot gas composed of roughly 75 percent hydrogen, 23 percent helium, and small amounts of deuterium and lithium. The essential point is that this gas filling the universe had extraordinarily low entropy.   And on page 173-174:     We have now come to the place where the buck finally stops. The ultimate source of order, of low entropy, must be the big bang itself. In its earliest moments, rather than being filled with gargantuan containers of entropy such as black holes, as we would expect from probabilistic considerations, for some reason the nascent universe was filled with a hot,   uniform, gaseous mixture of hydrogen and helium. Although this configuration has high entropy when densities are so low that we can ignore gravity, the situation is otherwise when gravity can't be ignored; then, such a uniform gas has extremely low entropy. In comparison with black holes, the diffuse, nearly uniform gas was in an extraordinarily low-entropy state.   In the first part of the chapter Brian Greene explains the concept of entropy with tossing the 693 pages of War and Peace in the air.  At first, the pages are ordered. The specific order they are in make sense and are required to recognize the pages as a readable book called War and Peace. This is low entropy. It is very highly ordered and there is no chaos.  Now, when you throw the pages in the air, let them fall and then pick them up one by one and put them on top of each other, the chances you get the exact same order as the initial state are extremely small. The chance you get another order (no matter what order, just not the one from the beginning) is extremely big. When the pages are in the wrong order, there is high entropy and a high amount of chaos. The pages are not ordered and when they are not ordered you would not notice the difference between one unordered state and another one.  However, should you swap two pages in the ordered, low entropy version, you would notice the difference.  So I understand low entropy as a highly ordered state with low chaos in which a reordering of the elements would be noticeable.  I hope I'm still correct here :-)  Now, what I don't understand is how a uniform mixture of hydrogen and helium can by highly ordered? I'd say you wouldn't notice it if some particles traded places. I'd say that a uniform mixture is actually in a state of high entropy because you wouldn't notice it if you swapped some hydrogen atoms.  Brian Greene explains this would indeed be the case when gravity plays no important role, but that things change when gravity does play a role; and in the universe right after the big bang, gravity plays a big role.  Is that because a reordering of the particles would change the effects of gravity? Or is there something else that I'm missing here? How to delete directories with path/names too long for normal delete Windows seems to have a length limit on file names when trying to delete, though it won't prevent those files from being created.  Our build process creates a number of temporary files (many build off of a WSDL) that run afoul of this limit.  Our ant script is somehow able to delete them when doing a clean, but sometimes I need to delete the workarea directory (where all the temp files go) without actually doing a full clean from ant.  This is the same errors , but the answers there don't really work for me as I'm dealing with a directory, not a file, and I don't always know what specific files or subdirectories are causing the problem.  And I'm trying to avoid any manual process (other than triggering a single command) to actually delete them.  If I try deleting the directory from Explorer I get the error   Cannot delete [file name]: The file name you specified is not valid or too long.   Specify a different file name   Trying Remove-Item in powershell gives the following error:  Remove-Item : The specified path, file name, or both are too long. The fully qualified file name must be less than 260 characters, and the directory name must be less than 248 characters. At line:1 char:12 + Remove-Item  &lt;&lt;&lt;&lt; -force -Recurse &lt;directory&gt;   Does anyone know of any tools or easy ways to get around this delete error without having to manually find the problem files and move/rename them?