Let $P$ be a $p$-group and let $N\triangleleft P$. Prove $N\cap Z(P)\ne e$ .  Here's what I know so far: $P$, being a $p$-group, is nilpotent and therefore is solvable. That means that it has its upper central series: $e=Z_0\triangleleft Z_1\triangleleft Z_2....Z_n...$ where $Z_1=Z(P)$ and that at some point, the series stabilizes at P forever... I also know that $Z_{i+1}=${ $x\in P|\forall y\in P: [x,y]\in Z_i$}. (In my notebook it's been defined that way: $Z_{i+1}=\pi^{-1}(Z(G$\ $Z_i)$ whichc confuses me a little. )  How can I show that the desired intersection is not empty?
I have a finite $p$-group $G$ and a normal subgroup $N$ which is not the trivial subgroup. I am asked to show that $|N \cap Z(G)| &gt; 1$.  There has been a similar question on MSE here:  Unfortunately, I cannot comment on other user's questions yet, so I made my own question.  I know that the trick is to use conjugation but I can't follow the logic. Can someone elaborate? Especially on the last part of the top answer.
Prove : $p$(n│even number of ODD parts)=$p$(n│distinct parts ,number of ODD parts is even ) What is a good complex analysis textbook, barring Ahlfors's? I'm out of college, and trying to self-learn complex analysis. I'm finding  text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.) Why does Linux require that a user be root/using sudo/specifically authorized per mount in order to mount something? It seems like the decision as to whether to allow a user to mount something should be based on their access rights to the source volume/network share and to the mount point. A couple of uses for non-root mounting are mounting file-system images to a user owned direction and mounting a network share to a user owned directory. It seems like if the user has control over both sides of the mount equation everything should be cool.  Clarification of access restriction:  I feel I should be able to mount anything that the user otherwise would have access to to a mount-point that the user is the owner of.  For instance, on my computer /dev/sda1 is owned by user root and group disk with permissions brw-rw----. Therefore non-root users can't mess with /dev/sda1 and clearly mount shouldn't allow them to mount it. However if the user owns /home/my_user/my_imagefile.img and mount point /home/my_user/my_image/ why shouldn't they be able to mount that image file on that mount point with:  mount /home/my_user/my_imagefile.img /home/my_user/my_image/ -o loop   As kormac pointed out there is a suid problem. So some restrictions would have to be added to prevent suid from being a problem as well as potentially some other issues. Perhaps one way to do this would be to make the OS treat all files as belonging to the user that did the mounting. However for simple read/write/execute, I don't see why this would be a problem.  Use case:  I have an account in a lab where my home space is restricted to 8GB. This is tiny and very very annoying. I would like to mount an nfs volume from my personal server to essentially increase the amount of room that I have. However, because Linux does not allow such things I'm stuck with scp'ing files back and forth to stay under the 8GB limit. Executing cmd.exe commands from Java Prove $\sin^2\theta + \cos^2\theta = 1$ How do you prove the following trigonometric identity: $$ \sin^2\theta+\cos^2\theta=1$$  I'm curious to know of the different ways of proving this depending on different characterizations of sine and cosine. I put some alias commands in my .bashrc file, so that they might be loaded everytime I open a new Terminal window. Yet this doesn't happen. I have to select run script: in the Terminal&gt;Preferences&gt;&quot;MyDefaultTheme&quot;&gt;Shell prefpane and add: source .bashrc &amp;&amp; clear for it to work... This seems odd since many tutorials only say you just have to add to the .bashrc file and all is good. Note that I don't run in bash when using the terminal, I like the other (default) one (don't know what it is) better, because it show me where I am all the time eg: &gt;&gt;d54c6b47b:~ romeo$  in stead of: &gt;&gt;bash$  In bash all is loaded as should be. So my question is, why didn't my .bashrc file load automatically and did I have to add the option to call it everytime? Also some tutorial told me to do something like: $alias la=’ls -la’ &gt;&gt; ~/.bashrc  which should write the alias to my .bashrc, this doesn't work either... Note that I'm a UNIX novice, so be gentle. Do Stack Exchange sites allow editing of comments and posts from other users? How can I use the "signature" function in Preview on Mac Mini with no camera? I have a Mac Mini with no attached camera. I would like to use the feature of Preview that allows you to insert a signature. This relies on taking an image of your signature using an isight/facetime/webcam. I do not have on.  I do have a MacBook pro on which I use the signature function. Is there a way to copy a file from the the MacBook Pro to the Mac Mini to enable this function on the Mini? How to prepare/augment images for neural network? I would like to use a neural network for image classification.  I'll start with pre-trained CaffeNet and train it for my application.    How should I prepare the input images?  In this case, all the images are of the same object but with variations (think: quality control).  They are at somewhat different scales/resolutions/distances/lighting conditions (and in many cases I don't know the scale).  Also, in each image there is an area (known) around the object of interest that should be ignored by the network.  I could (for example) crop the center of each image, which is guaranteed to contain a portion of the object of interest and none of the ignored area; but that seems like it would throw away information, and also the results wouldn't be really the same scale (maybe 1.5x variation).  Dataset augmentation  I've heard of creating more training data by random crop/mirror/etc, is there a standard method for this?  Any results on how much improvement it produces to classifier accuracy? it VS. this Vs. that What counts as an attack? I was looking over the class feature of the 'fiend' patron warlock in 5e. I'm a bit confused on what actions the player needs to take before they can use the 'Hurl Through Hell' ability on an enemy. The PHB states:     Starting at 14th level, when you hit a creature with an attack, you can use this feature to instantly transport the target through the lower planes.   I'm not clear on what an 'attack' means in this context. Does it have to be a physical attack like a sword or crossbow? Do magical attacks like Eldritch Blast that require an attack roll count? What about spells with no attack roll? All html canvas shapes are given the color of the last object added How come unpolarized light does not undergo destructive interference? Showing that $\Bbb R[x] / \langle x^2 + 1 \rangle$ is isomorphic to $\Bbb C$ question R: Trouble using SMOTE package "invalid 'labels'" Using SMOTE package from DMwR library. After loading the data frame, I try to perform sampling as follows :   crime_bal$target &lt;- as.factor(crime_bal$target) crime_bal &lt;- SMOTE(target ~ .,crime_bal,perc.under = 200, perc.over = 100)   But it always ends up to this error :   Error in factor(newCases[, a], levels = 1:nlevels(data[, a]), labels = levels(data[,  :    invalid 'labels'; length 0 should be 1 or 2 In addition: Warning messages: 1: NAs introduced by coercion  2: NAs introduced by coercion    Details of my dataset :   &gt; summary(crime_bal)      text               url            target    Length:6326        Length:6326        0:5994    Class :character   Class :character   1: 332    Mode  :character   Mode  :character   Why do I always end up with the error? If I attempt to change my password to nothing by opening "User Accounts", the "Change" button remains greyed out:    How do I change my password to be empty? I know you can set Ubuntu to automatically log you in, but I want my password to be empty, I never want to type in a password to authenticate myself as that user.  I know that there are reasons why this might not be a good idea, but I want to know if it is even possible. I'm using Ubuntu 12.10. Cardinality of cartesian square Given an infinite set $A$ -  does the cardinality of $A$ equal to the cardinality of $A^2$? I know that I can use something like cat test.txt | pr -w 80 to wrap lines to 80 characters  wide, but that puts a lot of space on the top and bottom of the printed lines and it does not work right on some systems  What's the best way to force a text file with long lines to be wrapped at a certain width?   Bonus points if you can keep it from breaking words. Google Chrome (Windows) not displaying text/encoding correctly on certain websites I experience a problem similar to    Here's a screenshot on the website     I've tried:   Removing fonts I have installed Setting the encoding to Western, UTF-8, Greek, Automatic Changing the fonts used in Google Chrome Removing Google Chrome, including all application data/user profiles Running CCleaner Reinstalling Google Chrome, twice Installing Chromium (fonts display fine in Chromium) Turning off Windows ClearType Changing Windows/Chrome languages from English (Australia) to US, UK Restarting Windows several times after making changes   All of which did not work. I do not connect over VPN/proxy.   Can anyone give me any tips?  Windows 7 x64 Ultimate Word for killing off or attempting to kill off an entire bloodline?