If $G$ is a finite $p$-group and $H\neq 1$ is normal in $G$, then $H\cap Z(G)\neq 1$ I know that there is an answer  but it uses composition series. My teacher gave me the following hint: $$G = Z(g)\cup (\cup G_{x_i})$$ and to see $H$ as $$|H| = |H\cap Z(G)| + \cdots$$ I think it has something to do with the class equation. If we let $G$ act on $H$ by conjugation, I know that the class equation becomes: $$|H| = |\{h | ghg^{-1}=h\}| + \sum_{i}|\{gh_ig^{-1}| g\in G\}| = \\ |\{h \ | \ gh =  hg\}| + \sum_{i}|\{gh_ig^{-1}| g\in G\}| = Z(G) + \sum_{i}|G_{x_i}| = Z(G) + \sum_{i}(G:G_{x_i})$$ (the last equality I'm just guessing... Shouldn't it have something to do with $H$?) I also feel that the conclusion should follow from $|H|$ being $&gt;1$ and divisibility, but I'm not quite sure. Could somebody help me and also verify if my class equation is right?
If $G$ is a finite $p$-group with a nontrivial normal subgroup $H$, then the intersection of $H$ and the center of $G$ is not trivial.
When using Alt+Tab, I can't seem to get back to Skyrim. When clicking its icon or Alt+Tab to get back, I just see an empty window. Windowed mode doesn't allow me to use it in a Windowed (Fullscreen) manner as seen with other games, so that doesn't seem to be an option either. Is there any way to fix this behavior? How do I synchronise two folders in real-time in Windows 7? I want to synchronise two folders in real time under Windows 7.  Basically, I want to monitor a folder and synchronise each change (new files, changed files, deleted files) to another drive. It has to be in real time, so it gets synchronised instantly when a change happens. A one-direction synchronisation is enough.  I tried Microsoft's , but it does only syncing by hand or scheduled (thus not satisfying the "real-time" requirement).  Can this be achieved with Windows 7 itself? If not, are there other methods of doing this? I've just made a set of armor by using a male body mesh model I downloaded and selecting certain faces from the model and duplicating them to create viable armor pieces.   Since the pieces are just duplicated from the same mesh male model, then it's still part of the same object.   How do I separate the armor piece from the model to make the pieces separate? Closed-loop function question How do I find the   How do I find the transfer function of Gd(s)?  I am able to find the transfer function of Gr(s) but I am not really sure about Gd(s)? and how do i determine which order is this closed system? Unable to upgrade pip Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1 How does one prove that a simple (steps of length $1$ in directions parallel to the axes) symmetric (each possible direction is equally likely) random walk in $1$ or $2$ dimensions returns to the origin with probability $1$?  Edit: note that while returning to the origin is guaranteed $(p = 1)$ in $1$ and $2$ dimensions, it is not guaranteed in higher dimensions; this means that something in a correct justification for the $1$- or $2$-d case must fail to extend to $3$-d (or fail when the probability for each direction drops from $\frac14$ to $\frac16$). Where is attribute table in ArcGIS Desktop when not visible? I've just updated to ArcGIS 10.1 for Desktop and when I go to open an attribute table, I can not see it anywhere.   I'm wondering if it is off my screen somewhere or if its just super tiny that I can't see it.   I'm using dual monitors and it's not popping up on either.  Is there a way I can find my attribute table? Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$? $\{\emptyset\}$ is a set containing the empty set. Is $\{\emptyset\}$ a subset of $\{\{\emptyset\}\}$?  My hypothesis is yes by looking at the form of "the superset $\{\{\emptyset\}\}$" which contains "the subset $\{\emptyset\}$". The output from uname:  root@debian:~ # uname -a Linux 5asnb 2.6.32-5-amd64 #1 SMP Mon Jun 13 05:49:32 UTC 2011 x86_64 GNU/Linux   However the /sbin/init executable shows up as 32-bit:  root@debian:~ # file /sbin/init /sbin/init: ELF 32-bit LSB executable, Intel 80386, version 1 (SYSV), dynamically linked (uses shared libs), for GNU/Linux 2.6.18, stripped   Other aspects of the system seem to contradict things as well:  root@debian:~ # echo $HOSTTYPE i486  root@debian:~ # getconf LONG_BIT 32 I read this book once when I was in middle school. A girl solves a murder or a mystery on a spaceship. It was a thin paperback books.   All I can remember is that there was an alien, and there was a movie star or actress or famous makeup artist on the ship traveling with another woman. By the end of the book the murderer was the woman the actress was traveling with who forced her to use her makeup skills to disguise her, and she was really some kind of alien. Matrix exponential for Jordan canonical form Let $X$ be a real $n \times n$ matrix, then there is a Jordan decomposition such that $X = D+N$ where $D$ is diagonalisable and $N$ is nilpotent.  Then, I was wondering whether the following is correct.  $e^{tX}(x) = \sum_{k=0}^{m} \frac{t^k N^k}{k!} \left(e^{t \lambda_1} \alpha_1 v_1+..+e^{t \lambda_n} \alpha_n v_n \right).$  Here $x = \sum_{i=1}^{n } \alpha_i v_i$ and $v_i$ are the eigenvectors of the diagonalisable matrix, $\lambda_i$ are the eigenvalues of $D$ and $m$ is the degree up to which $N^k$ is still non-zero.  Is this correct or am I doing something wrong? Cause I could not find a general equation for this matrix exponential, so I tried my best. (Thus, I am only asking for a verification or correction of this answer.)  If anything is unclear, please let me know. So, through typing several commands I've found that there's not only ls, but l and la too. There doesn't appear to be any man entries on Ubuntu 12.14. They all appear to do similar things with minor differences:  $ ls app        config     CONTRIBUTING.md  doc       Gemfile       Guardfile  LICENSE  MAINTENANCE.md  Procfile  Rakefile   script  tmp     VERSION CHANGELOG  config.ru  db               features  Gemfile.lock  lib        log      PROCESS.md      public    README.md  spec    vendor $ la app      CHANGELOG  config.ru        db   features  Gemfile       .git        Guardfile  LICENSE  MAINTENANCE.md  Procfile  Rakefile   .rspec  .secret     spec  .travis.yml  VERSION .bundle  config     CONTRIBUTING.md  doc  .foreman  Gemfile.lock  .gitignore  lib        log      PROCESS.md      public    README.md  script  .simplecov  tmp   vendor $ l app/       config/    CONTRIBUTING.md  doc/       Gemfile       Guardfile  LICENSE  MAINTENANCE.md  Procfile  Rakefile   script/  tmp/     VERSION CHANGELOG  config.ru  db/              features/  Gemfile.lock  lib/       log/     PROCESS.md      public/   README.md  spec/    vendor/   Just as a bit of trivia, are there more of these and what do they do? Is here any place to find this out? Unfortunately, google searching these commands gets ignored because they're so short. NSArray becomes NSCFArray when passed Complement of maximal multiplicative set is a prime ideal Let $R$ be a commutative ring with identity. I've been trying to prove the following:     If $S \subset R$ is a maximal multiplicative set, then $R \setminus S$ is a prime ideal of $R$.   I have spent some time in it, but nothing is coming to my mind. Any idea/solution will be appreciated. I have some Python code that makes Raster Attribute Tables for GeoTiffs using GDAL.    In ArcGIS I can see these attribute tables fine and they behave how I like (as classifications) but in QGIS the whole concept of an attribute table seems to be missing.    Is there any way to access a RAT in QGIS? I want to georeference jpg map without any digitizing data Recently the cranks on my bike started to creak. I checked with my usual "bike-bible" . They recommended dis-assemble, lubricate, re-assemble.  So I did that.  Remove cranks. Remove spindle. Clean. Lubricate. Reassemble.  No, the creak is still there.  (To find out the source-area of the sound I had a friend riding the bike slowly while I ran alongside listening. And the sound where coming from the area of the bottom bracket/cranks. I am happy no neighbor saw me...)   For each revelation of the cranks there are two creak-noises.  If I, when on the bike, push down with both feet I hear a noise. Turn 180 degrees without pressure, push down again and there is the noise again. If I am "too soft", low pressure there are no sound. Since I have clips I have tried to only work with one foot to keep a constant load during a revelation, there where no detectable sound then.  Anyone have any idea what is wrong on the bike? And how I should fix it?  /Leif How could I write my name, for example, to look like the \LaTeX logo? How could I write my name, for example, in a \LaTeX style? It is just interesting, as in Google all other things pop up when I try to search for an answer! Closed form for $\int_0^\infty\frac{1}{(1+x^2)^s}\,dx$ when $s\in (0.5,\infty)\setminus\mathbb{N}$ AirPort Time Capsule wifi is suddenly very slow: Boom network load