Show that if G is a group with $$|G| = p ^ n,$$ where p is prime and $n ∈\mathbb N$, then Z (G) (i.e. the center of G) must have at least p elements.   It is not an assignment question I only saw a similar exercise in a book and I want to know how it would be solved.  I think it can be done with the Orbit-Stabiliser Theorem because each orbit has size a power of p, and the orbit sizes sum to $$p^n$$ So,the number of elements not in Z(G) is divisible by p, whence the number in Z(G) is also. But there is at least one element in Z(G), namely the identity; so there are at least p such elements.Now, if g is an element of order p in Z(G), then  is a normal subgroup of G of order p. I do not know if it is so, thanks in advance.  I would like to know how it would be using the class equation...
I know this is a very common corollary of the class equation. And I know how to do it by using class equation.   But can you do it bu using group action, maybe find a nice set for G to act on?
I have a MacBook Pro Retina (2015). After updating to OS Mojave, Bluetooth has stopped working. I get an error message that "No Bluetooth Hardware Found".  I have already tried to do the following:   Go to the /Library/Preferences and delete the com.apple.Bluetooth.plist file Successfully resetting SMC. (pressing Shift+ctrl+alt + power button) Removing the Bluetooth icon from the menu bar    None of the above solutions has worked for me. Question: Can anyone shed some light on this formula? I can't find any information on it.  It has three corollaries that I also need to understand: Install windows 7 through virtual box Convert binary to ASCII and vice versa Natbib In-Text Citation displays (author?) I have taken a look through many other questions on this subject but cannot seem to fix my issue. Essentially while references display correctly I cannot get the author citation to appear in my text despite running pdflatex followed by bibtex followed by two more runs of pdflatex. Any help would be appreciated.  Sample tex file:  \documentclass{article} \usepackage{natbib}  \begin{document}  Blahblahblah \citet*{Test12}.  \bibliographystyle{plain} \bibliography{refs}  \end{document}   refs.bib:  @article{Test12, author = "John Smith", title = "Test Article", year = "2012", journal = "The Test Journal" }   Output: Texture messes up after extruding a modified plane I want to make some 3d models of pixel art. I started by adding a plane, adding a texture to it, added loop cuts to my plane, 15 loop cuts because it's a 16x16 texture, then I selected the blank faces and deleted them. As I extrude this plane, A &gt; E &gt; $0.1$, the texture messes up on the edge of the item, as shown below  Although not all faces mess up on the edge, a lot still do. I tried recalculating normals, unwrapping the object again, but I ended with the same problem, no change at all. Though I may have done it wrong, since I never did it before. I tried smart UV project and cube projection, but they just mess it up even more. Does anyone have a solution to this? Unknown site on Android app When I select the unknown site (i.e. the meta sites) the app crashes.   I have a Samsung Note 5 and version 1.0.89 of the app. in a file containing :  ... 18-11-2018:othercharacters 10-11-2018:othercharacters 03-10-2018:othercharacters 30-10-2018:othercharacters 27-09-2018:othercharacters 03-12-2018:othercharacters ...   the command :  sort -t- -k2 -k1    does not sort by date, what am I missing ? Proof of properties of injective and surjective functions. I'd like to see if these proofs are correct/have them critiqued.     Let $g: A \to B$ and $f: B \to C$ be functions.  Then:   (a) If $g$ and $f$ are one-to-one, then $f \circ g$ is one-to-one.   (b) If $g$ and $f$ are onto, then $f \circ g$ is onto.   (c) If $f \circ g$ is one-to-one, then $f$ is one-to-one?   (d) If $f \circ g$ is one-to-one, then $g$ is one-to-one?   (e) If $f \circ g$ is onto, then $f$ is onto?   (f) If $f \circ g$ is onto, then $g$ is onto?     (a)  Let $a,b \in A$.  If $(f \circ g)(a)=(f \circ g)(b)$, then $f(g(a))=f(g(b))$.  Since $f$ is one-to-one, we know that $g(a)=g(b)$.  And, since $g$ is one-to-one is must be that $a=b$.  Hence $f \circ g$ is one-to-one.    (b) Since $f$ is surjective we know that for all $c \in C$ there is a $b\in B$ such that $f(b)=c$.  Since $g$ is surjective, there is a $a \in A$ such that $g(a)=b$.  Hence, for all $c \in C$ there is an $a \in A$ such that $(f \circ g)(a)=f(g(a))=f(b)=c$.    (c)  This is false.  Let $A=\{1\}$, $B=\{1,2\}$, and $C=\{1\}$. Define $g(1)=1$ and $f(1)=f(2)=1$. Then $f \circ g$ is injective, but $f$ is not injective.    (d)  This is true.  Let $a,b \in A$ where $a \neq b$.  If $(f\circ g)(a)\neq(f\circ g)(b)$, then $f(g(a))\neq f(g(b))$.  Hence $g(a) \neq g(b)$.  (e)  Since $f \circ g$ is surjective, then for every $ c \in C$ there is an $a \in A$ such that $f(g(a))=c$.   Let $b = g(a)$.  Then $b \in B$ and $f(b)=g(f(a))=c$.  Thus $f(b)=c$.  Hence $f$ is surjective.  (f)  Using the same set up as part (c), we have that $g$ is not surjective since there is nothing in $A$ such that $g$ will go to 1. I want the QED symbol to be 1 line lower at the end of proof Just as my title says, I would prefer to have my qed symbol to be 1 line lower than the last line of my proof. By default its on the same line and aligned to the right.   How can I achieve this?  EDIT: See below code for an example:  \documentclass[11pt,a4paper]{article} \usepackage[margin=3cm]{geometry} \usepackage[english]{babel} \usepackage{amsmath, amsthm} \begin{document}  \begin{proof}     Hello there good chap! I'll put in another line here just to be sure its clear what I mean. \end{proof}    \end{document} I've just made a minor alteration to a file from a TeX package.  I copied it into my local texmf tree and made the change there.  When I tried kpsewhich &lt;filename&gt; to test that it was finding my local version, it returned the original one and not the new one.  In actual fact, the original file was from my local texmf tree already so I renamed the old one something simple.  Now kpsewhich &lt;filename&gt; returned the system version of the file.  To check that the file was there, I tried kpsewhich -all &lt;filename&gt; and got the correct two versions of the file: my modified one and the system one (in that order).  If I try this with another file that I know exists in both places, then kpsewhich &lt;filename&gt; returns my local file.  Here's the various invocations and their results:  ~% kpsewhich pgfsys-pdftex.def /usr/local/texlive/2012/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def ~% kpsewhich -all pgfsys-pdftex.def  /Users/stacey/Library/texmf/tex/latex/bzr/pgfsys-pdftex.def /usr/local/texlive/2012/texmf-dist/tex/generic/pgf/systemlayer/pgfsys-pdftex.def ~% kpsewhich tqft.sty               /Users/stacey/Library/texmf/tex/latex/bzr/tqft.sty ~% kpsewhich -all tqft.sty /Users/stacey/Library/texmf/tex/latex/bzr/tqft.sty /usr/local/texlive/2012/texmf-dist/tex/latex/tqft/tqft.sty   Is it something special about the .def extension?  Is there some command that I forgot to run?  Are there some other options to kpsewhich that will help me debug this? Cancelled cut and paste, what happens? When there are X New Questions, the views don't update untill you refresh the page Tangent lines to ellipse from external point Import PLY doesn't display vertex colors I'm trying to import a .PLY mesh in Blender. I've troubles with colors: the mesh appear in a grayscale palette. Other softwares open it correctly (eg. MeshLab).  This is the header of the PLY file:  ply format binary_big_endian 1.0 comment created by Geomagic Sculpt Exporter. element vertex 182488 property float x property float y property float z property uchar red property uchar green property uchar blue element face 364980 property list uchar int vertex_indices end_header   Any tips? Thanks! Meaning of syntax $N(\mathbf{y} \mid \mathbf{0}, \mathbf{K})$ (multivariate normal distribution) Show that $e^3 > 20$ how to write in TeX "computer zero" with slanted vertical line on it? In TeX, how is computing zero sign written with slanted vertical line on it? "Next tag badge" panel score and answers counter discrepancy On the "Next tag badge" panel of  for C# tag, I see:      92/100 score   88/20 answers     But, when I open the "Select your next badge" panel, there's:     97/100 score   90/20 answers       I'm not sure which one is correct. But I believe that both displays should show the same counts.  The discrepancy is not due to a caching, I've noticed it the first time many weeks ago.  For these particular current counts: I have a single C# answer with 3 votes only in the last 5 days, last upvoted 13 hours ago (2015-10-12 19:30:05Z). So it should not cause the 5 votes and 2 answers difference.  I believe that the counters on the "Select your next badge" are correct and the counters on the "Next tag badge" are wrong, as I have:     Asked 1 non-wiki question with a total score of 6. Gave 90 non-wiki answers with a total score of 97.       Today (2015-10-16), the "Next tag badge" got bumped to 97/100. But "Select your next badge" shows 98/100 already. I am looking for a YA book about a boy who learns to use powers from a mind-adept, in a seemingly abandoned house. The adept helps him open his mind by using a cube. One of his powers, I believe, is to go into dreams of people that  he knows. He helps his young female neighbor friend by doing this, opening a door in her mind. He tries to open the mind-door of a local older bully, but finds out the young man can't be helped because he has been abused mentally and physically. I don't believe the book I read is past 1995. It might have the word "adept" in it.