Existence of finite indexed normal subgroup for a given finite indexed subgroup.
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
Is it possible to run Android apps on Ubuntu? And I don't mean inside an emulator, but natively, as you would with any other application installed using the Ubuntu Software Center. Allow user to show / hide sites in the android app Feed I am a registered user in ,  and several other sites, only because I had to ask 1 or 2 questions. In general, I have no interests in these sites, so I don't want to see them on my Feed. Is it possible to hide them somehow?   For now, I want to see only 4 sites (could be meta or non-meta) on my Feed. Notifications are not an issue, because I normally don't get any notifications from those not-interested-in-sites; I rarely participate there. I have spent a whole day trying to solve the following problem. Please help if you know how to do this.  I am using natbib (\usepackage[sort&amp;compress,square,comma]{natbib}), and my current citation style looks like [1-4], [5, 8-9].  What is the easiest way to change the citation style to [1]-[4] and [5],[8]-[9]? The new software center in Ubuntu 16.04 shows no application data found This is the error I get when I open the software center. I've tried rebooting and running sudo apt-get update but I see no improvement. It worked earlier but I don't know what went wrong since then. I'm using Ubuntu 16.04 final release on a Dell Inspiron 15R. Any advice will be appreciated. Thanks in advance.  EDIT: The problem magically solved itself without me doing anything special (or maybe inadvertently doing something, I don't know). So I'd still like to get to the root of this. Does it make a difference to CP if you feed candies before or after evolving? Postfix Connection Timed out Let $A$ be a $n \times n$  invertible  matrix.Define a function $F:\mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ by $F(x,y)=\langle Ax,y\rangle$. Let $DF(x,y)$ denotes the derivative of $F$ at $(x,y)$ which is a linear transformation from $\mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R} $. Then      1) if $x\neq 0$,then  $DF(x,0)\neq 0$      2) if $y\neq 0$,then  $DF(0,y)\neq 0$      3)  if $(x,y)\neq (0,0)$,then $DF(x,y)\neq 0$      4) if $x=0 \; or \;y=0$,then $DF(x,y)= 0$   I don't even know how to start the solution and also how to take derivative in higher dimensions.  Please help!  Thank you. How can I pad an integer with zeros on the left? Tor browser not opening I have the following problem illustrated by a simple example: If I cite the paper of an author T. Coven in my document, the outcome is 'Coven (2004b)'. If I cite the appendix to this paper, the outcome is 'Coven (2004a)'. Now the appendix pops up much less frequently than the original paper, therefore I would like to reverse the order (exchange 'a' and 'b' in the citations). Is this possible?  I am using Biblatex. Here is the bib-file of the above example (named: lib.bib):  @article{co:2004, author = {T. Coven}, title = {Title A}, journaltitle = {The X Journal}, date = {2004} } @article{coapp:2004, author = {T. Coven}, title = {Appendix to Title A}, journaltitle = {The X Journal}, date = {2004} }   MWE:  \documentclass[a4paper, 12pt, headsepline, headings=small,]{scrreprt} \usepackage[backend=bibtex8,     style=authoryear-icomp,     dashed=false,     autocite=footnote,     maxcitenames=3,     mincitenames=1,     maxbibnames=100,     firstinits=true,     sorting=nty     ]{biblatex} \bibliography{lib}  \begin{document} \cite{co:2004} and \cite{coapp:2004} \end{document} Comma after introductory words, phrases, clauses: unacceptable, obligatory or optional? What can you use Python generator functions for? I'm starting to learn Python and I've come across generator functions, those that have a yield statement in them.  I want to know what types of problems that these functions are really good at solving. Group of order 15 is abelian How do I prove group of order 15 is abelian?  Is there any general strategy to prove that a group of particular order(composite order) is abelian? Strange black shading. Cannot fully light mesh. (Cycles) Here is the viewport in material view.  The node setup is just a PNG texture into a diffuse shader.    And this is in rendered view.  I've tried using environmental lighting as well as a directional light, and while the lit parts become brighter/washed out, the darker parts stay dark.  Normals are correct and there is no z-fighting.  Ambient Occlusion is not enabled.  Changing to Flat shading makes it worse.    Update:  Any modifier that alters the mesh (bevel, subdivide, etc) make the problem go away instantly. How to compute moments of log normal distribution? The computed moments of log normal distribution can be found . How to compute them? How can I turn on the Intel HD manually? For some weeks my MacBook 6.2 does not switch to the integrated graphics adapter (Intel HD). This is especially crucial when I work on battery (which happens quite often). I guess the runtime could be enhanced when it would take the HD adapter for display.  I checked that it automatically changes the graphics mode. I am running Snow Leopard 10.6.8.  I recently installed 8GiB RAM (not from Apple). Could this probably related to this issue? Actually I do not want to downgrade or use the expensive Apple RAM.  So, how can I manually turn on the internal graphics adapter? Finding Radii of Convergence for $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ Setting: Let $\sum a_n z^n$ have radius of convergence $R$.  We have that  $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$  via Hadamard's formula for the radius of convergence.  Question: What are the radii of convergence for (i) $\sum a_n z^{2n}$ and (ii) $\sum a_n^2 z^n$?  Attempt for $\sum a_n^2 z^n$:   For ease of notation, let $R_1$ and $R_2$ denote the radii of convergence for power series $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ respectively. From Hadamard's formula, we then have  $$ R_2 = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n^2 \right|^{1/n}} = \left( \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} \right)^2 = R^2 $$   But what about $R_1$? Editing Subject in new Gmail For , the answer given on 7 Feb about how to edit the subject line in the new format of Gmail does not work for me.   There is no box on the left with a down arrow to options to reply, forward or edit subject in my version. There is a book on the right offering options to    Reply Reply to all Forward Filter message Print Delete Report spam Report phising Show original Message text garbled Translate message Mark answered    But, no option to edit subject How force Chrome to use google.com search engine without https Zombies - Why do they need to feed on living animals/humans?