I am basically not able to understand how to start.  Can someone help me to see the picture (probably draw diagrams of cosets) instead of giving some formal proof.I want to see as to why this is happening
This is not for homework. (I am a grader for a class.)  The case in which $G$ is finite is trivial. (That is, use a corollary to Lagrange's Theorem, and set $[G:H] = \dfrac{|G|}{|H|}$, and similarly for $[H:K]$.) How do you prove this for when $G$ is infinite?   The proof of this statement that I know well tries to show that $[G:K]$ is finite iff $[G:H]$ and $[H:K]$ are finite, but I am not a fan of that proof. For pedagogical purposes, I'm looking for a better proof. (The course uses right cosets by convention.)  The course has just started covering normal subgroups, so things like the Isomorphism Theorems are not known yet.
Differential equation: autonomous system This isn't homework. I have no idea what theorems I should be looking at to solve this. Guidance, partial and total solutions are all welcomed.     Let $f$ be a locally lipschitz function in an open set $G\subseteq \Bbb R^n$. Consider the autonomous system $y'=f(y)$.      Let $y(x,\xi)$ be the value at point $x$ of the maximal solution that satisfies the initial condition $y(0)=\xi$.         Prove that the domain of $y(\cdot, y(s, \xi))$ is $I-s$, where $I$ is the domain of $y(\cdot ,\xi)$.   Prove that for all $s, t$ such that $y(s, \xi)$ and $y(t+s, \xi)$ exist, then $y(t,y(s,\xi))$ also exists and $y(t,y(s,\xi))=y(t+s, \xi)$.   If $y$ is a maximal solution and there exists $T&gt;0$ such that $y(0)=y(T)$ and $f(y(0))\neq 0$, then $y$ is a periodic solution and not constant.   If $y$ is a solution whose domain is $(a,+\infty)$, if $\eta:=\lim _{x\to +\infty}y(x)$ and $\eta \in G$, then $f(\eta)=0$.      EDIT  I found an alternative solution for 3. Please check my proof and give feedback in comments:  let $u$ be the restriction of $y$ to $[0,T]$, now let $\overline u$ be the periodic extension of $u$ to $\mathbb R$. It is easy to see that $\overline u$ is a solution to the given differential equation. But since $f$ is locally lipschitz, $\overline u$ must coincide with $y$ wherever they are both defined. Since $y$ is a maximal solution, it must be $\overline u$, so $y$ is defined on $\mathbb R$. I am asked to solve this ODE: $$yy''-y'y^2-(y')^2=0.$$ Can I get some guidelines on how I can solve it? I am not sure how to approach without initial conditions. Is there a gender-neutral word for "aunts and uncles"/"parents' siblings"? Removing control chars (including console codes / colours) from script output I can use the "script" command to record an interactive session at the command line.  However, this includes all control characters and colour codes.  I can remove control characters (like backspace) with "col -b", but I can't find a simple way to remove the colour codes.  Note that I want to use the command line in the normal way, so don't want to disable colours there - I just want to remove them from the script output.  Also, I know can play around and try find a regexp to fix things up, but I am hoping there is a simpler (and more reliable - what if there's a code I don't know about when I develop the regexp?) solution.  To show the problem:   spl62 tmp: script Script started, file is typescript spl62 lepl: ls add-licence.sed  build-example.sh  commit-test         push-docs.sh add-licence.sh   build.sh          delete-licence.sed  setup.py asn              build-test.sh     delete-licence.sh   src build-doc.sh     clean             doc-src             test.ini spl62 lepl: exit Script done, file is typescript spl62 tmp: cat -v typescript Script started on Thu 09 Jun 2011 09:47:27 AM CLT spl62 lepl: ls^M ^[[0m^[[00madd-licence.sed^[[0m  ^[[00;32mbuild-example.sh^[[0m  ^[[00mcommit-test^[[0m         ^[[00;32mpush-docs.sh^[[0m^M ^[[00;32madd-licence.sh^[[0m   ^[[00;32mbuild.sh^[[0m          ^[[00mdelete-licence.sed^[[0m  ^[[00msetup.py^[[0m^M ^[[01;34masn^[[0m              ^[[00;32mbuild-test.sh^[[0m     ^[[00;32mdelete-licence.sh^[[0m   ^[[01;34msrc^[[0m^M ^[[00;32mbuild-doc.sh^[[0m     ^[[00;32mclean^[[0m             ^[[01;34mdoc-src^[[0m             ^[[00mtest.ini^[[0m^M spl62 lepl: exit^M  Script done on Thu 09 Jun 2011 09:47:29 AM CLT spl62 tmp: col -b &lt; typescript  Script started on Thu 09 Jun 2011 09:47:27 AM CLT spl62 lepl: ls 0m00madd-licence.sed0m  00;32mbuild-example.sh0m  00mcommit-test0m         00;32mpush-docs.sh0m 00;32madd-licence.sh0m   00;32mbuild.sh0m          00mdelete-licence.sed0m  00msetup.py0m 01;34masn0m              00;32mbuild-test.sh0m     00;32mdelete-licence.sh0m   01;34msrc0m 00;32mbuild-doc.sh0m     00;32mclean0m             01;34mdoc-src0m             00mtest.ini0m spl62 lepl: exit  Script done on Thu 09 Jun 2011 09:47:29 AM CLT Show that, in any finite graph with at least two vertices, there must be two vertices with the same degree.  HINTS ONLY! How can I manually install a package on MiKTeX (Windows) I'm new to LaTeX, investigating using it for some work projects.  I'm using MiKTeX on Windows.  My employer's locked-down network blocks the application's automatic installation function.  I can take my laptop home and successfully install from there, but if I need a package in the middle of the day I'm stuck.  I am able to access the CTAN website and download the package files (.dtx or .ins?), but I don't know what do do with them.  How can I do a manual package installation? How to remove title entry in the bibliography contents? I use plain bibliography style for my book. When I compile it, by default, I get the details including the title in my reference. like this:     H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, a. Dinola, and J. R. Haak. Molecular dynamics with coupling to an external bath. J. Chem. Phys., 81:3685-3690,1984.   This is too long and I have more than 100 such references which makes the document too long. How can I force it to remove the title from this reference? something like this:     H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, a. Dinola, and J. R. Haak.  J. Chem. Phys., 81:3685-3690,1984.   Your answer will be greatly appreciated.  By the way, I still would like to use the plain bibliography style. How can a single slit diffraction produce an interference pattern? How can a light passed though a single slit produce a similar interference pattern to the double-slit experiment? How does the diffracted wave produce the points of cancellation and reinforcement, if there is only one wave? HTTP GET request in JavaScript? I need to do an  request in JavaScript. What's the best way to do that?  I need to do this in a Mac OS X dashcode widget. Virtual box ubuntu: Internet not detected on guest(Ubuntu) in host(winXp) Geocoding USA addresses that cannot be sent over internet? For a one-time project, I need to geocode a few thousand addresses.  In the past I've used various online resources for this kind of thing (e.g., Google Maps API), but the addresses I'm working with have to be kept confidential - which means no sending it over the Internet, unless there's some iron-clad guarantee of privacy.  What other options do I have? Create a shortcut for URL? Problem: James has a pile of n stones for some positive integer n ≥ 2. At each step, he   chooses one pile of stones and splits it into two smaller piles and writes the product   of the new pile sizes on the board. He repeats this process until every pile is exactly   one stone.      For example, if James has n = 12 stones, he could split that pile into a pile of size   4 and another pile of size 8. James would then write the number 4 · 8 = 32 on the   board. He then decides to split the pile of 4 stones into a pile with 1 stone and a   pile with 3 stones and writes 1 · 3 = 3 on the board. He continues this way until he   has 12 piles with one stone each.      Prove that no matter how James splits the piles (starting with a single pile of n   stones), the sum of the numbers on the blackboard at the end of the procedure is   always the same.   Hint: First figure out what the formula for the final sum will be. Then prove it using strong induction.  The formula I came up with is $n(n-1) / $2. It could be totally wrong...  My (Partial) Solution:  1) Base Case: (n=2) James has one pile of 2 stones. Suppose he takes the top-most stone and puts in one pile A, which now has size 1. He takes the remaining stone and places it into another pile B, which now has size 1. Then the product of the sizes is 1. The sum of all the products on his board is 1. Now, if he were to start again, and take the bottom stone first and put it in pile A, he has a pile of size 1, and then takes the top stone and puts it in pile B, he has a pile of size 1, and the product is 1, with sum of all products on the board is 1.   2) Inductive Hypothesis (Strong Induction): Suppose for some $k \geq 2$, $n$ stones can be split in $ 2 \leq n \leq k $ stones and $k-n$ stones.  3) Inductive Step:  Consider $n=k+1$. What do I do now? Could someone advise me on how to determine all entire functions $f$ such that $|f'(z)| \leq |f(z)|, \forall z\ ?$  Hints will suffice, thank you. Active vs passive voice with inanimate object as the subject I have asked the  in ELL Stackexchange forum. There all the answers and comments I received was really helpful, but still I was looking for something more. So I have decided to ask this same question here also.   I have seen some sentences like this -         The film was announced to be released on this coming Friday.   The film was announced to release on this coming Friday.      Both the sentences are used, and I know they are interchangeable and does bear the same meaning. But I have some doubt about the second sentence - why is it "to release", when the film cant release itself.   Some other sentences of similar structure that arises my confusion -         The product sells well.   The book reads well.   The glass broke.      The sentences are understandable, but in each of the sentences the subjects can't perform any job, as they all are inanimate object. The verbs are active, but they acts like they are used as passive.  Now my question is how to decide which verb to use this way and which not? Comma before "and" in compound sentences? I read this sentence on an educational website:     Now times have changed and you are ready for situations involving forces in two dimensions.   Shouldn't there be a comma before and, since the subject of the sentence has changed (things vs. you)?      Now times have changed, and you are ready for situations involving forces in two dimensions.   What is the exact comma rule for joining two clauses with and? Prove that $ S=\{0\}\cup\left(\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}\right)$ is a compact set in $\mathbb{R}$, but $\bigcup_{n=0}^{\infty} \{\frac{1}{n}\}$ is not a compact set.   (Can we use definition of compact set for this problem? ) JSON.parse: unexpected character Four tasks in parallel... how do I do that? I have a bunch of PNG images on a directory. I have an application called pngout that I run to compress these images. This application is called by a script I did. The problem is that this script does one at a time, something like this:  FILES=(./*.png) for f in  "${FILES[@]}" do         echo "Processing $f file..."         # take action on each file. $f store current file name         ./pngout -s0 $f R${f/\.\//} done   Processing just one file at a time, takes a lot of time. After running this app, I see that the CPU is just 10%. So I discovered that I can divide these files in 4 batches, put each batch in a directory and fire 4, from four terminal windows, four processes, so I have four instances of my script, at the same time, processing those images and the job takes 1/4 of the time.  The second problem is that I lost time dividing the images and batches and copying the script to four directories, open 4 terminal windows, bla bla...  How do that with one script, without having to divide anything?  I mean two things: first how do I from a bash script, fire a process to the background ? (just add &amp; to the end?) Second: how do I stop sending tasks to the background after sending the fourth tasks and put the script to wait until the tasks end? I mean, just sending a new task to the background as one tasks end, keeping always 4 tasks in parallel? if I do not do that the loop will fire zillions of tasks to the background and the CPU will clog. How do you easily horizontally center a  using CSS? I'm trying to horizontally center a &lt;div&gt; block element on a page and have it set to a minimum width. What is the simplest way to do this? I want the &lt;div&gt; element to be inline with rest of my page. I'll try to draw an example:  page text page text page text page text page text page text page text page text                -------                | div |                ------- page text page text page text page text page text page text page text page text