Let $G$ be a group with the following properties   G has a normal subgroup $N$ with $[G;N]=p$ where $p$ is a prime number; G has a subgroup $H$ such that $H$ is not contained in $N$   Prove   $$G=HN$$    attempt1  $\frac{|G|}{|N|} =p$ is a quotient group so $n=p|N|$   Theorem 8.7 hungerford $p$ is prime every group of order $p$ is cylcic and isomorphic to $Z_p$  call that $&lt;g_p&gt; =\frac{G}{N} \cong Z_p$  $$\frac{G}{N}=\{g_p, \dots , g_p^p\} $$  and   $$ \frac{G}{N} = g_1 N \cup \dots \cup g_p N= \cup ^{p}_{i=1} (g_p)^i N$$     lead 2  $K \not \geq H $ so $[G:HN]&lt;p$ and it divides $p$ so $$ \frac{|G|}{|HN|}=1$$ and the order of $|G|=|HN| $  that means that $HN=G$ since they have the same number of elements     need $HN$ to be normal?? anything wrong/corrections??
The full question: suppose N is normal in G and $[G:N]=p$ is a prime. Prove that for any $K \leq G$, either (1) $K \leq N$ or (2)$KN=G$ and $[K:K \cap N]=p$.  Since $N,K \leq G$, it is of course possible that $K \leq N$ could happen.  but for (2), we can show $KN \subseteq G$  but how can we show $KN=G$?  And what information does "$[G:N]=p$" give us?
Best WPF open source projects How does Mage Armor interact with Unarmored Defense? surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism Does there exist a surjective immersion $\mathbb{R}^2 \to \mathbb{R}^2$ which is not a diffeomorphism?  I tried to modify $\exp: \mathbb{C} \to \mathbb{C}$ to be surjective, but I find it hard to preserve the property of being immersion.  I also noticed that if we replace $\mathbb{R}^2$ with $S^2$ or any other compact simply connected manifold, no such immersion exists, because then it would be a covering map (as a local diffeomorphism from compact space, thus proper), but since the manifold is simply connected, it must be a diffeomorphism. this is my first question on PhysicsSE (I'm already an user of MathSE).   I'm a mathematics students trying to understand Faraday's law, that is   $$\varepsilon= -\frac{d \Phi_B}{dt}$$  where $\varepsilon$ means electromotive force and   $$\Phi_B=\iint \mathbf{B}\cdot d\mathbf{S}$$   means flux of magnetic field. As my textbook points out, there is an interpretation problem here: if the change in magnetic flux is due to movement of the conductor, then free charges in it are subject to Lorentz force, which then causes a current. On the contrary, if the conductor holds steady in a changing magnetic field, the induced current must be explained in terms of an electric field $\mathbf{E}$, described by the equations   $$\begin{cases} \nabla \cdot \mathbf{E}=\frac{\rho}{\varepsilon_0} \\ \nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t}\end{cases}.$$  Question Do those equations hold if we have a moving conductor in a stationary magnetic field? I guess not: this would mean $\mathbf{E}=\mathbf{0}$. How to solve this? Add review queues to Android app Left inverse implies right inverse in a finite ring Let $R$ be a finite ring with identity $1$, and assume $\exists x,y\in R$ such that $ xy=1$. How can I show it implies $yx=1$? Mortarboard badges being awarded as a result of bounties after recalc I am trying to write an if statement to test whether there are any files matching a certain pattern. If there is a text file in a directory it should run a given script.  My code currently:  if [ -f /*.txt ]; then ./script fi   Please give some ideas; I only want to run the script if there is a .txt in the directory. Filtered Review Queue doesn't show any audits What is the best way to generate a GUID/UUID from Apex code (such as a trigger)? Preferably in the following format:     nnnnnnnn-nnnn-nnnn-nnnnnnnnnnnnnnnnn   Example:     13219ec0-3a81-44c5-a300-de14b7d0235f How to get the IPV4 address for an interface from /proc Is there a clean, simple way to get an IP address for a network interface from /proc, similar to the way I can get the MAC address for a network interface? Ideally I would just type cat /proc/&lt;foo&gt;/{interface_name} and get the IPv4 address.  I'd rather not run anything other than cat. Can some one explain this terminology please I think I understand it but not completely sure.   I think pull down is where you place a resistor between +V and the other component and pull up is where you place the resistor between 0v and the component.    If I am completely wrong then let me know! Rigify generated rig, hand IK controller has unexpected rotation Have Kylo Ren's Force abilities been shown before? I'm only familiar with the movies, so in The Force Awakens I saw some use of the Force that was new to me.  Most notable were these two abilities:      freezing blaster bolts in mid-air, and directly reading someone's mind like he tried with Rey.   Have either of these happened before? Settings disappeared in Ubuntu 19.10 I usually go to the right top corner and at the drop down menu I select the tool icon to go to the settings.  However, 19.10 does not have a tool icon.  How can I go to settings now? For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous. This is from Apostol's Mathematical Analysis.  Let $f$ be a function defined in $[0, 1]$ such that for each real number $y$ either there is no value of $x \in [0, 1]$ such that $f(x) = y$ or there are exactly two such values of $x$. Prove that a) $f$ must be discontinuous on $[0, 1]$. b) $f$ has an infinite number of discontinuities in $[0, 1]$.  Long back I had tried to solve the problem. I reasoned for a) as follows: Suppose that $f$ is continuous and let $m, M$ be its minimum and maximum values on $[0, 1]$. Clearly if $m = M$ then $f$ is constant and clearly for every $x \in [0, 1]$ we have $f(x) = m = M$ which contradicts given property of $f$. So we must have $m &lt; M$. Clearly there are now four distinct points $a, b, c, d$ in $[0, 1]$ such that $f(a) = f(b) = m, f(c) = f(d) = M$. By checking possible linear orderings of $a, b, c, d$ I could find a find a value of $y \in (m, M)$ for which there exist more than 2 values of $x$ such that $f(x) = y$. However this approach looks very clumsy and I suspect much simpler and easier proof is possible. Part b) seemed bit complex and my only guess was that probably $f$ was discontinuous on each subinterval of $[0, 1]$ but I could not prove this claim (may be the claim is wrong!). Perhaps there is a simpler solution to this part of the problem. Apostol's also asks to provide an example of such a function. Any hints on the overall problem will be appreciated.  Update: After looking at Marc's answer, I think I have found a way to show that there are infinitely many discontinuities. Earlier the solution was a part of the question itself but it made the question look quite lengthy hence I moved it to a separate answer. Integration by partial fractions; how and why does it work? Could someone take me through the steps of decomposing  $$\frac{2x^2+11x}{x^2+11x+30}$$ into partial fractions?   More generally, how does one use partial fractions to compute integrals $$\int\frac{P(x)}{Q(x)}\,dx$$ of rational functions ($P(x)$ and $Q(x)$ are polynomials) ?    This question is asked in an effort to cut down on duplicates. See .  Also see . Leave bash script running on remote terminal while not logged in? I have a bash script that takes several hours to run.  While it's running, I would like to do other things, which may involve logging out or disconnecting from the internet (my script runs network tests on various computers).  I understand that there is a command that would allow me to run my tests from a remote terminal and logout of the terminal while it runs.  Does anyone know what this command is? Sometimes I need to upload, embed, or display inline an image in a question or answer. How can I do that? Upgraded to 13.10 now clock settings are all disabled and clock does not display I've upgraded to 13.10 and now I don't have the standard menu clock, which I need for work.  I checked and 'indicator-datetime' is installed.  I even uninstalled/reinstalled that applet with no luck.  Also my "clock" settings under System Preferences are all disabled.  I can't change anything.  Does anyone know how I can get the old menu clock back?  Alternatively is there another menu clock app I can download?  Edit 1  Thanks for the responses    I've restarted several times and that didn't fix the issue. I just ran the 13.10 update today.  But I ran it again a few minutes ago.  I got   about 200KB in random updates.  The issue is still present after a reboot. # apt-cache policy indicator-datetime indicator-datetime: Installed: 13.10.0+13.10.20131016.2-0ubuntu1 Candidate: 13.10.0+13.10.20131016.2-0ubuntu1 Version table: *** 13.10.0+13.10.20131016.2-0ubuntu1 0     500 http://us.archive.ubuntu.com/ubuntu/ saucy/main amd64 Packages     100 /var/lib/dpkg/status   Edit 2  I believe the issue is related to 'rarings' changing the systray somehow...   Or , although I haven't got that solution to work yet.  Edit 3  Because of a suggestion from RolandiXor I've check the binaries.  My system is missing 'libical.so.0' but it has 'libical.so.1'.    ldd /usr/lib/x86_64-linux-gnu/indicator-datetime-service libical.so.0 =&gt; not found   Does anyone know how to get the libical.so.0 binary that the datetime service needs but doesn't install?