Suppose that G is a group and N &lt; G a normal subgroup. Assume that there is no normal subgroup M &lt; G with N &lt; M. Prove that G/N is simple.  I am not really 100% sure where to start here. I feel like I may be missing something obvious but I'm not really sure. I know based off the information that N is the smallest normal subgroup of G, and that simple groups' normal subgroups are only the identity and themselves, so I can see the connection, I just can't see HOW to actually connect them
Suppose that G is a group and N &lt; G a normal subgroup. Assume that there is no normal subgroup M &lt; G with N &lt; M. Prove that G/N is simple.  This problem is really throwing me for a loop. The concept of quotient groups and how they relate to simple subgroups is confusing me, and so I am not really sure how to begin. If a simple subgroup's only normal subgroup is itself (and the trivial group), then G/N being simple would be proving that when multiplying the entire subgroup N by an element g for all g in G, that the collection of those sets cannot be broken into smaller normal subgroups?
$-1$ is a quadratic residue modulo $p$ if and only if $p\equiv 1\pmod{4}$ Let, $D=\{z\in \mathbb C:|z|&lt;1\}$. Which are correct?   there exists a holomorphic function $f:D \to D$ with $f(0)=0$ &amp; $f'(0)=2$. there exists a holomorphic function $f:D \to D$ with $f\left(\dfrac{3}{4}\right)=\dfrac{3}{4}$ &amp; $f'\left(\dfrac{2}{3}\right)=\dfrac{3}{4}$. there exists a holomorphic function $f:D \to D$ with $f\left(\dfrac{3}{4}\right)=-\dfrac{3}{4}$ &amp; $f'\left(\dfrac{3}{4}\right)=-\dfrac{3}{4}$ there exists a holomorphic function $f:D \to D$ with $f\left(\dfrac{1}{2}\right)=-\dfrac{1}{2}$ &amp; $f'\left(\dfrac{1}{4}\right)=1$.   With the help of Schwarz lemma &amp; its applications, we find that $(1)$ is false &amp; $(3)$ is true.  But, I can not think about options $(2)$ &amp; $(4)$. How can I check dependency list for a deb package What would be an appropriate opposite of "exceed"? When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The question is:     When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?   The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution?  Edit  I should add that $f:\mathbb R\to\mathbb R$ to the question. I am getting a Too many SOQL queries: 101 error on a trigger when I am trying to deploy another trigger.  I am posting the trigger which is causing the error.  trigger rollupAccountShowings on Account (before insert, before update) {    if (trigger.isinsert) {       for (Account a:Trigger.new){           a.No_of_Viewings__c = 0;        }    } else {      for (account a : [select                   id,                  (select id from pb__Showings__r)                  from account                   where id in:trigger.new LIMIT 1]) {           trigger.newMap.get(a.id).No_of_Viewings__c = a.pb__Showings__r.size();      } }   Can anyone correct me where I have gone wrong and suggest me the changes.  Thanks Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$. I can't seem to find a good way to solve this.  I tried using L'Hopitals, but the derivative of $\log(n!)$ is really ugly. I know that the answer is 1, but I do not know why the answer is one.  Any simple way to go about this? Let p be a prime. Consider the equation $\frac1x+\frac1y=\frac1p$. What are the solutions? Write down the set of distinct solutions and prove your list is complete. $x$ and $y$ are positive integers.  I have rewritten it as $\frac{(x+y)}{xy} = \frac1p$, but I don't understand where to go from here? failed to get D-Bus connection: Operation not permitted I'm trying to list services on my CentOS image running in Docker using  systemctl list-units     but I get this error message:  Failed to get D-Bus connection: Operation not permitted   Any suggestions what the problem might be? I would like to make some changes to the core toolbar module across all my sites, and I don't want to modify the actual core's css.  It looks like I should be able to override the core css found in modules/toolbar/toolbar.css by simply copying this file into my sites directory at /sites/modules/toolbar/toolbar.css.  I tried doing this, cleared the Drupal cache, but nothing happened.  What am I missing? I would like to convert tab to spaces in gVim. I added the following line to my _vimrc:  set tabstop=2   It works to stop at two spaces but it still looks like one tab key is inserted (I tried to use the h key to count spaces afterwards).  I'm not sure what should I do to make gVim convert tabs to spaces? The meaning of imaginary time What is imaginary (or complex) time? I was reading about Hawking's wave function of the universe and this topic came up. If imaginary mass and similar imaginary quantities do not make sense in physics, why should imaginary (or complex) time make sense?    By imaginary I mean a multiple of $i$, and by complex I mean having a real and an imaginary part, i.e., $\alpha + i\beta$, where $\alpha, \beta \in {\mathbb R}$. This is a homework problem for my real analysis class. It is problem 13.1 from Mattuck's Introdution to Analysis. The question reads:       Suppose F(x) is continuous on some open interval I and c is a maximum point inside this interval. Is it true that f(x) must be increasing immediately to the left of c and decreasing immediately to the right of c? Proof or counterexample. (Note: A constant function is considered to be both increasing and decreasing.)      My friend and I have been attempting to find a counterexample. We figured a function that begins oscillating infinitely many times near c but converges to c would work. We're having difficulty constructing such a function out of elementary functions so we think we will have to define it in another way, piece-wise using receding intervals. I have no clue how this would look. Any advice would be greatly appreciated. How do "pet" Walkers help ward off other Walkers? In both the comics and the show, Michonne first appears with two walkers that she essentially keeps as pets.    The comics sort of half-explain this - as per :     She notices walkers don't attack one another and hacks off the arms and lower jaws of Mike and Terry. She chains them up by their necks to use them as escorts for an escape that was successful.   In , we see that even if they still sort of smell like Walkers, the Walkers can smell/detect the living and will attack.  In ,      We see Andrea effectively making a new "pet" so that she can use to deter the Walkers that she comes across on her trip to the prison. Along the trip, she is shown to come within close range of Walkers, but they give her space when she points the "pets" in the right direction.   Why would the Walkers not just attempt to go around the "pets"? What about them creates an effective protective barrier? How to delete a file from Time Machine backups in Big Sur? Inset in Blender not coming out even Surjective morphism of affine varieties and dimension I'm trying to do the following exercise from the book An Invitation to Algebraic Geometry:     Show that if $X \to Y$ is a surjective morphism of affine algebraic varieties, then the dimension of $X$ is at least as large as the dimension of $Y$.   Could anyone give me a hint? I'm looking for a proof that avoids commutative algebra since it is not mentionned in the book at this point. I develop a windows application and am curious whether it works via Wine. The application runs fine with Wine but I get no sound. In fact, I'm not sure whether my ubuntu installation has MIDI capabilities. The sound output settings shows "LFE on Separate Mono Output" one with "no amplifier" and another with "amplifier". They both function. I am using Ubuntu 12.04 as a client on virtualbox that runs on windows 7.  Thanks for any suggestion!  EDIT: virtual box hosted by windows 7, ubuntu running as a client in virtualbox. Unable to install any updates through update manager/apt-get upgrade I have been running Ubuntu 12.04 for the past few weeks. I recently tried to download and install some files through the Update Manager. It located a few files, downloaded them, but then failed to install them. It gave the error:  installArchives() failed: (Reading database ...  (Reading database ... 80% dpkg: unrecoverable fatal error, aborting:  reading files list for package 'libasn1-8-heimdal': Input/output error   I tried using apt-get upgrade but a similar error was produced:  Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be upgraded:   icedtea-6-jre-cacao icedtea-6-jre-jamvm openjdk-6-jre openjdk-6-jre-headless   openjdk-6-jre-lib 5 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Need to get 0 B/34.6 MB of archives. After this operation, 1,024 B of additional disk space will be used. Do you want to continue [Y/n]? y (Reading database ... 80% dpkg: unrecoverable fatal error, aborting:  reading files list for package 'libasn1-8-heimdal': Input/output error E: Sub-process /usr/bin/dpkg returned an error code (2)   Any help would be greatly appreciated. Thanks in advance.  Edit: Here's some extra information that may be useful. My original problem was that the Ubuntu Software Centre would not open. I found that the Update Manager and Synaptic Package Manager also failed to open. They all gave an error along the lines of:   "installArchives() failed: dpkg: error: reading package info file '/var/lib/dpkg/available': Input/output errorError in function:"   After much googling I seemed to fix that problem as the package managers would open again. When I then tried to download and install any updates from the Update Manager I got the error at the beginning of this post. Calling PHP functions within HEREDOC strings In PHP, the HEREDOC string declarations are really useful for outputting a block of html.  You can have it parse in variables just by prefixing them with $, but for more complicated syntax (like $var[2][3]), you have to put your expression inside {} braces.  In PHP 5, it is possible to actually make function calls within {} braces inside a HEREDOC string, but you have to go through a bit of work.  The function name itself has to be stored in a variable, and you have to call it like it is a dynamically-named function.  For example:  $fn = 'testfunction'; function testfunction() { return 'ok'; } $string = &lt;&lt;&lt; heredoc plain text and now a function: {$fn()} heredoc;   As you can see, this is a bit more messy than just:  $string = &lt;&lt;&lt; heredoc plain text and now a function: {testfunction()} heredoc;   There are other ways besides the first code example, such as breaking out of the HEREDOC to call the function, or reversing the issue and doing something like:  ?&gt; &lt;!-- directly output html and only breaking into php for the function --&gt; plain text and now a function: &lt;?PHP print testfunction(); ?&gt;   The latter has the disadvantage that the output is directly put into the output stream (unless I'm using output buffering), which might not be what I want.  So, the essence of my question is: is there a more elegant way to approach this?  Edit based on responses: It certainly does seem like some kind of template engine would make my life much easier, but it would require me basically invert my usual PHP style.  Not that that's a bad thing, but it explains my inertia..  I'm up for figuring out ways to make life easier though, so I'm looking into templates now.