I'm trying to prove that an image of a self-normalizing subgroup of a finite group under a surjective homomorphism is again self-normalizing in the target group. I proved that if the following is true then the statement above is also true. If H is a self-normalizing subgroup of a finite group G and N is a normal subgroup of G, then NH is also a self-normalizing subgroup of G. However, I couldn't figure out why this holds. Also, these two statement is used in some papers such as &quot;Nilpotent self-normalizing subgroups of soluble groups&quot;, but I couldn't find the reference that contains the proof of them. Can anyone help?
Let $f:G\to K$ be a surjective group homomorphism and $H$ be a subgroup of $G$ satisfying $N_G(H)=H$. Then $N_K(f(H))=f(H)$. I'm trying to prove or disprove it. perhaps, this proposition is true. But, how can we prove it? Edit: I thought it is true because WIKI said it is true without proof
How to change the light grey text that many web sites seem to use these days? Of late, I've noticed that many web sites seem to be using light grey text for many sections of their sites. Some such sections include - the text (link, usually) that shows in the status bar at the bottom of the browser when you hover the mouse over a link on their page, the indicator text in form fields to be filled in (e.g. the word "email" in the email address field), and many other elements on the page.  That light grey makes it much less readable.    Not sure whether is a new trend, or something to do with my browser settings (I haven't changed them lately, AFAIK), or Google Chrome, which is my browser. This is on Windows 7 running on a Dell Vostro laptop. I added the info about the laptop because I suspect the problem may even be due to my hardware settings, since I found the default brightness to be too high after I bought it, and changed it via Control Panel.  Would like to know what are some of the things I can check or change to solve this.  Thanks. I use LINQ to Objects instructions on an ordered array.  Which operations shouldn't I do to be sure the order of the array is not changed? Unable to install Cisco Packet Tracer 7.3 on Ubuntu 20.04 I am trying to install Packet Tracer 7.3 (latest version) on Ubuntu 20.04 but do not succeed to so so because of two dependencies which are not installable ("libdouble-conversion1" and "qt-at-spi").  However I successfully installed the previous version (7.2.2) which was available in an Ubuntu installer format (.run file) but I nothing happens when I run it from my gnome desktop.  I am quite a newbie in Linux so I appologies if answers are obvious...  Thanks in advance ;-)  Denis Page number position I am working on a long document in LaTeX with documentclass book. I need the page number to always be in the upper right corner of each page, even if that page is the first page of a chapter (right now on 1st pages of chapters, the page number is bottom-centered, on all other pages it's top-right).  I control the position of the page number with fancyhdr:  \usepackage{fancyhdr} \pagestyle{fancy} \lhead{} \chead{} \rhead{\thepage} \lfoot{} \cfoot{} \rfoot{} \renewcommand{\headrulewidth}{0pt}   Also, I don't know if the problem is related but my chapters do not start from the top of the page. There is a white area, then comes chapter X, then a newline with the chapter line. What I also want is the chapter to start from the top of the page.  The main question here is how I can get the page number to always appear in the upper right corner, I mention the thing with the chapter title position only in case that might be related. If you are root, and you issue   rm -rf /    Then how far can the command go? Can you recover data from this kind of an action? Even after the binaries are gone, would the running processes still be active? What would it take to make the same physical machine boot again? What files would you need to restore to make this happen?  I could try this on a VM and see, but I want to know the rationale behind what to expect if I do this. Automatically Locate Included Images ok, ive a class and i pass an object as property.  the object that i pass is a List&lt;X&gt;  in my class im trying to access the Object index by reflection BUT I CAN'T!!!   Example:  this class works i just wrote down the part i want to show you and i need help.  class MyClass {     private object _recordSet;     public object RecordSet     {         get { return _recordSet; }         set { _recordSet = value; }     }      public string Draw()     {         system.reflection.Assembly asem = system.reflection.Assembly.getAssembly(_dataSource.GetType());          object instance;          instance = asem.CreateInstance(_dataSource.GetType().UnderlyingSystemType.FullName);          //to access de Count of my List         int recordcount = int.Parse(_dataSource.GetType().GetProperty("Count").GetValue(_dataSource,null));          //i need to do a          for(int cont = 0; cont &lt; recordCount; cont++)         {             _dataSource[cont].Name; // &lt;-- THIS PART IS NOT WORKING!!! because i cant access the Index Directly.... WHAT TO DO!! ???         }     } } After the therapy, eight children (43%) became able to crawl/move on their back.   Or should I use "on their backs"?   Singular because each child only has one back, or plural because we're dealing with eight backs? Normalizing the final weights vector in the upper bound on the Perceptron's convergence The convergence of the "simple"  says that:  $$k\leqslant \left ( \frac{R\left \| \bar{\theta} \right \|}{\gamma } \right )^{2}$$  where $k$ is the number of iterations (in which the weights are updated), $R$ is the maximum distance of a sample from the origin, $\bar{\theta}$ is the final weights vector, and  $\gamma$ is the smallest distance from $\bar{\theta}$ to a sample (= the margin of hyperplane).  Many books implicitly say that  $\left \| \bar{\theta} \right \|$ is equal to 1. But why do they normalize it ? What is the purpose of -e in sed command? I can't find any documentation about the sed -e switch, for simple replace, do I need it?  e.g.  sed 's/foo/bar/'   VS  sed -e 's/foo/bar/' Is there a way to solve this integral?  $$\int \frac{dx}{\sqrt{-ax^2 + bx +c}}, \;\;\;\;\; \forall\; a,b,c \in (0,+\infty) \;in\;R$$  I have tried it by changing of variable and by parts, but no result. I don't know what else to do.  Thanks in advance. Prove that, for $p > 3$, $(\frac 3p) = 1$ when $p \equiv 1,11 \pmod{12}$ and $(\frac 3p) = -1$ when $p \equiv 5,7 \pmod{12}$ $(\frac 3p)$ is the Legendre symbol here, not a fraction, if that wasn't clear.  This is what I have so far:  We know by Gauss's Lemma that $(\frac qp) = (-1)^v$ where $$v = \#\left\{1 \le a \le \frac{p-1}{2} \mid aq \equiv k_a \pmod{12}),\space -p/2 &lt; k_a &lt; 0\right\}$$  And we also know that multiples of $(\dfrac{p-1}{2}) \times 3$ must be between $0$ and $p/2$, between $p/2$ and $p$, or between $p$ and $3p/2$.   Representing $p$ as $12k + r$ seems like the next step, but I'm not sure where to go from there. In AVR datasheets under the Electrical Characteristics section you will typically find a graph like this (this one is from the ATMega328):    I've seen designs that seem to "work" but operate outside the shaded envelope. Specifically, I've seen 3.3V (Arduino) designs that run the clock from an external 16MHz crystal. Clearly, this is out of spec. What are the practical negative consequences of running outside this envelope? How do the radian have a unit? The radian is defined as the ratio of the circumference and the radius. Both are measured in meters. So there should not be a unit for that. But we use 'rad' as the unit of the radian value.   The coefficient of static/kinetic friction also the same, it is a ratio of both forces. Therefore it doesn't have a unit.  So, is there a special reason to have a unit for the radian value? Is it true that $||a|^p - |b|^p| \le C_p|a - b|^p$ for $1 < p < \infty$? If $a, b \in \mathbb{C}$, then we have the standard triangle inequality for the difference:  $$||a| - |b|| \le |a - b|.$$  I am wondering if this inequality generalizes to exponents greater that one.     My question is, for $1 &lt; p &lt; \infty$, does there exists a constant $C_p$ such that $||a|^p - |b|^p| \le C_p|a - b|^p$ for all $a, b \in \mathbb{C}$?   I am aware of the "standard" triangle inequality for $p &gt; 1$:  $$|a +b|^p \le 2^{p}(|a|^p + |b|^p),$$  and that $2^p$ may not be sharpest constant possible. If I try to use this estimate to prove, say, that $|a|^p - |b|^p \le C_p |a - b|^p$, I get stuck with an extra term that I'm not sure what to do with.  $$|a|^p - |b|^p \le 2^p|a -b|^p + (2^p -1)|b|^p.$$  A resolution on this matter is greatly appreciated. Delete all files except files with the extension pdf in a directory We have two sets: set No.1 and set No.2 as in this picture:   The observer is fixed to set No.1 . He sees set No.1 motionless and observes set No. 2 approaching with velocity 100,000 m/s.   Each set has one lamp and two, so called, touchers. Each set is designed so that if both touchers are touched simultaneously the lamp is turned on otherwise it remains off.  Set No. 2 is approaching set No. 1 so that each toucher in each set will be touched twice by the touchers of the other set.  The observer on set No. 1 observes the distance between touchers in each set 10 meters. He thinks the lamps on set No. 1 will be turned on because touchers in it will be touched simultaneously.  My question: Will the lamp on set No. 2 be turned on? How if an electrical current flowing from one toucher in a set to the other proves simultaneous touching? Every Artinian ring is isomorphic to a direct product of Artinian local rings Proposition. Let $R$ be commutative ring with $1_R$. We assume that $R$ is an Artinian ring and $M_1,\dots,M_n$ its maximal ideals. Then         $R/\mathrm{Jac}(R)\cong (R/M_1)\times \dotsb \times (R/M_n)$.   The ring $R$ is isomorphic to the direct product of a finite number of Artinian local rings.      Proof. 1. It's obvious that $M_i+M_j=R,\ \forall 1\leq i \neq j \leq n$. So, from Chinese Remainder Theorem, we have  $$R/\bigcap_{i=1}^{n}M_i = R/\mathrm{Jac}(R) \cong (R/M_1)\times \dotsb \times (R/M_n),$$ as we wanted.    Since $R$ is an Artinian ring, we have $\mathrm{Jac}(R)^m=\{0_R\}$, for some $m\in \Bbb Z^+$. But,  $$\{0_R\}\subseteq M_1^m\dotsb M_n^m=(M_1\dotsb M_n)^m\subseteq (\bigcap_{i=1}^{n}M_i=\mathrm{Jac}(R))^m=\{0_R\}.$$ So, if we apply CRT we will take  $$R\cong R/\{0_R\}\cong R/M_1^m\dotsb M_n^m\cong (R/M_1^m)\times \dots \times (R/M_n^m).$$   Questions.   1) Are these thoughts complete and correct?  2) Why $R/M_i^m$ are artinian local rings?  3) Could you please elaborate some examples as an application? How are questions in the 'hot' tab on a site selected? I know questions on the "hot" tab of a site homepage are based on some sort of views/velocity criteria, but does anyone know the actual algorithm used?  Background: I was considering a feature-request about the "hot" tab, but I don't have enough information to speak about it intelligently. Is there a clipboard changed or updated event that i can access through C#?