Why is this quotient group not cyclic? Consider $D_4$ generated by the &quot;reflection&quot; element, $\sigma$, and the &quot;rotation&quot; element $\rho$.  $D_4 = \{\mathbb{1}, \rho , \rho^2 , \rho^3 , \sigma, \sigma \rho ,  \sigma \rho^2 , \sigma \rho^3\}$ Now consider the (cyclic) group of order $2$,  $\mathbb{Z}_2[\sigma] = \{\mathbb{1} ,  \sigma \}$ Then $D_4 / \mathbb{Z}_2[\sigma] = \{ \{ \mathbb{1},\sigma\} , \{ \rho ,\sigma\rho\} , \{ \rho^2,\sigma\rho^2\} , \{\rho^3,\sigma\rho^3 \} \} = \{\{ \mathbb{1},\sigma\} , \rho\{ \mathbb{1},\sigma\} , \rho^2\{ \mathbb{1},\sigma\}, \rho^3\{ \mathbb{1},\sigma\} \}  $ Which is surely cyclic? But I have been told that it is not. Could anyone clear something up for me Thanks :)
Find all Quotient (or Factor) Groups of $D_4$ (Dihedral Group of order $8$) I need to be able to find all of the quotient groups for dihedral group 4 with $D_4 = \{ e,R,R^2,R^3,V,H,D,D'\}$. I know I have to start by finding the normal subgroups, which are $\{e,R^2\}$ $\{e,R,R^2,R^3\}$ $\{e,R^2,V,H\}$ $\{e,R^2,D,D'\}$. Then I need to find the sets defined by $G/H=\{ aH : a \in G\}$ with operation $aH  bH = abH$. I am stuck here; could someone please show me how to find all of the factor groups for D4?
As small as it may be, does every 'thing' have a gravitational pull?  That is, something with mass or energy. No matter how obsolete or negligible it may be, is it there? If so, how is it calculated? What does 'it' affect? Why didn't Thor lose his power? In Thor: Ragnarok, Valkyrie told Thor this about Hela:  Her power comes from Asgard the same as yours  Thor's plan was to unleash Surtur who destroys Asgard, which weakens Hela allowing Surtur to kill her. But why didn't that weakened Thor as well? We saw him later in Avengers: Infinity War very powerful. Question: For every polynomial $P(x)$ of degree at n we have $$\sum_{i=0}^{n+1}(-1)^i\binom{n+1}{i}P(i)=0$$  Well I was inducting on degree of the polynomial, for $n=0$ its true and I got a bit confused in the inductive step, actually I cant figure the inductive step.   Is this property (if true) a famous one?  Thank you for your help. The bottom bracket of my bike was completely stripped to the point that when you push the left side crank arm, the right crank moves and you can see the bottom bracket lock move out the frame. The bike shop tried putting Teflon tape but it's temporary. It's a 26' xc hard tail. For all I know both sides of the bottom bracket(square tapered) are stripped clean hence the Teflon tape put by the previous owner. Can I fix the problem by changing into a hallow tech cranks because it works like a thread less bb, or should I just buy a thread less square tapered bb? Any suggestion on what to do? 407 Proxy Authentication Required When I try to install a software using Ubuntu Software center I get:  Failed to download repository information Check Your Internet connection   When I try to do a apt-get install something, I get:  407  Proxy Authentication Required   I use a proxy server that requires a user-name and a password. I have set my systems proxy manually, by plugging in the required numbers in the Networks proxy and applied it system wide. I guess the problem now is plugging in my user-name and password.  When I use INTERNET via Mozilla, it specifically asks me for my user-name and password. I am wondering if the explanations correct and whether what is the difference between different &quot;seems&quot;.   He seems happy. He seems a happy fellow. He seems to be a happy fellow. He seems to work hard. He seemed to be happy   &quot;Seem&quot; also enters into an 'impersonal' construction which employs 1) the 'dummy' subject it + an active tensed form with &quot;seem&quot;, followed by 2) a that clause (sometimes called a 'content' clause) which expresses the assertion which is hedged or qualified.   It seems that he is happy (but he may not be).  It seemed that he was a happy fellow (but he wasn't).  It will seem that he is working hard (but he will in fact be spending his time answering questions on ELL instead of working).  He is seemed to be something. Is there a formal proof for $(-1) \times (-1) = 1$? It's a fundamental formula not only in arithmetic but also in the whole of math. Is there a proof for it or is it just assumed? Distribution of a ratio of uniforms: What is wrong? Suppose that  $X$ and $Y$ are two i.i.d. uniform random variables on the interval $[0,1]$  Let $Z=X/Y$, I am finding the cdf of $Z$, i.e. $ \Pr(Z\leq z) $.  Now, I came up with two ways of doing this. One produces a correct answer consistent with the pdf here: , the other does not. Why is the second method wrong?  First Method  $\newcommand{\rd}{\mathrm{d}} \Pr(Z\leq z) = \Pr(X/Y\leq z) = \Pr(X\leq zY) = \int^{1}_{0}\int^{\min(1,zy)}_{0} \rd x \rd y = \int^{1}_{0}\min(1,zy)\ \rd y$ $  = \left\{      \begin{array}{lr}        \int^{1/z}_{0}zy\ \rd y + \int^{1}_{1/z} \rd y&amp; : z &gt; 1\\        \int^{1}_{0}zy\ \rd y &amp; : z \leq 1      \end{array}    \right. $ $  = \left\{      \begin{array}{lr}        1 - \frac{1}{2z} &amp; : z &gt; 1\\        \frac{z}{2} &amp; : z \leq 1      \end{array}    \right. $  This appears correct.  Second Method  $\Pr(X/Y\leq z) = \Pr(X \leq zY\ |\ zY \geq 1)\Pr(zY \geq 1) + \Pr(X \leq zY\ |\ zY &lt; 1)\Pr(zY &lt; 1) $ by total probability  $ = \Pr(X \leq zY\ |\ zY \geq 1)\Pr(Y \geq 1/z) + \Pr(X \leq zY\ |\ zY &lt; 1)\Pr(Y &lt; 1/z)$  Taking $z&gt;1$ yields $ (1)(1-\frac{1}{z}) + (\int^{1/z}_{0}\int^{zy}_{0} \rd x \rd y)(\frac{1}{z}) = 1-\frac{1}{z} + (\int^{1/z}_{0}zy\ \rd y)(\frac{1}{z}) = 1-\frac{1}{z} + \frac{1}{2z^{2}}$  This is already different. Why is this wrong?  Thanks! How can I create a /dev/null-like "blackhole" directory? I would like to create a "/dev/null" directory (or a "blackhole" directory) such that any files written to it are not really written, but just disappear.   I have an application that writes out large temporary files to a directory. I have no control over the name of the files and I don't really care about the content of these files. I could write a script that periodically clobbers these files, but the files are written out very quickly and fill my disk. I'm looking for something cleverer. I want the application to "think" that it is writing out these files, when in fact, the writes are just being discarded at the other end.  Also see  old related thread. How to add new item in existing string array in C#.net?  I need to preserve the existing data. I'm looking for a way to insert a &lt;style&gt; tag into an HTML page with JavaScript.  The best way I found so far:  var divNode = document.createElement("div"); divNode.innerHTML = "&lt;br&gt;&lt;style&gt;h1 { background: red; }&lt;/style&gt;"; document.body.appendChild(divNode);   This works in Firefox, Opera and Internet Explorer but not in Google Chrome. Also it's a bit ugly with the &lt;br&gt; in front for IE.  Does anyone know of a way to create a &lt;style&gt; tag that   Is nicer Works with Chrome?   Or maybe   This is a non-standard thing I should avoid Three working browsers are great and who uses Chrome anyway? SharePoint Migration 2007 to 2013 Let $M_n$ denote the vector space of all $n\times n$ real valued matrices and for any matrix $A\in M_n$ let $A^t$ denote its transpose. Define the linear map $T:M_n\rightarrow M_n$ as $$T(A)=A+A^t.$$ What is the minimal polynomial of $T$? What are the eigenvalues and the corresponding eigenspaces of $T$? What is the characteristic polynomial of $T$?  I learned about minimal polynomials of operators from $\textit{Linear Algebra Done Right}$ by Axler. In his description he explains how in order to find the minimal polynomial you must find the smallest positive integer $m$ such that $T^m$ is a linear combination of $I, T, T^2, \ldots, T^{m-1}$. That is, the smallest integer $m$ such that $$T^m = a_0I+a_1T+a_2T^2+ \cdots +a_{m-1}T^{m-1}.$$  As for the eigenvalues...since we are working with real-valued matrices, then $A$ and $A^t$ have the same eigenvalues, so if $\lambda \in \mathbb{R}$ is an eigenvalue of $A$ (and hence of $A^t$) wouldn't $2\lambda$ be an eigenvalue of $A+A^t$? I apologize if I am saying utter nonsense... What's the projective limit of these polynomial rings ? Define an inverse system of polynomial rings over a commutative ring $k$ by the canonical projection $k[x_1,...,x_n] \to k[x_1,...,x_m]\;(m&lt; n)$.   Question: What is the projective limit $\varprojlim_n k[x_1,...,x_n]$ ? Proving $\int_{0}^{\infty} \mathrm{e}^{-x^2} dx = \frac{\sqrt \pi}{2}$ Reference to undeclared entity exception while working with XML I am trying to set the innerxml of a xmldoc but get the exception: Reference to undeclared entity  XmlDocument xmldoc = new XmlDocument(); string text = "Hello, I am text &amp;alpha; &amp;nbsp; &amp;ndash; &amp;mdash;" xmldoc.InnerXml = "&lt;p&gt;" + text + "&lt;/p&gt;";   This throws the exception:      Reference to undeclared entity 'alpha'. Line 2, position 2..    How would I go about solving this problem? Is it OK to reduce the charging current for a Li-ion 18650 battery? I am using NCR18650 batteries with the tp4056 charging module.  When I connect the charger at 2.8&nbsp;V (fully discharged), the tp4056 IC is getting hot at a 1&nbsp;A charge current. Is it OK to reduce this charge current by changing the charge current selecting resistor? So the tp4056 IC heat would be reduced, wouldn't it?  For my application I don't need a quick charge. With a low current, charging time will be increased, but it does not matter for my application.  Will reducing the charge current affect the battery's lifetime? Or will it damage the battery?  I am trying to reduce the heat of the tp4056 IC by reducing the charging current, maybe from 1&nbsp;A to 500&nbsp;mA or less. The 18650 battery never heats for a 1&nbsp;A charge current. How do I pass have a Javascript script request a PHP page and pass data to it? How do I then have the PHP script pass data back to the Javascript script?  client.js:  data = {tohex: 4919, sum: [1, 3, 5]}; // how would this script pass data to server.php and access the response?   server.php:  $tohex = ... ; // How would this be set to data.tohex? $sum = ...; // How would this be set to data.sum? // How would this be sent to client.js? array(base_convert($tohex, 16), array_sum($sum)) Which pseudo-$R^2$ measure is the one to report for logistic regression (Cox & Snell or Nagelkerke)? I have SPSS output for a logistic regression model. The output reports two measures for the model fit, Cox &amp; Snell and Nagelkerke.  So as a rule of thumb, which of these $R^²$ measures would you report as the model fit?  Or, which of these fit indices is the one that is usually reported in journals?    Some Background: The regression tries to predict the presence or absence of a bird (capercaillie) from some environmental variables (e.g., steepness, vegetation cover, ...). Unfortunately, the bird did not appear very often (35 hits to 468 misses) so the regression performs rather poorly. Cox &amp; Snell is .09, Nagelkerke, .23.    The subject is environmental sciences or ecology. Limiting sessions to one IP at a time On a site with premium user account subscriptions, I'd like to be able to limit logins to one computer at a time. The most straightforward way to accomplish this would be to limit by IP, but I haven't had any luck finding a plugin to accomplish this. Does anybody know of one I can use to get this functionality?  Related: