I have the following practice homework where I have run out of ideas: Let $G$ be a group and let $N$ be a normal subgroup in $G$ with index $p$ (prime). Let $H$ be a subgroup in $G$ which is NOT contained in $N$. Show that $G=HN$ and $[H:H \cap N]=p$ I am thinking about maybe I need to apply an isomorphism theorem but I can't get anyone to fit.
Let $G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then is it true that $G=HK$ ? ( I know that the fact is true if $p=2$ , but I don't know for the general case . Please help . Thanks in advance )
Inequality similar to the Schwarz-Pick theorem How can I insert a newline in a framebox? At the moment, I have created a code that retrieves data from the database and displays it. However, for some reason, I cannot see the files I want to retrieve on my page. My goal is that the data gets retrieved from the database, and is displayed on the webpage. I do not need to make a connection with the database since Wordpress does that automatically.  My code:  &lt;?php  global $wpdb; // this adds the prefix which is set by the user upon instillation of wordpress $table_name = $wpdb-&gt;prefix . "wpex_programma"; // this will get the data from your table $retrieve_data = $wpdb-&gt;get_results( "SELECT * FROM wpex_programma; " ); print_r($retrieve_data); ?&gt; &lt;ul&gt; &lt;?php foreach ($retrieve_data as $retrieved_data){ ?&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;column_name;?&gt;&lt;/li&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;another_column_name;?&gt;&lt;/li&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;as_many_columns_as_you_have;?&gt;&lt;/li&gt; &lt;?php  } ?&gt; &lt;/ul&gt;   My question: the data is not shown and I believe it is not retrieved. How can I fix that? The table name is correct, and I have data in my database tabel. The print_r($retrieve_data)returns the following: Array ( [0] =&gt; stdClass Object ( [id] =&gt; 35 [naam] =&gt; /public_html/wp/wp-content/plugins/AbonneerProgrammas/FilesUpload/Country Kickin 1.mp3 ) )  My query log:    My database structure:   New:  &lt;?php  global $wpdb; // this adds the prefix which is set by the user upon instillation of wordpress $table_name = $wpdb-&gt;prefix . "wpex_programma"; // this will get the data from your table $retrieve_data = $wpdb-&gt;get_results( "SELECT * FROM wpex_programma; " ); print_r($retrieve_data); ?&gt; &lt;ul&gt; &lt;?php foreach ($retrieve_data as $retrieved_data){ ?&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;id;?&gt;&lt;/li&gt; &lt;li&gt;&lt;?php echo $retrieved_data-&gt;naam;?&gt;&lt;/li&gt; &lt;?php  } ?&gt; &lt;/ul&gt; Imagine you're trapped inside a free falling elevator. Would you decrease your impact impulse by jumping during the fall? When? Find   $$\lim _{x\to \infty }x\Big(\ln\left(x+1\right)-\ln x\Big)$$  Here's how I do it:  $$x\Big(\ln(x+1)-\ln x\Big) = x\Bigg(\ln(x(1+\frac{1}{x}) - \ln x\Bigg)$$ $$x\Big(\ln x + \ln(1+\frac{1}{x}\Big)-\ln x =x\ln\Big(1+\frac{1}{x}\Big)$$ $$\ln\Bigg( \Big(1+\frac{1}{x}\Big)^x \Bigg) \rightarrow \ln 1^\infty = 0$$  What am I doing wrong? The answer is supposed to be $1$, but I get $0$. What is the difference between HAVING and WHERE in an SQL SELECT statement? EDIT: I have marked Steven's answer as the correct one as it contained the key bit of information on the link:  When GROUP BY is not used, HAVING behaves like a WHERE clause  The situation I had seen the WHERE in did not have GROUP BY and is where my confusion started. Of course, until you know this you can't specify it in the question. What is the difference between sourcing ('.' or 'source') and executing a file in bash? What's the difference between executing a script like this:  ./test.sh  and executing a script like this:  . test.sh?  I tried a simple, two-line script to see if I could find if there was a difference:  #!/bin/bash ls   But both . test.sh and ./test.sh returned the same information. Is there any way to include an animated GIF directly in either PDFLaTeX or XeLaTeX? I realize the animate package can include animations in a PDF, but it doesn't support animated GIFs and you have to split them up manually into EPS or PNG files as far as I can tell. What is the name of the first decade in a century? 80s: the "Eighties"   90s: the "Nineties"   00s: the ???   For that matter, what is the second decade called? The "tens" just doesn't sound right. Among other sources,  states that triple DES using three seperate keys (k1, k2, k3) is vulnerable to meet-in-the-middle-attacks, while triple DES using only two keys (k1, k2, k1) is not.  Why is this attack possible if all the keys are different, but not if the first and the third are the same? How can I restrict a user on the SSH server to allow them only the privileges for ? i.e. So they cannot run commands even if they log in via SSH.  My Linux servers are Ubuntu 11.04 and OpenWrt. Is $(\mathbb{R}^2, lexicographic) \cong (\mathbb{R}, discrete)\times (\mathbb{R}, usual)$? I have the following question:  Is $(\mathbb{R}^2, lexicographic) \cong (\mathbb{R}, discrete)\times (\mathbb{R}, usual)$?  Note: the symbol $\cong$ denotes a homeomorphism between topological spaces  What I think:  I think that in fact there is a homeomorphism between those spaces, because is possible to express the lexicographic order as union of vertical lines with open extremes, I mean this is a basis for the lexicographic order. If I consider the Identical function I have the homeomorphism. Am I right? Let $Z_i,Z_2,\ldots$ be IID Random Variables with density $f$. Suppose that $P(Z_i&gt;0)=1$ and that $\lambda=\lim_{x \to 0+} f(x)&gt;0$. How can I show that $X_n=n \times \min\{Z_i\}$ has a limiting exponential distribution with mean$1/\lambda$?  I know that the CDF of the first order Statistic is for $t&gt;0$  $$F_{min \{ Z_i \}} (t)= 1- \left[ 1-F(t) \right]^{n}$$  Convergence in distribution requires that $F_n (t) \to F(t)$ for some distribution function $F$ but I cannot quite take the limit of the above, with $t$ replaced by $x/n$, as the distribution is not known. I guess I have to insert the second piece of information somewhere but I do not recognise where.  Could you please give me a hint or two?  Thank you. Why can't I destroy or place blocks? Proving that the triangle inequality holds for a metric on $\mathbb{C}$ Show that $(X,d)$ is a metric space where $X =\Bbb C $ and the distance function is defined as:   $$d(x,y) = \frac {2|x-y|}{\sqrt {1+|x|^2} +  \sqrt {1 + |y|^2}}, \text{ for } x,y \in \Bbb C.$$   I have done the proof of the first two propositions for being a metric, but I'm having a problem in proving the triangle inequality. With \setbeamercovered{transparent} covered parts of a slide are displayed transparently. If there is a (small) picture among the text, it is shown as it is. Beamer has only this picture, so it shows this picture. My question: How can I give a gray version of the picture to beamer such that in covered parts of a slide, this gray version is used? Fonts larger than \Huge? How can I make text that is larger than the size of the output of {\Huge ...}?   I would like to be able to make text arbitrarily large (even if that is done by some suboptimal scaling routine). $\mathbb{R}P^4$ and $\mathbb{R}P^6$ do not admit fields of tangent $2$-planes See the related question . This is the second part of question 4-C in Milnor and Stasheff's book on characteristic classes. In the solution to the first part, we rely on the fact that having a field of tangent $1$-planes allows us to split the tangent bundle into a direct sum. It doesn't seem like this should be true for a field of tangent $2$-planes (otherwise the question wouldn't just be asking about $P^4$ and $P^6$), so I'm not sure exactly where to start.  edit: actually, since $P^n$ can be given a Riemannian metric, any subbundle should be a Whitney summand; this doesn't answer the question as to why they only ask for $P^4$ and $P^6$ though, so I'm still confused. How many ways can $r$ nonconsecutive integers be chosen from the first $n$ integers? Can I apply for a Schengen visa of the first port of entry or does it need to be the country of the longest duration? I'm planning to travel to the Netherlands, Spain and Bulgaria. My longest stay will probably be in Bulgaria. Now, Bulgaria doesn't issue a Schengen visa but allows people with a Schengen visa to visit. Which visa should I be applying for? Bulgarian Visa AND Schengen visa or just the Schengen visa for Netherlands or Spain?