Given the finite group $(G, \cdot)$ with $2n$ elements, $n \in N^*$ and also H a subgroup of G with $n$ elements.   Prove that if $x \notin H, y \notin H$ then $xy \in H$.  I'm not really sure how to do this since I've barely started working with subgroups. Could I have some hints on how to approach this type of exercise? Thank you.
Let $G$ be  a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.   $a,b\notin H\implies aH,bH\neq H\implies $ $aH=bH$  since $H$ has only two distinct left cosets in $G$.  $\implies a^{-1}b\in H$  How to show that $ab\in H$ from here?
I'm new to Linux. I got the "Unix And Linux System Administration Handbook" It speaks of several verisons of linux and unix, and the commands for viewing linux man pages and not specific to ubuntu.   How do I view the manpages? What should the standard spelling be - British or US? My 12.04 x64 system won't boot into GUI, but I can still use virtual console with my own account, so apparently it's the Windows Management System that's causing it, how can I reinstall it from tty, what's the apt-get command? sometimes when writing mathematical expression with partial derivatives, one needs to write a vertical bar afterwards and write the value at the which the partial derivatives are being computed as subscripts. See for example the bottom post here:    However, the \mid symbol doesn't produce a tall bar that corresponds to the size of the partial derivatives term. Is there a way to have a taller bar?   PS: Some other posts here mention \middle| but this doesn't compile on my computer.  Thanks in advance.... After T-2, why not send back another T-800? Does convergence in $L^p$ imply convergence almost everywhere? If I know $\| f_n - f \|_{L^p(\mathbb{R})} \to 0$ as $n \to \infty$, do I know that $\lim_{n \to \infty}f_n(x) = f(x)$ for almost every $x$? I am getting an 'access is denied' error when I attempt to delete a folder that is not empty. I used the following command in my attempt: os.remove("/folder_name").   What is the most effective way of removing/deleting a folder/directory that is not empty? Mouse gestures for Unity desktop We have a few cases of chat accounts remaining without valid parent account:  and  for example. What should mods do when that happens:  delete the chat account notify the SE team so they can reattach or clean it up leave it as is: it's a normal situation Is there any way to specify a suggested filename when using data: URI? 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When we set our profile in PS4, we set the console as the primary console.  Now we want to buy PSN Plus subscription, and I want to know if the same subscription can be used in both profiles, or  only in one.   In case the same subscription can be used in both profiles, how can we set it up? Or is automatic? I can't not be pregnant How can I create an executable JAR with dependencies using Maven? I want to package my project in a single executable JAR for distribution.  How can I make a Maven project package all dependency JARs into my output JAR? Duplicate user in reputation league So I understand that this has happened in the past: , but all of the &quot;duplicate&quot; posts are marked as  so here is a new bug report: On the , I am listed twice:  While that's fun and all, it isn't exactly right. Let $C$ be a finitely generated $R$-algebra where $R$ is unital commutative ring. Let $S$ be a simple $C$-module. Show that the annihilator of $S$ in $R$ is a maximal ideal. What I have tried is: When $C=R$ then it is a direct consequence of the simplicity of $S$ as an $R$-module. Further, since $S$ is simple then $S$ is cyclic (i.e. $S=Cy$ for nonzero $y\in S$). Consider the map $$f: C \to S$$ $$c \mapsto cy$$ Then $\ker(f)$ is the annihilator of $S$ in $C$ and $\ker(f) \cap R$ is the annihilator of $S$ in $R$. I have shown that it is an ideal of $R$ but I have no idea to continue showing the maximality. Is it a true direction or is there anything else I should consider to show the maximality? Can I use one Apple ID for Apple Music and another for iTunes & App Store? Let us suppose I want to use a shared Apple ID for logging into iTunes &amp; App Store, in order to share music and app purchases with a friend.  But let us also suppose that, despite this, my friend and I have very different tastes in music, so I don't want to share Apple Music playlists and "like" assertions with my friend. In other words, I don't want Apple Music to think I like the music my friend likes, and vice versa.  Is there a way that my friend and I can share an Apple ID for logging into iTunes &amp; App Store, but use different Apple IDs for Apple Music?  I suspect the answer is no, but I just wanted to ask. How can continue the equation of the rectangle red in next line?   \documentclass[11pt,a4paper,twocolumn]{article}  ...  \begin{equation*}     \left| \dot{r} \right| =     \sqrt{{- \, {e}^{-t} \left( \cos t + \sin t \right)}^{2} \, + \, {{e}^{-t} \left( \cos t + \sin t \right)}^{2} \, + \, {(- \, {e}^{-t})}^{2}} \end{equation*} Why are there both Static and Kinetic Friction? When dragging an object, there is a greater start-up force than the force it takes to keep it moving. Why is this? Why are there two different values for friction?