PROBLEM     Let $H$ and $K$ be subgroups of a group $G$.  Prove that $HK = \{ab | a \in H, b \in K\}$ is a subgroup of $G$ if and only if $HK = KH$.   WHAT I KNOW  (1) If $H = \{e, \alpha, \alpha^2\}$ and $K = \{e, \beta, \beta^2\}$ are subgroups of $S_4$, one can show that $HK \neq KH$.  We can compute $HK$ and $KH$ to verify that:  In general, the product of two subgroups of a group $G$ is not a subgroup of $G$.  (2) If $H = \{h_1, h_2, \ldots, h_r\}$ and $K = \{b_1, b_2, \ldots, b_p\}$ are subgroups and one of $H$ or $K$ is normal in $G$, then (a) $HK = KH$, and (b) $HK$ is a subgroup of $G$.  (3) If $H$ and $K$ are normal subgroups of $G$, so also is $HK$.  QUESTION     How do I then solve the problem at hand, given that what I know seems to require that at least one of $H$ or $K$ is normal in $G$?  That is, how does one prove the statement in the PROBLEM section in full generality (without requiring that one of $H$ or $K$ is normal in $G$)?
Let $H$ and $K$ be subgroups of a group $G$, and let $HK=\{hk: h \in H, k \in K\}$, $KH=\{kh: k \in K, h \in H\}$. How can we prove that $HK$ is a subgroup iff $HK=KH$?
How to completely remove virtual packages? Computing $\int_{0}^{\pi}\ln\left(1-2a\cos x+a^2\right) \, dx$ Integrable function whose Fourier transform is not integrable Moving Grub to a new drive to get rid of Windows I have some section where user can upload CV and profile images,  I'm using the native attachment functions,  $attachment_cv_id = media_handle_upload( 'candidate_cv', 0 ); if ( ! is_wp_error( $attachment_cv_id ) ) {     // We link this file to the post      // add_post_meta( $result , 'wpcf-cv-file', wp_get_attachment_url($attachment_cv_id)); } else {     $submit_candidate_errors[] = lang("Hubo error subiendo la CV"); }   Is there any way I can hide this files from the media section in the wp-admin? ( perhaps changing its folder )  Thanks  -EDIT-  i'm trying like Dave suggested, adding a meta: 'rrhh-file'  $attachment_id = media_handle_upload( 'apply_file', 0 );  if ( ! is_wp_error( $attachment_id ) ) {     // The image was uploaded successfully!     add_post_meta($attachment_id, 'rrhh-file', true); }   And even that it has the attribute:  var_dump(get_post_meta(1200)); // last id I uploaded die();   Prints out:  array(4) {   ["_wp_attached_file"]=&gt;   array(1) {     [0]=&gt;     string(20) "2017/06/image001.jpg"   }   ["_wp_attachment_metadata"]=&gt;   array(1) {     [0]=&gt;     string(666) "a:5:{s:5:"width";i:523;s:6:"height";i:122;s:4:"file";s:20:"2017/06/image001.jpg";s:5:"sizes";a:2:{s:9:"thumbnail";a:4:{s:4:"file";s:20:"image001-150x122.jpg";s:5:"width";i:150;s:6:"height";i:122;s:9:"mime-type";s:10:"image/jpeg";}s:6:"medium";a:4:{s:4:"file";s:19:"image001-300x70.jpg";s:5:"width";i:300;s:6:"height";i:70;s:9:"mime-type";s:10:"image/jpeg";}}s:10:"image_meta";a:12:{s:8:"aperture";s:1:"0";s:6:"credit";s:0:"";s:6:"camera";s:0:"";s:7:"caption";s:0:"";s:17:"created_timestamp";s:1:"0";s:9:"copyright";s:0:"";s:12:"focal_length";s:1:"0";s:3:"iso";s:1:"0";s:13:"shutter_speed";s:1:"0";s:5:"title";s:0:"";s:11:"orientation";s:1:"1";s:8:"keywords";a:0:{}}}"   }   ["rrhh-file"]=&gt; // &lt;-- here   array(1) {     [0]=&gt;     string(1) "1"   }   ["_edit_lock"]=&gt;   array(1) {     [0]=&gt;     string(12) "1498731595:1"   } }   With this action this attachment still shows up in the media library  add_action( 'pre_get_posts', 'wpse_hide_cv_media_list_view' ); function wpse_hide_cv_media_list_view( $query ) {     // Bail if this is not the admin area.     if ( ! is_admin() ) {         return;     }      // Bail if this is not the main query.     if ( ! $query-&gt;is_main_query() ) {         return;     }      // Only proceed if this the attachment upload screen.     $screen = get_current_screen();     if ( ! $screen || 'upload' !== $screen-&gt;id || 'attachment' !== $screen-&gt;post_type ) {         return;     }      // Modify the query.     $query-&gt;set( 'meta_query', [         [             'key'     =&gt; 'rrhh-file',             'compare' =&gt; 'NOT EXISTS',         ]     ]);      return; }   Any idea what i'm missing? Generating random results by weight in PHP? I know how to generate a random number in PHP but lets say I want a random number between 1-10 but I want more 3,4,5's then 8,9,10's. How is this possible? I would post what I have tried but honestly, I don't even know where to start. Lots of ram. How to use it? My PC has 8 Gb of ram. Is there any way to make ubuntu use most of it? I mean rarely drop caches and keep once opened programms in ram longer, preload apps on boot, etc. Why does a neon lamp indicator on a power strip switch flicker in the dark? Neon lamp indicators embedded in power strip switches indicate whether the switch is ON or OFF. When the neon lamp is ON in a dark environment (lights of the room are off), the lamp begins flickering and it sometimes goes off for a few seconds. When I switch the lights of the room on, it stops flickering and remains bright. Even when I unlock my phone in the dark near the lamp, it stops flickering! Is the light-making-mechanism in neon lamps affected by the light intensity of the surrounding environment?  UPDATE: I captured my observations with my iPhone and uploaded it to YouTube. Here is the link of the video:  All the lights of my room were turned off. There were no plugs connected to the power strip. The only source of light was the screen of my MacBook Pro, which was not plugged in.  I increased and decreased the brightness of the computer screen during the video.  I then produced two additional light environments using my phone. Map matching links and ideas? I'm using OpenStreetMap and its vectorial road network and I'd like to implement a map matcher algorithm.   Currently I'm able, for each GPS position, to retrieve the nearest road segment and calculate the projection of this position to that segment, like on this image (Red pin is the pure GPS position, in blue the mapped segment and in Green the mapped position):    However, due to the lack of accuracy of the GPS, sometimes the mapped position jumps from segment to another and can provide some inconsistent mapped position from time to time.  My current algorithm is very basic : from the pure GPS position, I get the nearest segment and decide that the mapped matched position is on this one. I know that this can be really improved.  I can imagine that taking the vehicle direction into account will improve the map matching but do you know any other approach that would enable me to improve my map matcher ?  I seek any link, and/or open source software? Minimal sufficient statistic of $\operatorname{Uniform}(-\theta,\theta)$ I wanted to know if there is difference in terms of charging the battery when a MacBook Pro 16" is connected to two monitors via USB-C, both with Power Delivery capability.   In this specific case I am talking about the LG 27UK850-W monitors. The MacBook Pro 16" will be connected via USB-C to both of them and use them as external monitors. Since both can charge it as well, will the MacBook Pro  charge faster while being connected to both or will just one monitor be able to deliver the charge at a time? GUI/Unity crashing in 16.04 LTS after updates 2018-01-04, compiz segfaults The latest updates (4/1/2018) make my laptop constantly crash the GUI/Unity:  (Intel Pentium su4100 processor)  syslog:  Jan  5 11:38:53 1810 kernel: [ 1850.327738] compiz[11911]: segfault at 0 ip 00007f6c7baf0c16 sp 00007ffd5ec71c60 error 4 in i965_dri.so[7f6c7b513000+82d000]   After login I end up with an empty desktop, just a wallpaper, GUI/Unity are in a crash loop:   No panel.  No dash.  Can't start a program. Launcher disappears. Sidebar disappears. System crash. Windows key not working. The key combination Alt+Tab not working.   Simply, the computer is not usable. Same for guest login. The only thing working: open up a terminal CTRL+ALT+T to examine log files.  I somehow forced  to let me log in and file the bug, but still lots of problems:   Dash not working Alt+Tab not working Sometimes windows all disappear for a few seconds     Any ideas for a solution or workaround?   UPDATE     If you have the same/ very similar problems, follow the link to my    report and mark there that you're affected too. Or file your own. Please leave a comment   if you wish, or upvote - this is a nasty problem - but an answer should   offer a fix or a workaround so users can use their computer again.    Without , Ubuntu can't fix issues. Show that a norm on an inner product space satisfies parallelogram law. Hence use the parallelogram law to show that the space of continuous real functions defined on the interval $[a,b]$ is not a Hilbert space. Here I did the first part. Let $X$ be an inner product space and $x,y\in X$ consider $\Vert x+y\Vert ^2$ and $\Vert x-y\Vert ^2$ Then on adding I get $\Vert x+y\Vert ^2+\Vert x-y\Vert ^2=2(\Vert x\Vert ^2 +\Vert y\Vert ^2)$ For the second part I don’t know how can a parallelogram law prove $C[a,b]$ is not a Hilbert space, help me please. How can I test whether my camera is working after it was dropped? Issue : How can I evaluate my cameras performance synthetically?  Scenario : Bought a Nikon D3100 an year back. Photos looked good enough. The camera fell, zoom lens effected.  But works. Cleaned lens using soap, water etc.  Carefully. Now photos look much worse.  How do I know if Photos are really bad?  Is there a tool?    Sample Photo  Now (Bad)   Original (Good)   Please help me realize whether my camera has gone bad.  Can you tell your personal opinion also? $M /K \land L /K$ algebraic $\implies ML/K$ algebraic? How to install TP-LINK Archer T4U driver? I bought the wireless adapter to use with a computer without internet access but I can't figure out how to install it with just a USB with the driver on it. I've read a few articles on how to install the driver but they all require me to have internet access and I don't have internet. I'm completely new to Ubuntu. If you can help that would be great. Thanks!  Ps: I have the latest version of Ubuntu as of 7-5-16. Why is the light reflected at the same angle from mirror? From the school physics I know that the material objects bounce from the plane surface at the same angle, losing some kinetic energy. In the same school I was taught that the light (and waves in general) obeys this principle too.  Obviously, in the case of light the plane surface should be a perfect mirror. But I can't understand how should this work from the quantum point of view. Let's assume that our mirror consists of a single silver atom.  Why should the electrons of this atom re-emit consumed photons at some specific angle? I put some alias commands in my .bashrc file, so that they might be loaded everytime I open a new Terminal window. Yet this doesn't happen. I have to select run script: in the Terminal&gt;Preferences&gt;&quot;MyDefaultTheme&quot;&gt;Shell prefpane and add: source .bashrc &amp;&amp; clear for it to work... This seems odd since many tutorials only say you just have to add to the .bashrc file and all is good. Note that I don't run in bash when using the terminal, I like the other (default) one (don't know what it is) better, because it show me where I am all the time eg: &gt;&gt;d54c6b47b:~ romeo$  in stead of: &gt;&gt;bash$  In bash all is loaded as should be. So my question is, why didn't my .bashrc file load automatically and did I have to add the option to call it everytime? Also some tutorial told me to do something like: $alias la=’ls -la’ &gt;&gt; ~/.bashrc  which should write the alias to my .bashrc, this doesn't work either... Note that I'm a UNIX novice, so be gentle. linux: touch date precision Air friction at supersonic velocities