Let G be an abelian group and let H and K be finite cyclic subgroups with |H| = r and |K| = s. Show that G contains a cyclic group of order lcm(r, s).
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
MariaDB won't start on Ubuntu: Can't lock aria control file I am in 1.13 Pre-Release 7 trying to make a one command creation and I need an item with a specific name. The command I use is /give @p diamond_sword{display:{Name:"Sword of Power"}} and it gives my a diamond sword, but it doesn't have the name.   This is me running the command.   This is the command not working. Word for killing off or attempting to kill off an entire bloodline? I would like to know if there is a word or phrase for a systematic campaign to annihilate every descendant of a particular ancestor.   We have the word 'genocide' for killing off an entire race, and plenty of words for killing people based on how they are related to you (fratricide, filicide, et. al.), but looking over the  and the , I couldn't find a term for systematically killing an entire bloodline.   My question is prompted by a US news story about the horrific murder of 8 members of the same family, but those murders are not an example of what I'm thinking of because the children of the victims were left alive, so the motive for the killings was not to kill off the bloodline. The sense that some group had coordinated an attack against just one family is what made me wonder what the "not an entire race, just one bloodline" version of 'genocide' would be called.   Searching the Internet for "kill off" and bloodline sends me straight into the land of vampires, Illuminati, reptilian overlords, and, naturally, Priory of Sion nonsense so I haven't had much luck finding a thread to pull.   It may be that there is no word for it in the sense of murdering a human bloodline, because we just don't attempt kill off our royals down to the last heir often enough to coin a term for it. Maybe there is a scientific term for culling a line from a test population as part of biological research? I would like to know the model of my graphics card. I think it may be an ATI, but I want to be sure!   I have Ubuntu 11.10 (32 bit) and an Asus A6 VA laptop. What is ubuntu 14.04 doing while showing "stopping early crypto disks"? I created a virtual machine to install ubuntu 14.04 on VirtualBox 4.2.12.*. I created virtual CD to install it with fixed-size virtual disk of 200GB. After successful installation, it restarted as usual. But after showing message of below, it hangs. Actually, I created virtual machines to install ubuntu at Virtual Box successfully several times. It's very easy. The installed ubuntu versions ranged from 9.10, 13.04 and even 14.04 before. I don't know why this time it hangs at restarting after successful installation. The only difference between this failure restart and previous successful ones is that the virtual disk is much larger, 200GB. It failed three times.  * Deactivating swap... [OK] * Stopping remaining crypto disks... [OK] * Stopping early drypto disks... [OK] Fast way of finding lines in one file that are not in another? Survey database design: associate an answer to a user I'm doing the conceptual model for a survey database.  The goal is store the answers given by users (it's going to be an Android app).  I have three entities: user, question and option.  A question will have one or more options (for example: How many employees do you have? 1-40, 40-1000, +1000).  Options will have a text (1-40) and a value (the value selected by user).  User will select one (or more) of these options.  My conceptual design is:    I don't know how to associate an answer with an user.  How can I represent that relation? Do I have another entity to represent option value?  This model will store questions and pre-made answers (offered answers) and allows them to be re-used in different surveys.  I have to represent question like this one:    This question is related to this one: I have a large amount of duplicate records (25.000) in an even larger table (250.000 records). I've done the detection using the reporting function. The criteria I used to detect the duplicates were 3 fields (unique account identifier, unique user identifier and date). Based on a list of criteria, I would like to either merge or delete these records. I've got a handle on the logic associated with the merging and the deleting but what I'm having trouble with is defining the duplicate detection process.   I'm looking at either creating a custom object to take care of the detection or dumping the records into an SQL database and taking care of it offline. Any thoughts on which one of the two is the best approach?  Edit:  After reading the first comment, I think the more appropriate question would be "How do I approach the problem in SOQL".  The object I'm working with is the Event object and I'm really only looking at the standard fields, AccountId, OwnerId and ActivityDate. From what I've found online, there's no easy way to find records that have the same value for these three fields. I think it could be accomplished by nesting SOQL queries but I'd like to use a different approach if at all possible. How to get rid of banding in illustrator while using gradient? How to get Rid of Banding in illustrator while using Gradient How do I add .e00 data into my QGIS project? I define a shell keybinding in my ~/.bashrc like so:  bind -x '"\e\C-w":"/usr/bin/reset"'   This does make Ctrl-Alt-w launch the reset executable. However, it leaves the terminal in a state that resembles the one you get into by pressing Ctrl-v. So, for example, if immediately after resetting in this way I press Ctrl-p, instead of invoking the action of that shortcut (repeat previous command) I get ^P printed.  Any idea what's causing this problem and how to fix it?  EDIT: I am attaching a Python program that reproduces the messed-up terminal state I am trying to fix with the reset. I encounter this condition very often while debugging multi-threaded applications:  import pdb, thread, time  def interrupt():     time.sleep(2)     pdb.set_trace()  thread.start_new_thread(interrupt, ()) raw_input('? ')   To reproduce the messed-up state, just run the code above, and press Enter when you see the debugger invoked (it will print a bunch of text, ending with pdb.set_trace(), about two seconds after you run the program and the initial prompt appears). Will an object always fall at an infinite speed in a black hole? Is it possible in Leaflet that popup opens on mouse-over, not on click?  This is working for just one marker at a time, but I need it for a bigger number of markers:  marker.on('mouseover', function(e){     marker.openPopup(); }); I know and have observed that the the division algorithm can be used to convert any number in the decimal system to the binary system. However, I have tried searching for an intuition of why this method works, and I just can't seem to see anything, although I know what type of intuition I am trying to make out.   So, my question is why we can use the division algorithm to convert between the decimal bases and any other number bases?   An example from the decimal system to binary would be appreciated. I noticed that $f(x) = x\phi(x)$ seems to be one-to-one, where $\phi(x)$ is Euler's Phi function.  In particular, I'm writing some numerical python code and the line I have looks something like  sorted([n*phi(n) for n in range(1,1000)])   and there are no duplicates in the list.  First, is it one-to-one?  Second, if it is, is there a simple proof sketch? Galois group of irreducible cubic equation Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$. Prove that there are no positive integers $x$ and $y$ such that $x^3 + y^3 = 10^3$.   This is a homework question, and I understand that its part of Fermat's Last Theorem, but when I looked that up to try to figure out the homework, I realized that it is way further than what we've learned in class so far. We have to prove this, but the only way I can think to do it is by exhaustion, which would be extremely lengthy. Any little shove in the right direction would be appreciated. Thanks! Why doesn't the tilde (~) expand inside double quotes? Which characters played the parts of the lineup discussed by Rusty and Daniel? In planning the heist, Rusty names a number of roles that are going to be needed to pull off the job.      Off the top of my head, I'd say you're looking at a Boesky, a Jim Brown, a Miss Daisy, two Jethros, and a Leon Spinks. Not to mention the biggest Ella Fitzgerald ever.   Which characters in the film lined up with these named roles? A significant amount of zombies were detected in your city Can anyone please help and tell me if this sentence is correct?  &quot;A significant amount of zombies were detected in your city.&quot;  In my opinion it should be &quot;A significant amount of zombies was detected in your city.&quot; because &quot;amount&quot; should be in agreement with &quot;was&quot; even though amount usually refers to something plural. Are both (was and were) acceptable in this case? Edit1: Yes, &quot;number&quot; would be the correct one in this case. So allow me to ask one more thing.  &quot;A significant amount of purchases were detected...&quot;  Does the use of &quot;number&quot; apply in this case also? As purchase is not exactly a thing/person. And if no, is it correct to say &quot;A significant amount of purchases were detected...&quot;? Edit2: So, please tell me what is the correct phrasing?  &quot;A significant amount of purchases were detected...&quot; &quot;A significant amount of purchases was detected...&quot;  Or, since purchases are countable, is the use of &quot;number&quot; instead of &quot;amount&quot; correct here, as in:  &quot;A significant number of purchases were detected...&quot; &quot;A significant number of purchases was detected...&quot;  Or would amount be correct since a purchase involves money and it refers to the amount of money? Would greatly appreciate an answer, and if possible a detailed explanation.