Assume two natural numbers $m,n$ are coprime (this means $(m,n)=1$) such that for each $a ,b \in G$ we have $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$. Then $G$ is an abelian group.
Let $(G,.)$ be a group and $m,n \in\mathbb Z$ such that $\gcd(m,n)=1$. Assume that $$ \forall a,b \in G, \,a^mb^m=b^ma^m,$$ $$\forall a,b \in G, \, a^nb^n=b^na^n.$$  Then how prove $G$ is an abelian group?  Some context: Some of these commutation relations often imply that $G$ is abelian, for example , or . This looks like another example of this phenomenon, but the same techniques do not apply.
Is there a word to describe a highly desirable cursed treasure? Is there a single word to describe an object or idea that is so desirable that everyone wants to attain it but once they have it they are immediately cursed?  The idea is often used in literature—some examples:   In Lord of the Rings, the 'one ring' is a perfect example of this. In Guy de Maupassant's story , the necklace might be considered to be this. There are many stories of people making deals with the devil (see question: ), or being granted three wishes by a Djinn or fairy, which inevitably turn out bad. In the Twilight Zone episode , Bemis's desire to be left alone with his books would be an example of this word.   'Bane' covers the cursed part, but no one desires it.  Is there a single word to describe a most highly desired treasure that (knowingly or unknowingly) comes with a curse? Why is Jupiter so sharply defined? In photographs of Jupiter, . Being a gas giant, my naive self thinks that the atmosphere should have a more gradual cut off, creating a hazy effect similar to that on Titan. I am aware of Jupiter's immense size, so each pixel in the linked photograph is likely tens or hundreds of kilometers. Even so, wouldn't there be some small smudge? Does perhaps Jupiter's terrific gravity have any effect on the apparent edge of its atmosphere? GRUB does not detect Windows I have finally installed Ubuntu on my second drive. When I start my computer GRUB only offers me to boot Ubuntu, not Windows 7. What needs to be done so that I can choose between Ubuntu and Windows in GRUB?  When I press F12 for boot menu at startup and I choose Windows Boot Manager it boots into Windows 7.    I ran command sudo fdisk -l and here is log ():  WARNING: GPT (GUID Partition Table) detected on '/dev/sda'! The util fdisk doesn't support GPT. Use GNU Parted.   Disk /dev/sda: 1000.2 GB, 1000204886016 bytes 255 heads, 63 sectors/track, 121601 cylinders, total 1953525168 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 4096 bytes I/O size (minimum/optimal): 4096 bytes / 4096 bytes Disk identifier: 0xc3ffc3ff     Device Boot      Start         End      Blocks   Id  System /dev/sda1               1  1953525167   976762583+  ee  GPT Partition 1 does not start on physical sector boundary. Anyone know a way to immediately show the seconds of a file's date modified property in the GUI? So if you create a file, any file in any directory, right-click and choose Properties, the date modified (if it's recent) will say something like "dd/mm/yyy hh:mm, one minute ago" - reminder this is in Windows 7. Windows XP did it normally. Then they changed something.  If you wait a while, eventually you'll see the seconds, I'm not sure how long a while is, but this is incredibly annoying if you want to troubleshoot something that relies on the seconds of timestamps... is there a setting? registry key I can change perhaps?  I'm literally using Chrome, pasting in the path of the directory to be able to see the seconds quickly (as a workaround) but would be nice to be able to use Win7. How to shop for a lens filter? What should I consider when shopping for a lens filter?    I see filters that range from $15 to $115 and more.  What are the differences between the $15 and the $115 filter?    I'm asking in the generic sense but if you need specifics, I'm looking to buy a UV filter and a polarizing filter for a Panasonic GF1 w/ 20mm pancake lens.  EDIT: I've read some of the other posts and some refer to multi-coating but doesn't explain what is this and why it matters.  What I've gathered from reading related posts is that price is generally the only indication of quality in the case of choosing a filter.  Is this right? How does TeX actually work? Theorem. Let $X$ and $Y$ be sets with $X$ nonempty. Then (P) there exists an injection $f:X\rightarrow Y$ if and only if (Q) there exists a surjection $g:Y\rightarrow X$.  For the P $\implies$ Q part, I know you can get a surjection $Y\to X$ by mapping $y$ to $x$ if $y=f(x)$ for some $x\in X$ and mapping $y$ to some arbitrary $\alpha\in X$ if $y\in Y\setminus f(X)$. But I don't know about the Q $\implies$ P part.  Could someone give an elementary proof of the theorem? How to run scripts without typing the full path? Randomize a List What is the best way to randomize the order of a generic list in C#? I've got a finite set of 75 numbers in a list I would like to assign a random order to, in order to draw them for a lottery type application. How to perform a logistic regression for more than 2 response classes in R I want to predict the impact of oil price over a Colombian oil company's stock price. I plan to use a logistic regression for this with a categorical variable (Up or Down given the direction of the stock price). Here is part of my dataset:  Minute  ecopet  profit  sum_profit   direccion  cl1_chg   sum_cl1    direccion_cl1 571     2160     0       10           Up         -0.03     0.00      Down 572     2160     0        0           Neutral     0.07    -0.03      Down 573     2160     0       -5           Down       -0.08     0.04      Up 574     2160     0       -5           Down       -0.07    -0.04      Down 575     2160     5       -5           Down       -0.08    -0.11      Down 576     2165     0       -25          Down        0.00    -0.19      Down 577     2165     0       -25          Down       -0.05    -0.19      Down 578     2165     0       -15          Down       -0.17    -0.24      Down 579     2165     5       -15          Down       -0.06    -0.41      Down 580     2170     0       -20          Down        0.03    -0.47      Down 581     2170    -10       0           Neutral     0.04    -0.44      Down   My dependent variable is 'direccion'. But as you can see it has 3 response classes. I know that to implement a binary logistic regression in R the code is:  glm.fit=glm(direccion~direccion_cl1, data=datos, family=binomial)   I am working with intraday information and plan to predict what happens when the oil moves up/ down (in the previous 10 minutes) and how it impacts the stock price in the next 10 minutes.  Could anyone tell me how could I perform this? I don't really know how to perform the logistic regression with 3 response classes. Thanks! Move the period outside the quotes on the site-analytics page If you don’t have 25k reputation, the  contains a stub text:        As you can see on the image, the dot should be moved outside the quotes. Also, on ruSO we need to  from “” to «», but this is not possible via Transifex, because there is no full sentence available to translate. How to resolve a Java Rounding Double issue How to best respond to a request to send a passport scan by unsecured email? It is not uncommon for travel or holiday accommodation companies to ask for a scan of your passport by email. Sometimes local laws require them to keep a record of the passport numbers of all visitors. I'm happy to provide that information but you usually have to send it in an unsecured email, which may have consequences .  Could these companies be more responsible, and provide a public cryptographic key at least?  Are there other steps you should take to protect yourself, if you do send by unsecured email? Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice? iPhone screen randomly turns on After installing iOS 10, I found that the screen on my iPhone 6s turned on at seemingly random times, and it slightly annoyed me. I now know that this feature does what it is supposed to (turn the screen on when the phone enters certain positions), but I want to disable it.  Is there a setting can I use to do so? When a user without editing privileges suggests an edit to the post, a banner gets displayed until it is reviewed. Can the banner also provide a link to the Suggested edit? For example, I made an edit to a post on Unix&amp;Linux.SE. Can the banner also link to my ?     I know I can access the link through my Profile → Activity → Suggestions. But linking it directly on the face of the post would have immense benefits:  I have been a prolific editor on Ask Ubuntu (my first encounter with Stack Exchange Network) with a lot of Rep derived through Suggestions. I didn't get to know about these suggested edit links until very late (I guess I already had got about 400 edit suggestions approved by the time this came to my knowledge).  This means I didn't know about  when it mattered the most.  I never got to know why my suggestions got rejected because of my ignorance.  This would help reduce the barrier between Reviewers and Editors.  This would also help in case of Anonymous edits. Right now, there is no way an anonymous editor can access it unless the edit gets approved and he comes back to go through the Revision page of that particular post in order to access the Suggested Edit Review link. If the edit gets rejected - it's the end of the road for that anonymous editor. I just made an anonymous edit on one of the SE sites where I don't have an account and already feel like it's the end of the road for me since I just can't access the review link unless it gets approved.    Possibly also expand the banner to mention that they can also access the link to the suggestion via Profile → Activity → Suggestions. How can I lock a folder so that it CANNOT be erased? On my computer running Windows XP Pro SP3 I have a folder on my desktop that is very important. How can I lock it so that I can alter anything in it, but the folder itself cannot be removed without resetting a (lock/unlock?) switch? A man wonders why perforations don't work. He decides nothing is the strongest thing in the universe. He exploits his finding. how implement the jqueryUI in visualforce LaTeX to plain text for e.g. generation of statistics I would like to convert a large LaTeX project (i.e. spanning multiple files) into plain text. The purpose is generation of statistics, so representing mathematics is not an issue. In fact, all mathematics is ideally ignored.  I have found  but could not get it to run. It seems unfinished, but is exactly what I am looking for otherwise.