How to prove this binary operation? Let (G, ∗) be a group and a,b € G.   Show that (a✻b)^-1 = a^-1✻b^-1 if and only if a✻b=b✻a  I could not solve this, how can ve prove it?  a^-1 means inverse of a
Shorter proof for some equvalences Let $(G,\cdot)$ be a group show that  A) $G$ is abelian  B) For all $x,y\in G: (xy)^{-1}=x^{-1}y^{-1}$  C) For all $x,y\in G: (xy)^{2}=x^{2}y^{2}$  D) There existst an $n\in \mathbb{Z}$ such that for all $x,y\in G$  $\displaystyle (xy)^{n-1}=x^{n-1}y^{n-1}$ and $\displaystyle (xy)^{n}=x^{n}y^{n}$ and $\displaystyle (xy)^{n+1}=x^{n+1}y^{n+1}$  are equivalent.    I showed $A\iff B,A\iff C$ and $A\iff D$. That was not really hard, except maybe $D\Rightarrow A$. I would like to do a circular argument, but i have no idea how.  Any hints?  Thanks in advance!
I've been working through a bit of Ireland and Rosen's Number Theory for fun. Problem 4.11 of Ireland and Rosen asks     Prove that $1^k+2^k+\cdots+(p-1)^k\equiv 0\pmod{p}$ if $p-1\nmid k$, and $-1\pmod{p}$ if $p-1\mid k$.    My work is leading where I didn't expect to go. Here I assume $p$ id odd. First, if $p-1\mid k$, then we can wrike $k=(p-1)j$ for some $j$. But then $$ 1^k+\cdots+(p-1)^k\equiv 1^{(p-1)j}+\cdots+(p-1)^{(p-1)j}\equiv \underbrace{1+\cdots+1}_{p-1\text{ times}}=\frac{(p-1)p}{2}\equiv 0\pmod{p} $$ since $p-1$ is even.   Secondly, I suppose $p-1\nmid k$. WLOG, I can assume $0&lt;k&lt;p-1$, Since if $k=(p-1)j+r$ with $0&lt;r&lt;p-1$, then  $$ a^k\equiv a^{(p-1)j+r}\equiv a^r\pmod{p}. $$  Then if $g$ is a primitive root, $$ 1^k+\cdots+(p-1)^k=\sum_{i=1}^{p-1}(g^k)^i=\frac{g^k(1-g^{k(p-1)})}{1-g^k}. $$ Since $g$ is primitive, $p\nmid 1-g^k$, but $p\mid 1-g^{k(p-1)}$, so the above sum is again congruent to $0\pmod{p}$.  So in all cases, I get the sum is congruent to $0$ regardless. Have I messed up, or is there a typo? Thanks. I deleted the "/etc/passwd" file and cannot log in I was trying to change my username and also my home directory (/home/username) and my system started to crash. I deleted the passwd file but I had a backup named passwd_bkp. I tried to rename this passwd_bkp as passwd and it did not work. No commands were being executed... I was in a terminal window.   I restarted my system and now I cannot log in. GRUB gives two options: Linux and recovery mode.   I tried to open a sessions as root but it says the filesystem is corrupted. I cannot access my files.  Did I lose all my files? I'm trying to run an interactive command through paramiko.  The cmd execution tries to prompt for a password but I do not know how to supply the password through paramiko's exec_command and the execution hangs.  Is there a way to send values to the terminal if a cmd execution expects input interactively?  ssh = paramiko.SSHClient() ssh.connect(server, username=username, password=password) ssh_stdin, ssh_stdout, ssh_stderr = ssh.exec_command("psql -U factory -d factory -f /tmp/data.sql")   Does anyone know how this can addressed? Thank you. I am looking for the title of a novel that I only caught part of on audio. It takes place within our current time frame and deals with the Moon getting struck and fragmented into (initially) 7 pieces by a high-speed asteroid that they called "The Agent".  Within a few weeks of this episode, scientists figure out the the Moon will continue to break down in a cascade reaction over the next couple of years until the planet gets bombarded by so much debris that it sterilizes Earth. The governments of the world frantically start to build a series of ships using swarming concepts to evacuate the planet as much as possible... and that's all I have heard so far.  Thanks for any info. Is it okay to say "Yes you're." instead of "Yes you are."? Possible Duplicate:    I was having an SMS conversation with a friend and somehow &quot;Yes you're&quot; came into play in retaliation to a comment. Example:  Person 1: &quot;You are bad at English&quot;. Person 2: &quot;No I are not.&quot;. Person 1: &quot;Yes you're&quot;.  Is that acceptable? I would assume that it is. Think of &quot;don't&quot;. You can say:  &quot;No, don't.&quot;  or  &quot;I don't.&quot;  and it is a contraction like &quot;you're&quot;. So, is it okay? Is cancer caused by vitamin B17 deficiency? Favourites Page on iOS App I would like to have a page listing all the questions I marked as favourite, same as the .  Also, I think the desktop version, the star, is much more intuitive and easier to find. Thanks! is a typical example for , though it is said that long time no see has been accepted by the mainstream English speakers.  Recently, there is a  movement in the , and a few new Chinglish words have been invented to demonstrate the characteristics of the Mainland.  The most outstanding examples (in my opinion) are:   shitizen - a citizen without citizen rights freedamn - the freedom for shitizens (no freedom) democrazy - the democracy for shitizens (no democracy) smilence - the speeches under the freedamn of speech () z-turn - to make effort in vain ( (Pinyin))   Here are my questions. Has there ever been any neologism movement in the history of English? What are the criteria to adopt new words into English? Ubuntu on 32-bit UEFI (only) based tablet pc I've created a bootable USB of Ubuntu 16.04 64-bit with Rufus, setting the drive to be a GPT for UEFI. Then I booted my tablet PC and I ran the UEFI BIOS, from which I turned secure boot off. I then restarted and went again on the BIOS to boot from my key, but all I got was a black screen that lasted few seconds and then it brought me back to the BIOS. I read about a bootia32.efi file and other procedures that can be followed, but I didn't find a good explanation of what I need to get done. My tablet PC runs an Intel Core Atom Z3735F and has 2 GB of RAM. If you have the answer please let me know. Running macOS Sierra(Version 10.12.2 16C67).  I installed  version wine-1.8.6 (through a .pkg file). I opened up a game and it worked fine.  The next day when I downloaded Origin installer, it wouldn't open. When I double clicked on it, the Wine icon showed up in the dock and said it was open, but no windows showed up. When I clicked on another program(or on the desktop), Wine closed. Now when I try to start that same game I opened when I first installed Wine, the same thing happens that happened when I started Origin installer.  When I go to the terminal and type wine {path of .exe} I got this:   I installed wine tricks using brew install winetricks and it installed fine, yet it didn't change my situation.  winecfg works properly, and so does sh winetricks. Search with Auto-Complete (like Gmail / Intellisense) An idiom for deriving pleasure from another's suffering I believe it is what the Germans call "". English itself has no such equivalent word. (Although it has been adopted as a .)  Does an idiom exist that describes it? I have a follow up question to this question:   My question is: when should you add a comma when starting a letter/email with     Hello John,      It was fun to see ...   I am pretty sure that a comma goes after the name, but lately I have seen several people (academics) write      Hello, John,       It was fun to see ...   with a comma after the Hello.  I am wondering when a comma there is appropriate.    I see this question  but I am wondering about the situation where you have a comma after the name also. "Where" in careers' search seems to be broken Let's say I want to find a company in Ontario, Canada. Ok, I navigate to company pages and type ON, Canada to Where in search:        but there are no companies in there. So why if I type just Canada it finds companies in ontario; and if I type specific area (ON, CA)  it doesn't find anything ? How do I match any character across multiple lines in a regular expression? Why does the road look like it's wet on hot days? Often, I'll be driving down the road on a summer day, and as I look ahead toward the horizon, I notice that the road looks like there's a puddle of water on it, or that it was somehow wet. Of course, as I get closer, the effect disappears.  I know that it is some kind of atmospheric effect. What is it called, and how does it work? When should I use \input vs. \include? There are two different commands to incorporate another file into the source of some document, \input and \include. When should I use one or the other? What are the differences between them? Are there more things like them to be aware of? What is the word for when someone is overly nice and actually isn't nice at all because of the unnecessary added information/comments? For example, they will say things like "Your hair looks so good that wayyyy omg you should do it that way ALL the time! Looks way better than how you usually do it." And the word I'm looking for is not passive aggressive and it is not sarcasm. There is a word for these types of people. What is it? These people think they're being kind but actually blunt and rude. Another example would be for someone to say "hey you're beautiful, but maybe if you didn't wear harsh makeup you would look better" and they intend to sound like they're being nice when you know they don't like you and are saying in out of spite. For which values of $x$, will $f (x) . f '(x) < 0$? Unexpected AdjacencyMatrix output