How does $(ab)^{3}=a^{3}b^{3}$ and $(ab)^{5}=a^{5}b^{5}$ for all $a,b \in G$ implies that $G$ is abelian? I know that the first criteria alone isn't enough because there's a counterexample. What could we say if in those equation instead 3 and 5 we had two other relatively prime numbers?
Let G be a group, where $(ab)^3=(a^3)(b^3)$ and $(ab)^5=(a^5)(b^5)$. Prove that $G$ is abelian group.  Thank you in advance. Any help is appreciated.
Failed to fetch jessie backports repository I'm using a docker image as a base for my own development that adds the jessie backports repository in its Dockerfile and uses that to install a dependency. This image uses the following command to add the repository:  echo "deb http://ftp.debian.org/debian jessie-backports main" &gt;&gt; /etc/apt/sources.list   The problem is that fetching packages from the backports repository now  fails with the following error (this used to work previously):  W: Failed to fetch http://ftp.debian.org/debian/dists/jessie-backports/main/binary-amd64/Packages 404  Not Found  W: Failed to fetch http://deb.debian.org/debian/dists/jessie-updates/main/binary-amd64/Packages  404  Not Found   I looked on that server, and those paths are indeed not present there.  I tried to figure out on the Debian backports site whether this particular repository should still be available, and I didn't find any indication that this was deprecated or something like that.  Is this a temporary issue with the repository, or is the jessie-backports repository not available anymore? And if this is not a temporary issue, what options do I have to use this or an equivalent repository without upgrading to the newer Debian stable version? I need help with this:  If a is an element of $\mathbb Z_n$ and $\gcd(a,n) &gt; 1$ , then a is not invertible.   First you show that if $a$ is an element of $\mathbb Z_n$ and $\gcd(a,n)&gt;1$, then there  is an element $b$ of $\mathbb Z_n$ and ($b$ is not equal to the zero element) from which $ab = 0$.   The second part I cant get. i.e. show if $b$ is not the zero element  and $ab=0$ then $a$ is not invertible. Stop terminal auto executing when pasting a command Right, this is a slightly annoying feature because something this happens, and other times it doesn't, very odd.  I paste a command into the terminal with a view to tweaking its arguments, and sometimes it executes the second I paste it. Sometimes it doesn't.  This ranges from embarrassing ( posting nonsensical comments on an IRC ) to dangerous.   How can I shut this feature off for good? I never, never, never want the terminal to auto execute when I paste to it. Word for one who likes to share knowledge when asked i want to compile several pgfplots pictures with lualatex, since it dynamically allocates memory for compiling (in contrast to pdflatex).  In some plots, like my MWE.tex given below, i want to use a contour plot. This is typically done with gnuplot. To compile a document like this i need to use --shell-escape, such that gnuplot command can be executed while compiling the document.  \documentclass[a4paper]{article}  \usepackage[utf8]{luainputenc} \usepackage{pgfplots} \pgfplotsset{compat=newest} \begin{document} \begin{tikzpicture}   \begin{axis}     \addplot3[contour gnuplot]      {x+y};       \end{axis}   \end{tikzpicture} \end{document}   If i use  pdflatex --shell-escape MWE.tex   everything works fine, and the log says something along the lines  This is pdfTeX, Version 3.14159265-2.6-1.40.17 (TeX Live 2016) (preloaded format=pdflatex 2016.6.12)  13 JUN 2016 10:46 entering extended mode  \write18 enabled. .... Package pgfplots notification 'compat/show suggested version=true': document has been generated with the most recent feature set    (\pgfplotsset{compat=1.13}).  \openout3 = `MWE_contourtmp0.dat'.  \openout3 = `MWE_contourtmp0.script'. runsystem(gnuplot "MWE_contourtmp0.script")...executed.   If i use     lualatex --shell-escape MWE.tex i get the following error:  This is LuaTeX, Version 0.95.0 (TeX Live 2016)  (format=lualatex 2016.6.12)  13 JUN 2016 10:52  system commands enabled. ... Package pgfplots notification 'compat/show suggested version=true': document has  been generated with the most recent feature set (\pgfplotsset{compat=1.13}).  Package pgfplots info on input line 10: Using 'lua backend=false' for plot 0 (type 'contour external'): plot handler does not support LUA.  \openout3 = MWE_contourtmp0.dat  \openout3 = MWE_contourtmp0.script gnuplot "MWE_contourtmp0.script"  ! Package pgfplots Error: sorry, plot file{MWE_contourtmp0.table} could not be opened.   The .table file exists and looks identical to the one created by pdflatex, but it says that it cannot be executed.  (I just installed texlive 2016 yesterday, so everything should be up to date.) Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$ Show a connected metric space (X,d) is $\epsilon$-chainable for $\epsilon &gt;0$       $\epsilon$-chainable Definition      (X,d) is $\epsilon$-chainable if given any two points $a,b \in X$, $\exists$ a finite set $\{ x_0,x_1,...,x_n\}$ of points in X with $x_0 = a$, $x_n=b$ and $d(x_{n-1},x_n)&lt;\epsilon$, $\forall$ n      Clopen Definition      A set is clopen $iff$ its boundary is empty      Connected Metric Definition      Let $(X,d)$ be a metric space. It is connected $iff$ it has no proper subsets which are clopen      The following are equivalent:         $X$ is connected   The only subsets of $X$ that are clopen are $\emptyset$ and $X$   $\nexists$ non-empty, disjoint, open sets $A$ and $B$ such that $X=A\bigcup B$   $\nexists$ non-empty, disjoint, closed sets $A$ and $B$ such that $X=A\bigcup B$         Theorem      Let $(X,d)$ be a metric space, and let $S$ be any subset of $X$. Then $S$ is closed $iff$ $S^c$ is open.     So how would someone show this exactly?  Let the set $A$ be the set of all elements in $(X,d)$ that can be reached from $a\in X$ in a finite number of steps with length less than $\epsilon$.  Does this mean that there should be no isolated points? (an isolated point would be at distance infinity from any other point, or distance zero from itself, since $\epsilon &gt;0$ we don't want this?)  Knowing that $X$ is connected $\implies$the only subsets of X that are clopen are $\emptyset$ and $X$  Also, by construction of A;   A is the set of all $\epsilon$-chains between any element and $a$  And, so $A^c = \emptyset$ if $A$ was $\epsilon$-chainable (because A has to have a path from all elements to a, for this to be true, the complement wouldn't have anything in it)  Leaving only $A$ as the remaining clopen set  Does this make any sense? Or should an alternative be used, and how to formalize it.  Thanks. I am trying to prove the following identities:  a. $$\sum_{k=0}^n(-1)^k{n\choose k}^2 = \bigg\{^{0 \ \text{if k is odd}}_{(-1)^m{2m\choose m} \ \text{if n = 2m}}$$ b. $$\sum^k_{i=0} {n+i \choose i}={n+k+1\choose k}$$  For part $b$ I know I can use induction. It is clearly true when $k=0$ and then assuming for $k-1$ we get $$\sum^k_{i=0}{n+i\choose i}={n+k\choose k-1}+{n+k \choose k}={n+k+1\choose k} \ ...$$ Does anybody know of a method to trigger an event in Prototype, as you can with jQuery's trigger function?  I have bound an event listener using the observe method, but I would also like to be able to fire the event programatically.  Thanks in advance We, as high school students have been taught that-because Bohr's model of an atom assigns specific orbits for electrons-that it is better than Rutherford's model. But what Rutherford failed to explain was why electrons don't emit EM waves &amp; fall into the nucleus. I don't see how the introduction of 'atomic orbitals' overcame this defect. Can't it still radiate EM waves? I like having my hyperref hyperlinks a different colour, so it's obvious that they are links. But when I print a copy of the PDF, I don't want the links to be coloured, because you can't click them, so it doesn't matter and it's distracting.  Other than just having a separate "for printing" version of the PDF where I turn off the hyperlinks, is there a way to have the printed version come out without the coloured links? How to get text under mouse in any open window Which email was the verification email sent to? I started my app today and it said that I needed to verify my email. But I don't know the email I chose. I logged on every email in but there was never a verify email. Is there a way to know your email on Pokemon Go? How to Count Duplicates in List with LINQ How/when to become independent in research as a graduate student? I am in the last months of my MSc and I have done my thesis. I have published some conference papers but most of them were literature reviews.  My MSc thesis is about solving wave equations by means of method of potentials and boundary elements. I am interested in pursuing research in another field (differential equations) but I have no idea how to start any topic that may lead to a journal publication.    By asking this question, I want to understand how a research student, whether a recently graduated MSc student or a PhD student, can start searching for a new research topic and develop some researches based on his own interests. Specially, I would like to know:   To what extent should a graduate student be dependent on faculty members? How can he be more independent and start his own research activities? Is it a good idea that the graduate work on another topic with the faculty member or work independently on a more recent research topic? If the person is in a research group or at the university, is it a good idea or is it ethical that he develop his own research interests, besides to the topics which are being developed in that institute? How should a graduate student manage this and make some progress in the research fields of his own interest? (In particular, sometimes the faculty members are not interested in the topics you are interested in and the person has to start his own research activity.)      And also, if this research is part of his independent research, at the time of publishing what affiliation should be used in the publications. Should it be "Independent Scholar" as mentioned in  or something else? This is important in academic publications, because sometimes, the person is in a research group, but he is publishing a paper as part of his own researches not the things he is doing in that institute; in this case, has he mention that institute's affiliation or use his own one? So please make it clear that how the affiliation would be when the person does research independently in the cases like when he is not working in a research group or he is working with a professor or research group but his publication is output of his own research activities? It is a little unclear. Stein's lemma condition (Apologies if I break some conventions, this is my first time posting!)  I am working on proving Stein's characterization of the Normal distribution: for Z $\sim N(0,1)$ and some differentiable function $f$ with $E[|f'(Z)|] &lt; \infty$, $$E[Zf(Z)] = E[f'(Z)]$$ Writing the LHS expression in integral form and integrating by parts, I eventually obtain: $$E[Zf(Z)] = \frac{1}{\sqrt{2\pi}} \left[ -f(z) \cdot \exp \left\{ \frac{-z^2}{2} \right\} \right] \Bigg|_{-\infty}^{\infty} + E[f'(Z)]$$ Now I need to show that the first expression on the right hand size is zero. Intuitively, this seems clear because of the exponential term, but I am having trouble explicitly applying the condition on $f'$ to prove this rigorously. Any ideas? What is the fastest way to get from Courmayeur/Aosta to Milan? Convert a Bitmap to GrayScale in Android Multiple Linked Objects with unique Materials / Group Materials override How to escape text for regular expression in Java Show that given seven real numbers, it is always possible take two of them, such that $\left\vert\frac{a-b}{1+ab}\right\vert<\frac{1}{\sqrt{3}}$ Show that given seven real numbers, it is always possible take two of them, such that $$\left\vert\frac{a-b}{1+ab}\right\vert&lt;\frac{1}{\sqrt{3}}$$   The "Pigeonhole principle" states that if $n$ items are put into $m$ containers, with $n &gt; m$, then at least one container must contain more than one item, but how I an define my "container" such that meets the condition of the problem?