Let G be a group s.t $g^3=e$ for all g in G. Prove that $G$ is abelian. Let G be a group s.t $g^3=e$ for all $g$ in $G$. Prove that $G$ is abelian.  From here I have got $(ab)^2=(b^2)(a^2)$ &amp; $(ab)^3=a((ba)^2)b$ Then what??
If a group satisfies $x^3=1$ for all $x$, is it necessarily abelian? I know that any group satisfying $x^2=1$ for all $x$ is abelian. Is the same true if $x^3=1$? I don't think it is, but I can't find a basic counterexample.
Round a double to x significant figures Evaluating the infinite product $\prod\limits_{k=2}^\infty \left ( 1-\frac1{k^2}\right)$ Evaluate the infinite product $$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$  I can't see anything in this limit , so help me please. Execute a Java program from our Java program I used   Runtime.getRuntime().exec("_____")   but it throws a IOException as below:  java.io.IOException: CreateProcess: c:/ error=5   at java.lang.Win32Process.create(Native Method)   at java.lang.Win32Process.&lt;init&gt;(Win32Process.java:63)   at java.lang.Runtime.execInternal(Native Method   I don't know whether I have the problem with specifying the path or something else. Can anyone please help me with the code. I am getting a warning when trying to include the .net 3.5 sp1 prerequisite for my setup project.  The warning states Prerequisite could not found for bootstrapping.  Any suggestions?  Thanks I want to convert .mov to .mp4! I saved the video from iMovie but it is in .mov format! How do I convert it to .mp4?  I'm using a MacBook and I want to share it to my phone, but my phone can't open .mov format video. Today I was informed by VFS that my tourist visa for UAE got approved. I'm travelling to UAE on 16th December 2014 and returning on 7th Jan 2015. But when I checked the "valid until" date on the visa it states 30 December 2014.    Does this date mean I should return before December 30th? Should I get it corrected from VFS? How do I install HP P1102W printer driver How do I set my DNS when resolv.conf is being overwritten? Most of the info I see online says to edit /etc/resolv.conf, but any changes I make there just get overridden.  $ cat /etc/resolv.conf  # Dynamic resolv.conf(5) file for glibc resolver(3) generated by resolvconf(8) #     DO NOT EDIT THIS FILE BY HAND --  #     YOUR CHANGES WILL BE OVERWRITTEN nameserver 127.0.1.1   It seems that 127.0.1.1 is a local instance of dnsmasq. The dnsmasq docs say to edit /etc/resolv.conf. I tried putting custom nameservers in /etc/resolv.conf.d/base, but the changes didn't show up in /etc/resolv.conf after running sudo resolvconf -u.  FYI, I don't want to change DNS on a per-connection basis, I want to set default DNS settings to use for all connections when not otherwise specified.  UPDATE:  I answered this question myself:   I think it's the best solution since:   It works. It requires the least amount of changes and It still works in conjunction with dnsmasq's DNS cache, rather than bypassing it. This problem occurred to me when I came across a similar problem where the radii were taken over only the primes. That question was unanswered, but it seems to me infinitely many circles of radius $1/2, 1/3, 1/4...$ can fit into a unit disk. The area of all those circles would be $\pi \sum_2^\infty 1/n^2 = \pi^3/6 -\pi$, which is less than the area of the unit disk $\pi$. But can the circles actually be packed with no overlaps? How to answer a negative question without ambiguity? I faced a problem to answer a negative question, for example, when someone asks you:      Don't you have any money?   It's a yes/no question but how should one answer the question without ambiguity?  When you answer "yes", does it mean "yes, I don't have any money"? or the other way "Yes, I do have money"? the questioner may think you are agreeing to the negative, that yes, you do not have any money.  I know it's better to answer with "on the contrary" but is it possible to answer yes or no? Does it matter what I choose for serialVersionUID when extending Serializable classes in Java? What is the default root password? Lufthansa airline offers the &quot;Economy Flex&quot; fare. The listed advantages of this fare are:  Rebooking possible plus fare difference Refundable except 70 EUR plus fare difference  What does the &quot;plus fare difference&quot; mean? Imagine I buy a ticket for 200 €. Later I decide to cancel. They keep 70 Euro, so I get 130€ back. What's with the fare difference? Fare difference to what? As a comparison, the &quot;Economy classic&quot; states:  Rebooking: 70 EUR plus fare difference Refund not possible Peggy would like to prove to Victor that she knows the discrete logarithm of $y$ based $g$; that is, she knows $x$ such that $y = g^x \bmod p$. One round of the interactive proof protocol consists of the following steps.   Peggy picks random $k \in \mathbb Z/(p−1)\mathbb Z$, computes $t = g^k \bmod p$, and sends $t$ to Victor. Victor picks random $h \in \mathbb Z/(p−1)\mathbb Z$ and sends $h$ to Peggy. Peggy computes $r = (k − hx) \bmod (p − 1)$ and sends $r$ to Victor. Victor verifies that $t = g^r y^h \bmod p$.   The interactive protocol can be converted into a noninteractive zero-knowledge proof by choosing and making public a collision-resistant hash function $H$, and changing the second step of the interactive protocol to the following: Peggy computes $h = H(y, t)$. Then the noninteractive proof consists of $(t, h, r)$, which can be verified as follows: $$h = H(y, t), \qquad t \stackrel?= g^r y^h \bmod p.$$   What is the problem if in the non-interactive proof the hash $h$ depends only on $y$? That is, $h = H(y)$, and the proof consists of $(t, h, r)$, which can be verified as follows: $$h = H(y), \qquad t \stackrel?= g^r y^h \bmod p.$$ What is the problem if in the non-interactive proof the hash $h$ depends only on $t$? That is, $h = H(t)$, and the proof consists of $(t, h, r)$, which can be verified as follows: $$h = H(t), \qquad t \stackrel?= g^r y^h \bmod p.$$ ${!FOO} and zsh ${!FOO} performs a double substitution in bash, meaning it takes the (string) value of FOO and uses it as a variable name. zsh doesn’t support this feature.  Is there a way to make this work the same in bash and zsh?  Background:  I’ve got a list of environment variables, like  PATH MAIL EDITOR   and want to first print the variable names and afterwards their values.  This works in bash but not zsh:    for VAR in LIST do         echo $VAR         echo ${!VAR} done   It should be somehow possible “the old way” with eval, but I can’t get it to work:  for VAR in LIST do         echo $VAR         echo `eval \$$VAR` done   I’m never going to understand why I can’t simply do arbitrary deep substitutions like ${${VAR}} or even ${${${VAR}}} if need be, so an explanation for that would be nice, too. Will wolves attack and kill ocelots in Minecraft? I know they kill sheep. I just don't want my cats being eaten. Sometimes when adding an object to the scene I see that all options (on toolbar or in window after pressing F6) are disabled. Can anyone tell me when can this happen? What is causing it and how to revert it - make it enabled again. It worked for me when I restarted Blender and reloaded the scene. However I would really like to know what can cause it? "It is they who lied" or "it is them who lied?" Which is the correct usage of the third person, plural pronoun?     It is they who lied.   It is them who lied. Difference between “with”, “at” and “for” with a business title and a company name How to find normal and lognormal moments, given partial information? $Y=\ln(X)$. $X$ is lognormal and $Y$ is normal. If all I know is the arithmetic mean of $Y$ and the standard deviation of $X$. What is the formula to calculate the arithmetic mean of $X$ and the standard deviation of $Y$?