True or false? $(\mathbb{Z} \setminus \left\{0\right\}, \cdot)$ is a   group   I need to know this because it's from an old exam and it might be asked in my exam, or a task like that.  I'm not entirely sure but I would say that the statement is false. The associative property will work, there will be an identity element too. But we will fail when we are looking for inverse elements as these would require rational numbers.  (So even if we had something like $(\mathbb{Z}, \cdot)$, it wouldn't be a group still.)    I hope I did it alright?
I am teaching myself information about groups, but don't really understand how to work through this problem. Here is what I have been thinking so far (please note that I do not need to work through showing any closure): $\mathbb{Z}^*$ denotes the nonzero integers. Prove or disprove: $(\mathbb{Z}^*, \cdot)$ and/or $(\mathbb{Z}^*, \div)$ is a group.  For $(\mathbb{Z}^*, \cdot)$:  Group. Associative Law: Let $a,b,c \in \mathbb{Z}^*$. Then $(a\cdot b)\cdot c=a\cdot (b \cdot c)$, and the associative law holds. Existence of Identity: Let $a \in \mathbb{Z}^*$, then $a \cdot 1 = 1 \cdot a = a$, and the identity holds. Existence of Inverses: Let $a \in \mathbb{Z}^*$. Then the inverse of $a^{-1}=-a$, and the inverse holds. Note that this only works because we are ommitting zero.  Hopefully this looks like I am on the right track. I don't know to what extent I can explain logic vs showing it with mathematical operations, so I would appreciate guidance in this respect.  For $(\mathbb{Z}^*, \div)$: Not a group. Disprove by counterexample.  Yeah, I dunno about this one. I don't think it's true, because we will be introduction rational expressions that won't allow this to hold, but I'm not sure where I can use this information to disprove the statement.
Authenticate to Stack Exchange using GitHub I've seen sign in methods using  as an authentication provider, for example : (AWS Cloud9)        Is it possible to authenticate into Stack Exchange using this method? At the moment there's other methods such as OpenID, Google, Yahoo and Facebook, but it would be interesting to also have the GitHub authentication. Can I have my sudo/su password different from my desktop login password? Why was the alien threat in The Next Generation episode "Conspiracy" never followed up? In the Star Trek The Next Generation episode  Picard and crew are drawn into an alien conspiracy to take over the Federation. This ends with Picard and Riker destroying an alien "mother" beast, but not before it manages to send a signal to an uncharted sector of space. The feeling at the end of the episode is that this will be a threat that the intrepid crew of the Enterprise would be facing again at some point, but this never happens.   Why was the implied threat at the end of this episode never brought up again on the show? EntityFieldQuery display pager How do I display pager using EntityFieldQuery for fetch results? I can't find complete example.  I am doing simmilar to:   But I miss part to output simple pagination links (1,2,3... for every next page). $\chi(G)+\chi(G')\leq n+1$ How to prove, that the sum of chromatic numbers of graph and it's complement is smaller then the number of vertices incremented by one?  $\chi(G)+\chi(G')\leq n+1$  The notes from my classes say to color the first graph using sequential algorithm with vertices ordered by their degrees (the ones with lowest appear first). We would get $k_1$ colors, then we color the graph $G'$ in the backwards order and get $k_2$ colors, and the sum $k_1+k_2$ in my notes is magically smaller than $n+1$, but I just don't see why... Resetting theorem counters: \section and (missing) \subsection Weird line in my mirrored mesh How do you add an ActionListener onto a JButton in Java Where does the word "torsion" in algebra come from? Why do we use $x - y$ rather than $x + y$ in the definition of the convolution? Is it just convention? (If we are thinking of convolutions as weighted averages, for instance against "good kernels," it should make no difference.)  Why $(f * g) (x) = \int f(y) g(x - y) dy$ rather than $(f * g) (x) = \int f(y) g(x + y) dy$?  Edit: I'm finding it really hard to choose a best answer. There are at least three very good ones here. What is difference between int.class and Integer.TYPE in java? Disable Narrator shortcut key in Windows I've install Windows 8 on a BootCamp macbook air The problem is that the command key located differently from the winkey in a normal keyboard, therefor, I keep clicking Winkey+Enter which starts the narrator (which is really annoying!)  Is there a way to disable this shortcut? How do you join or create a server in multiplayer? How can I buy apps in the App Store without a credit card? While I'm waiting for my credit card to arrive, I still want to be able to buy some apps for my iPad!  Are there alternative ways to pay for apps, without resorting to someone else's credit card? Specifically, define the sub-gaussian norm for a r.v. X as  $$ \|X\|_{\psi_2}=\inf\{t&gt;0:e^{X^2/t^2}\leq 2\}. $$  How do we prove that the centering inequality with $C=1$ does not hold in general? i.e. $$ \|X-\mathbb{E}X\|_{\psi_2}\leq C\|X\|_{\psi_2}.  $$ A counter example would work too. Thanks in advance! Downgrading MacBook Pro 16” to Mojave Did anyone succeed in installing Mojave on the MacBook Pro 16”? It ships with Catalina but my corporate requires Mojave to connect to the corporate network.  I  but once I select booting from the Mojave external drive I get a “forbidden” icon and the system shuts down. For example an electron radiates when accelerated. So does a positron. But is the radiation emitted by accelerated positronium the sum of the radiation emitted by each separately? If not, why not? If so, does this provide a way of testing whether a given neutral particle is composite? For example, does a neutron bremsstrahlung when decelerated? Inform new users if they can upvote an answer they accepted I surprisingly often stumble upon answers which were accepted but clearly not upvoted by the asker. For example, the answer has not received any upvotes at all or the asker’s vote count is zero. While I acknowledge that , most such cases happen with new users, who have only recently received the privilege to vote and who upvote after a friendly reminder.  Given the anonymity of votes and that I do not check the asker’s profile for every question with an accepted answer I stumbled upon, I assume that there is a huge dark figure for this phenomenon. Also, I naturally most often spot this on my own answers, which makes reminding the asker of their privileges somewhat awkward.  I thus propose that users which accept an answer are reminded by a pop-up that they can upvote it, if:   They haven’t already upvoted it. They have the reputation required for voting. They have cast less than a handful of votes.   I consider the latter criterion to be better than an upper reputation for which the message is displayed, since I know at least one user with several thousand reputation points, who has cast much less upvotes or votes on answers than he has accepted answers on some sites (not that his acceptance rate or general feedback on answers is spectacular). SF Story about human adapting to alien environment This is a story that was read on BBC radio about thirty years ago, and this is about as right as I can remember it: The human pilot of a spacecraft is doing some reconnaissance on an alien planet. He crashes but survives. For some reason, no one will know where to look for him. He sees something in the distance that looks like a building of some sort. When he gets there, it seems to be a home. Very much like a human home. There's furniture, but it's strange and uncomfortable . There's food, but it's inedible. There's a faucet but what comes out of it seems more like acid. Everything looks as though it should be right, but it's not. He leaves the house, wanders and after a few days is desperate for water. But there isn't any water anywhere. He's close to death, crawling on the ground and then something happens. There are cracks in the soil and there appears to be water bubbling up from perhaps some underground stream. The water revives him. He goes back to the house and tries the food again. It's delicious and he eats enough to gain back his strength. Then he decides to try the water again. It's pure and wonderful. He takes a shower and the last line of the story is how refreshing the water is on his skin and something about him lifting his tail. How to Tail -f output with colors using only Awk and show the rest of the output? I was able to do a tail -f output with colors using the awk command. But I'm unsure how to output the rest of the log as it seems to be filtering only those 2 lines. I also want to see the other lines as the log is written.  What do I need to put in the awk command to show the rest of the output?  Image of my issue. The red text above is without using awk and displays the entire tail -f log. And below that is the same command with the awk color filtering.