Problem : Let $S$ be the relation G~H iff G is isomorphic to H.   Show reflexive, transitivity and symmetric.  First show G is automorphism, which will imply G~G. So the identity mapping gives us this automorphism.  So G~G. My question is, is this the right process for showing reflexiveness?  Now we show symmetry. Assume G is isomorphic to H. Thus there exists a mapping $\phi:G -&gt;H$.  We must show H is isomorphic to G. Since $\phi$ is this mapping, consider $\phi^{-1}.$ This is a bijection since $\phi$ is a bijection. How do I show it is homomorphic?  Finally show its transitive. This I have done in a previous problem.  This is different than the duplicate as suggested because previously I had done it incorrectly. I know now I must do it this way.
Let $S$ be the collection of all groups. Deﬁne a relation on $S$ by $G \sim H$ iﬀ $G ≈ H$. Prove that this is an equivalence relation. So $S$ is partitioned into isomorphism classes.   Proof: Let $S$ be a relation on S defined by $G \sim H$ iﬀ $G ≈ H$. Show reflexive, symmetric, transitive. It is obvious that $G≈G$ so $G\sim G$.  Assume $G\sim H$, thus $G≈H$, which implies that $H≈G$, so $H\sim G$, thus symmetric.  Assume $G\sim H$ and $H\sim K$, for $K$ a group in $S$. Thus $G≈H$ and $H≈K$. (I have already shown that if $G≈H$ and $H≈K$, then $G≈K$). From earlier this implies that $G≈K$, so $\sim$ is an equivalence relation.  Thus $S$ is partitioned into isomorphism classes. QED.
What command do I need to unzip/extract a .tar.gz file? I received a huge .tar.gz file from a client that contains about 800 mb of image files (when uncompressed.) Our hosting company's ftp is seriously slow, so extracting all the files locally and sending them up via ftp isn't practical. I was able to ftp the .tar.gz file to our hosting site, but when I ssh into my directory and try using unzip, it gives me this error:  [esthers@clients locations]$ unzip community_images.tar.gz Archive:  community_images.tar.gz   End-of-central-directory signature not found.  Either this file is not a zipfile, or it constitutes one disk of a multi-part archive.  In the latter case the central directory and zipfile comment will be found on the last disk(s) of this archive. note:  community_images.tar.gz may be a plain executable, not an archive unzip:  cannot find zipfile directory in one of community_images.tar.gz or community_images.tar.gz.zip, and cannot find community_images.tar.gz.ZIP, period.   What command do I need to use to extract all the files in a .tar.gz file? In Windows you have a C-drive. Everything labeled beyond that is with the following letter.   So your second drive is D, your DVD is E and if you put in a USB stick it becomes F and the following drive G. And so on and so forth.  But then, what and where are A and B-drives? How can I see flags I cast from all sites? This morning I flagged something on one of the sites where I have fairly low rep. Now I can't remember which site it was. I've checked my flag history on about half of them, even editing the URL for the site where I have no helpful flags, and I can't find this flag. I've been to my network profile and looked at the Activity tab, but it only shows public things like asking, answering, commenting, editing, and reviewing.  Is there a way to see my flagging history across sites? Can we have such a feature, if it's not possible? How to delete kexts in Catalina? I have a 2014 MacBook Pro experiencing . Catalina does not allow kexts to be deleted via this method. Unless I am able to delete or disable this kext my laptop will shut down randomly. Anyone know how to remove kexts in Catalina? Answers to a  mention that it is illegal for a US citizen to enter/leave their home country on a foreign passport. However this raises the following questions:   What is the maximum prescribed penalty for violating these rules? Are US immigration officials known for actually applying the penalty? E.g. the Cuba embargo banning visits to Cuba was almost never enforced, so it's possible this law is ignored as well. How do I display PHP code in HTML? Note: This is not the same as , which is about the moderator tool. I am talking about the flag history page that is accessible through the user profile at .  I tend to have a pretty good hit rate for flags, well above 90% helpful on several sites. I also semi-regularly review the flag history page, both to see the status of any outstanding flags as well as see how flags I raised have been handled. Particularly, whenever a flag of mine is declined, I look over the post I raised the flag against with this new information in mind, to try to determine if I should change my flagging behavior in some way to better match the specific site's scope.  However, comment flags are difficult in this regard. Most often, there is only a binary status: helpful or declined. Helpful comment flags nearly always lead to the comment being deleted, so that obviously could present a bit of a problem, but that's not really relevant because a helpful flag means you should just keep doing what you've been doing. Declined comment flags, on the other hand, usually mean the comment is kept. It's the declined comment flags that I want to learn more about. But there's no way to!  All the flag says is which post it was on, how I flagged a comment, that I was the one to raise the flag (which frankly is rather useless information when you are already looking at your own flags summary), when the flag was raised and whether it's pending, helpful or declined. In a very few cases the moderator handling the flag may leave a helpful return comment.  This can get even more confusing if I raise multiple flags in quick succession against comments on the same question including any answers to it, because they basically all get bundled together.  Even in the case of a declined flag where the comment remains, there is no way to find your way back to the specific comment you flagged! Thus it is very difficult to recognize the error of your ways. Because comment flags (with few exceptions) are handled by diamond moderators only, the lack of feedback about which comment's flag was declined means it's easy to keep wasting the moderators' time without even intending to do so.  Hence my feature request: add a link to the specific comment that the specific flag was raised against, ideally only if that comment remains. Interaction with the site on the app should count towards Enthusiast/Fanatic If I do any of the below, which would normally constitute "activity" on any Stack Exchange site (including mobile):   View my profile Vote on any question or comment Visit one or more questions   ...then Stack Exchange considers me to have "visited" the site.  If I do any of the below on the mobile app:   Vote on any question or comment Visit one or more questions   ...then Stack Exchange does not consider me to have visited the site.  If one wants the app to have feature parity with the mobile site, then shouldn't my participation within the app also count towards the Enthusiast/Fanatic badge? Return matched elements' attribute ar array using jQuery If $p$ and $q$ are distinct primes , prove that $$p^{q-1}+q^{p-1} \equiv 1 \pmod {pq}$$     Using Fermat's Theorem we can get $$p^{q-1} \equiv 1 \pmod q, \qquad q^{p-1} \equiv 1 \pmod p.$$  After that I have no idea at all. Can anyone help me please Why are only certain shutter speeds or apertures available? For the exposure control function DSLRs there are specific shutter speeds available. For example there are 1/4, 1/100, 1/500. Why, even in high end DSLRs, is there no option to use a custom shutter speed (such as 1/19)?   If the camera can do 1/4000, why can't it do 1/33?  Similarly, why are the aperture choices limited?  Why is it a technical challenge to add extra shutter speeds or aperture settings? This question attempts to collect a community-maintained list of quality books on the  programming language, targeted at various skill levels.  C is a complex programming language that is difficult to pick up on-the-go by reading online tutorials. A comprehensive book is often the best way to learn the language, and finding a good book is the first step. It is important to avoid badly-written books, and even more importantly, books that contain serious technical errors.  Please suggest edits to the accepted answer to add quality books, with an approximate skill level and a short blurb/description about each book. (Note that the question is locked, so no new answers will be accepted. A single answer is being maintained with the list)  Feel free to debate book choices, quality, headings, summaries, skill levels, and anything else you see that is wrong. Books that are deemed satisfactory by the C community here will stick around on the list; the rest will be regularly removed.  For books that have reviews by the Association of C and C++ Users (ACCU), a link to those reviews should be added along with the book.  See also:   Other C-related resources in the   A similar list for :      This question was discussed on  as part of the .The consensus was to keep it undeleted and actively maintained. Does a preprint include revisions made in response to peer review? This is related to but not answered in .  After an author has submitted a manuscript to a journal for peer-review, very often the journal asks the author to revise (and resubmit) the manuscript as a precondition for its publication. I am wondering whether the term preprint refers to the manuscript before or after its authors made these revisions.  If a preprint includes revisions made in response to peer review, then there is only a slight difference, mostly regarding layout and style, between the preprint and the article that is published in the journal. From a reader’s perspective, there would be no point in paying for access to the article, since they can simply download the preprint free of charge. A publisher would probably want to prevent their authors (by publication agreements etc.) from publishing preprints that differ from the respective journal article only in form.  As a side question, what kind of preprints do publishers tolerate, as a rule? I am new to data analysis and now working on a Mixed Effects Logistic Regression Model.  Currently, I have the following data frame (model_data):  Road Id Vehicle_id  entry_time           exit_time          1        1        2017-01-31 00:00:00  2017-01-31 00:00:00 last_veh_time  vehicle_type     status 300            4 wheel          0    Likewise, I have the information (nearly a million rows) of all the vehicles entering a place and exiting the same place.  Status can be either $1$ or $0$ for all of these.  last_veh_time is the last vehicle passed the same place before how much time.    I want to know the probability of a getting status $0$ for a vehicle based on the entry_time and last_veh_time. 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I do not want to format my phone then load back on WeChat for instance, yet I still want to be able to browse my old WeChat chats and voice clips (using a PC).  Thanks tables with lots of columns I have a table that lists various attributes of an individuals personal development plan; objective, time to complete objective, comments etc. The table is editable so the user can directly add comments without having to go to an intermediate form. The problem is that the client wants to add more columns and this is messing up the visual display.   The alternatives I have are:   allowing columns to expand out on click keeping essential attributes in table, rest in 'advanced edit option form' or somesuch I don't know - have you any ideas?   One more thing to add: I think the addition of extra columns is based on user feedback ...  edit: here is what I have gone for Database Encryption scan for database  was aborted. Reissue ALTER DB to resume the scan Proof that the Casimir invariant of a representation commutes with everything