Number of homomorphisms $f:S_3 \to S_3$ when $\ker(f)=(1)$.
What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?
Is there a word for a person who gives out too many extraneous details? Exact Sequences of Modules and Rank Suppose that:  $0 \rightarrow M_{1} \rightarrow M \rightarrow M_{2} \rightarrow 0$  Is an exact sequence of $R$-modules (for $R$ commutative integral domain). Show that $\mathrm{rk}(M) = \mathrm{rk}(M_{1})+\mathrm{rk}(M_{2})$    I have shown the inequality $\mathrm{rk}(M) \geq \mathrm{rk}(M_{1})+\mathrm{rk}(M_{2})$, however for the opposite direction I'm lost. I tried to take $N$ as a linear independent subset and tried to decompose it into $N_{1}$ (the elements in $\mathrm{Ker}(g)$) and $N_{2}$ (the elements not in $\mathrm{Ker}(g)$), but I haven't got far, since $N_{1}$ gives us a linearly independent set in $M_{1}$, however $N_{2}$ can give us a linearly dependent set in $M_{2}$.   We define $\mathrm{rk}(M)$ in this case as the supremum over all cardinalities of the linearly independent subsets of $M$ (where the only finite linear combination equaling $0$ implies trivial coefficients for $r \in R$).  Would this be the proper way to show the result? I think I'm missing something important. Remove blank attributes from an Object in Javascript In school, I have always seen sets. I was watching a video the other day about functors, and they started talking about a set being a collection, but not vice-versa. I also heard people talking about classes. What is their relation? Some background would be nice.   It has to do with something called Russell's paradox, but I don't know what that is.   I think that the difference between a family and a set is that the former is a function and the latter is a set. Is this right? Getting tabular statistics from table using QGIS? Leaflet Customer Marker Changes Position with Scale Why can´t we assign a new string to an char array, but to a pointer? I have a JavaScript object. Is there a built-in or accepted best practice way to get the length of this object? const myObject = new Object(); myObject[&quot;firstname&quot;] = &quot;Gareth&quot;; myObject[&quot;lastname&quot;] = &quot;Simpson&quot;; myObject[&quot;age&quot;] = 21; in questions, answers, and comments. Can we get something like this going on Stack Overflow?  I think it’d be appropriate as I at least pretty often want to write something like n^2 and would benefit greatly from prettier markup. windows bat file error How do I use 'chmod' on an NTFS (or FAT32) partition? Alignment issue in Area 51 main site - site switcher, inbox and achievements Proof Check: Every Cauchy Sequence is Bounded I am planning to visit London for 6 days from the US. I have given my biometrics and when I was preparing to mail my documents to the UK visa processing center in the US, I used this link from the UK government website to cross-check the documents to be sent to the processing center.     (Guide to supporting documents).  Something that really surprised me was in Section 4 (documents that should NOT be sent along with the package) was hotel bookings and flights bookings.     Section 4: documents you should not send unless specifically requested      This page provides guidance on the types of documents that are not required to consider your application.         hotel bookings   flight bookings      Typically I have always been asked to provide my travel plans, accommodation plans and proof of them. This is the first time that travel plans and accommodation plans are not asked to be sent. What might be the reason behind it? This really confuses me. Any help in clearing my confusion would be greatly welcome. Scientific visualization - How to replicate instances around a sphere? I want to replicate a protein structure around an icosahedron. Basically a vesicle of protein structure is what we want:  When I am instancing the protein structure using each vertex of the icosahedron it looks like:  However, the tails are supposed to be directed towards the center. How can I make that work? The rules of Transfiguration: examples of making the inedible edible? Information from the official website of J.K.Rowling is generally considered canon. I remember at least one case when she 'changed her mind' after a particular rule has been  published on her website (what happens with a secret if a Secret Keeper dies). Nevertheless, the information from the website can generally be considered reliable.  A rather obscure piece of information is J.K.Rowling's Wizards' Ordinary Magic and Basic Aptitude Test () published on her website in 2007 and now accessible with a Time Turner.   In particular, we know that only one of the following options is correct (as the software allows you to choose only one option):     Which of the following is CORRECT?            a. Food can be conjured out of thin air.       b. Any object can be Transfigured into food.       c. Foodstuffs can be increased, Transfigured, summoned from a distance and magically cooked.       d. It is impossible to make the inedible, edible.   e. Food-related Charms are some of the simplest forms of magic.      W.O.M.B.A.T. Grade 3, 13 June 2007, question 8   The test appeared on J.K.Rowling's website before the release of Book 7, but we can almost safely assume that there was no time for J.K.Rowling to change her mind after compiling the test (though she must have submitted these questions weeks before they were eventually published).  As Hermione explained in Harry Potter and the Deathly Hallows:      Your mother can’t produce food out of thin air, no one can. Food is the first of the  to . It’s impossible to make good food out of nothing! You can Summon it if you know where it is, you can transform it, you can increase the quantity if you’ve already got some...   It means that the correct option is c, and therefore all other statements are false. So, apparently, you can make food out of something else, though not any object can be transfigured into food. Can you come up with any example of such transformation mentioned in the books? All  I just install a fresh new Ubuntu 16.04 on my new desktop which has a 8th generation Intel CPU i5 8600K. When OS installed, I found there is only one option in display setting: 1024 x 768 (4:3),  My monitor is a HDMI monitor with 1920x1080 resolution  Do anyone have any idea how to make the resolution setting right? Thanks.  BTW. I do not install any graphic card, I use the internal GPU of Intel Mask and inherit the same systemd unit Suppose a package provides whatever.service file so certain events (including whatever.socket) cause a daemon to start with a certain configuration.  I want to disable this completely. In particular, I want the old socket and any other unit files that might refer to this service by name to be unable to start the daemon.  Instead, I want to create my own unit files for this daemon with a different .socket unit and different environment variables. I also want my override to automatically take account of possible upstream updates to the original unit file (which my current solution of copying whatever.service as whatever-modified.service and editing it unfortunately does not provide).  If I create the following symlinks and files: /etc/systemd/system/whatever.service → /dev/null /etc/systemd/system/whatever-modified.service → /usr/lib/systemd/system/whatever.service /etc/systemd/system/whatever-modified.service.d/fix.conf  systemd considers the new service to be masked as well and refuses to start it. How to avoid this? One very convenient property of $\mathbb{R}$ as a ring is that there is an upper bound for the degree of irreducible polynomials in $\mathbb{R}[x]$, as      If $f\in\mathbb{R}[x]$ has degree larger than $2$, then $f$ is reducible.   However, the proof as I know depends highly on the fact that $\mathbb{C}$ is algebraically closed, and the very nice property: $z\in\mathbb{C}$ is in $\mathbb{R}$ if and only if $z=\bar{z}$.  This makes a generalization to other integral domains rather difficult. So the problem is      For what kind of integral domain $R$, we have a finite upper bound on the degree of irreducible elements in $R[x]$?   Some most familiar examples are ruled out: in $\mathbb{Z}[x]$, $\mathbb{Q}[x]$ and $\mathbb{F}_{p}[x]$, there is not such a bound. Unfortunately these exhaust all integral domains about which I have a working knowledge.  Another result might help is , which says if we can find a prime ideal $\mathfrak{p}$ in $R$ such that $\mathfrak{p}^2\neq\mathfrak{p}$, then by picking $a\in\mathfrak{p}^2\backslash\mathfrak{p}$ we have an irreducible $a+x^d$, where $d$ can be arbitrary, and hence the upper bound is not possible.  So we only need to focus on domains where $\mathfrak{p}^2=\mathfrak{p}$ for all prime ideals. This seems to be a quite strong restriction but I am not sure what to make of it.  Can someone give a hint?  Thanks! After offering a bounty to , I realize that I cannot comment on the question because I no longer have the required reputation. Is it possible to change this behavior, so that the bounty owner can comment only on the post with the bounty irrespective of their reputation? While bounty owners are told that they'll lose privileges at the time they place a bounty, it makes sense to allow them to interact with potential answerers/commentators to help solve the problem, or at least provide additional information.