This is a true or false question. Apparently, it is false, but I don't follow.  Clearly, if it contains an odd permutation, and an even/odd permutation is defined by the number of transpositions it could be broken into, then every permutation is a set of transpositions. At least, that's what I had gotten out of it. Am I wrong? I'm very confused with the answer.
Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition?    Here $S_9$ denotes the group of all permutations (i.e. bijections with itself) of the set $\{1,2, \ldots, 9 \}$ under the binary operation of composition of functions.   By a transposition is meant a permutation that leaves all elements except two fixed, mapping either of these two onto the other.   A permutation can only be expressed either as a product of an even number of transpositions or as a product of an odd number of transpositions (but not both) and as such is defined to be even or odd, respectively.   Reference: Fraleigh p. 95  Question 23d. in A First Course in Abstract Algebra
Different dependencies for different build profiles Insert image from Google Drive into Google Sheets How do I insert/link to images existing in a folder in Google Drive into a Google Sheets spreadsheet?  There is the image() function, I imagine using it something like this:  image("googledrive://somefodleringoogledrivewithmypics/pic_1.png")   How?  Update: Some people noted that by uploading pictures to Picasa, we can then get the URL of each picture and use that. It works. The downside of this is that we have to copy each and every URL of all images. I put that in a separate sheet, then use VLOOKUP to get each image URL per index. It works but the URL copying part is a hell of a job if you have lots of images of course. A better solution is needed. Converting case of all values in all fields of table using ArcPy & Python? In Lieb and Loss's Analysis, I saw that they mentioned  $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$ (dense wrt the $L^2$ norm, I think). But I didn't find its proof in the book. So I wonder why it is?    Does this conclusion hold for $L^1(\Omega, \mathcal{F}, \mu) \cap L^2(\Omega, \mathcal{F}, \mu)$ for any measure space $(\Omega, \mathcal{F}, \mu)$?  Are there similar statements if replace $L^1$ with $L^p$, and  $ L^2$ with $L^q$ for $p \leq q \in (0, \infty ]$ or $\in [1, \infty]$?  Thanks and regards! Mysql resolving 127.0.0.1 to hostname $(x^3+xy^2-y)dx+(y^3+x^2y+x)dy=0$ I tried to find on Wolfram Alpha but it showed only the solution not in step-by-step, I know that the factor is $\frac{1}{x^2+y^2}$ but how to find it? What was Dolores Umbridge's blood status? Dolores Umbridge's dislike of Muggles, Muggle-born witches and wizards and half-breeds is quite obvious. In Order of the Phoenix she takes a particular dislike to Hagrid due to his giantess mother (and his allegiance to Dumbledore), as well as to the centaur Firenze who takes over as Divination teacher after Professor Trelawney is sacked. And, of course, during Deathly Hallows she is made Head of the Muggle-Born Registration Commission.  In Deathly Hallows, when Hermione (under the effects of Polyjuice potion) comments on the locket she's wearing, Umbridge says this:     "That's - that's pretty, Dolores," she said, pointing at the pendant gleaming in the ruffled folds of Umbridge's blouse.      "What?" snapped Umbridge, glancing down. "Oh yes - an old family heirloom," she said, patting the locket lying on her large bosom. "The S stands for Selwyn... I am related to the Selwyns... Indeed, there are few pure-blood families to whom I am not related. ..."   That seems a rather large and contrived lie to tell just to explain her possession of the locket, considering her blood status had never been questioned by the Ministry. I don't remember it being covered anywhere in the books (which I just recently re-read, so I'd hope I'd remember if it had been!), but I was wondering if there was any other evidence (interviews, etc) to say whether Umbridge was Pure-blood, Half-blood or Muggle-born? What is the best time to ask questions? ERD for a web development company I am working on an ERD which represents the relationships between entities in a Web development company behavioral model. There I have entities like:   Customer Service ----------- (like: web development, web design, web consultation, ...) Order-item ------- (like: web development, web design, web consultation, ...) Order Invoice Project Project-phase Task --------------- (like: web development task3, web design task7) Developer Delivery ----------- (like: the developed web app, web consultation session, ...)   I have this diagram so far, though I know it still has major problems, and it is just for a start point:    Main concept   Customer makes order-items on services, combined to form an order which relates to an invoice. Each project has multiple project-phases with each project-phase related to an order. Each project-phase has multiple tasks, each of which is assigned to a developer. Each project-phase has a delivery, which will be delivered upon payment of the related orders invoice.   Questions   Are has and contains completely the same? If not, how to choose between them for relationships?! what is the measure?! Are there any rational problems with the Main concept section? Any helpful ideas or suggestions to improve this model?   Any help with this, would be really appreciated.  Update 1  Refined version of the diagram     Update 2  Considering an analogy of a Fast food restaurant would be helpful:   Customer ---------- A hungry person Service ------------- Making a burger, Home delivery, Mobile delivery, ... Order-item --------- A burger, ... Order ---------------- A home-delivered burger Invoice -------------- The printed invoice Project -------------- Serving the customer for a complete year (like a contract) Project-phase ----- Serving the customer for a single day Task ----------------- Preparing the kitchen, process the meat, ... Developer ---------- The cookie Delivery ------------- A burger in wrapped in wax papaer and a drink with the tray   Update 3  ()     Project seems to be unrelated to Customer in your diagram. Is this what you intended?   I think Order is and Project should not be related, as I think it is something off-the-counter and customer has nothing to do with it. she/he is only after the delivery and is concerned about time and money, nothing else. Please tell if I am wrong.     Are all Orders part of a Project or can you have orders that aren't in a project at all?   All Orders are part of a project, except those new ones which are waiting in the process queue.     Similarly, you relate order item directly to customer, but also to order. Are orders split between customers?   Please think of Order-Item as these:   web design web design consultancy domain name consultancy ...   So, customer choose some of these items, collected as an Order which has an invoice and could be paid. Order-Items are directly related to offered Services. I don't understand what do you mean by Are orders split between customers?.     If not you don't need a ternary relationship between customer, order item and service. Rather, an order item is an intersection between order and service. Order items belong to customers by inheritance from order (or even project?)   This is somehow true: Order items belong to customers by inheritance from order, but I think it is more rational to say it like this:     Order-Items, inherited from Service, belong to Order which itself belongs to Customer. I'm using TeXLive 2014. Here's a MWE:  \documentclass{article} \usepackage{tkz-fct}         % Must appear before {thmtools}  \usepackage{thmtools} \declaretheorem{theorem}  \begin{document}   \begin{tikzpicture}     \tkzInit  % No need to even plot anything!   \end{tikzpicture}    \begin{theorem}[A Silly Theorem]     Let $a = 1$. Then $a &gt; 0$.   \end{theorem} \end{document}   The error message is:  ! Extra \endcsname. \kv@processor@default ...fincsname\@xa \endcsname                                                   \csname iftrue\endcsname \... l.12   \begin{theorem}[A Silly Theorem]   If the optional theorem name is removed, all is well. Adding \protect{} doesn't help.  Any idea how to remove this conflict?  Thanks! This is the views section of the site analytics visible to 25k users on one of my sites:    That's, uh, not correct.  Our site has had its usual activity level during this time.  I checked a couple other sites, and the pattern is consistent.  This looks like a problem in the analytics data that's coming in; did Google change their API?  Update: Same site a week later; note that in addition to the brief spike of accuracy, page views have dropped to (near-)zero for the whole period, unlike the earlier report.  (I know this question has been closed as a dupe, but I thought having the two screen shots together might help somebody.) Standard library sort and user defined types How can I start automatically a command on Ubuntu 18.04? System log full of session and slice messages Suppose you have a physical . It is true that as amplitude increases, the period increases. Can we demonstrate this fact without explicitly finding the period (which is pretty involved and pretty messy) in:    an intuitive fashion,  rigorously? Determine the number of lines within a text file Is there an easy way to programmatically determine the number of lines within a text file? On the first puzzle of the quest Under Saarthal (the one with the six pillars), I enter the correct combination, but the door still doesn't open, and arrows are still shot at me. I've reloaded the game a couple hundred times, I've left the room, and it still doesn't work. What do I do? Boot failure : No DEFAULT or UI configuration directive found! It seems that I'm missing something about this.  First of all, the series is convergent: $\lim_{n\rightarrow\infty}\frac{2^{-n-1} (n+1)^2}{2^{-n} n^2}=\frac{1}{2}$ (ratio test)  What I tried to do is to find a limit of a partial sum $\lim_{n\rightarrow\infty}S_n$ as follows: $S_n=\frac{\frac{1}{6} n (n+1) (2 n+1)}{\frac{1-\left(\frac{1}{2}\right)^n}{2 \left(1-\frac{1}{2}\right)}}$. Still, the limit is $\infty$ and I'm clearly doing something wrong. What are complex numbers, actually?