Let $G$ be a group where $o(G)=15$. If $G$ has only one subgroup of order $3$ and only one of order $5$, prove that $G$ is cyclic. Generalize to $o(G)=pq$, where $p$ and $q$ are primes.
This is question 2.9 #9 from Topics in Algebra by Herstein:     If $o(G)$ is $pq$ where $p$ and $q$ are distinct prime numbers and if $G$ has a normal subgroup of order $p$ and a normal subgroup of order $q$, prove that $G$ is cyclic.   Let $N$ denote the normal subgroup of order $p$ and $M$ denote the normal subgroup of order $q$. Here are a couple things I noted while exploring the problem:   $N$ and $M$ are cyclic, since they have prime order. $G = NM$ [1].   Any hints where to look next?    [1] Since $N,M$ are normal, $NM \le G$. Then $o(NM) \mid o(G)$, so $o(NM)$ is either $1$, $p$, $q$, or $pq$. The first three don't work because they force one of $N$ or $M$ to be $(e)$, so we have $o(NM) = pq$. Therefore, $NM = G$.
In English, especially in a poetic description, ships and cars are  referred to as "she".   Maybe their owners compare their beauty and elegance with those of women, but what if a female owner wanted to describe her yacht or car?   Would she also say, "She is a real beauty", or it might be "he" then?   Does the choice of a pronoun depend on whom she is giving the description of it to — a man or a woman?  What about a man-o’-war or a merchantman? Being ships, are they also referred as "she"?  P.S.  on the use of "she" in reference to ships, but there seems to be no answer to my question there.  P.P.S. My question wasn't pulled out of a thin air. I came across a book by a prominent Russian linguist, G.Veichman, written in 1990, (Novelties in The English Grammar is my translation of the title), where I read that the rule of personification of vehicles may be added regarding a female owner talking about her car using the preposition "he". Period. No further explanation or reference examples. Integrity constraints in a relational database - should we overlook them? I’m in a permanent discussion with the developers of the company where I work because they say it is better to get rid of relationship enforcement (via FOREIGN KEY constraint definitions) in a relational database in order to speed up large queries and to gain better performance. The platform under consideration is MySQL 5.x, and no FOREIGN KEY has been set up, even some PRIMARY KEY constraints of the relevant tables are missing which, at least for me, is not reasonable. Maybe they’re right and I’m wrong, but I don’t have enough arguments to discuss about this situation. This has been the preferred approach for three years now. I’m new in this company (only one month) but, as the product “works”, there is hesitation to enhance the database; nevertheles, the first thing I noticed is one page taking 1 minute to load (yes, 60 seconds!). One of the claims behind the current state of affairs is that a “denormalized” database is faster than a normalized one, but I don’t believe that’s true. Most of the relevant queries include JOIN operations, which makes them run very, very, very slow with large amounts of data (the database contains millions of rows). Commonly, the handling of “CRUD” operations is implemented at the application program code level; for example, in order to DELETE some data FROM, let’s say, TableA:  it is necessary to first check on the fly if there is some relationship between the rows of TableA and TableB, in case that said relationship is “detected”, then the app program code won’t allow to DELETE the pertinent row(s), but if for some reason the app program code fails, then the DELETE operation will “succeed”, no matter if there is any relationship regarding the involved rows and tables.  Question Could you help me to elaborate a good, accurate and solid answer to enrich the debate?  Note: Maybe something like this has been asked (and answered) before, but I couldn’t find anything by means of Google. Add an image directly from the clipboard I've tried to find other questions covering this, but (surprisingly) I did not find any.  When I want to add a screenshot to a post, as far I know the only way to do it is to save it as a file and then upload the file, using Stack Overflow's image hosting. OK, much better than finding a hosting site yourself, but sometimes I wish I could just paste the image from the clipboard.  It's not as straightforward as it may seem, I know. , and it needs a Java applet to get it done. Suppose we have two twins travelling away from each other, each twin moving at some speed $v$:    Twin $A$ observes twin $B$’s time to be dilated so his clock runs faster than twin $B$’s clock. But twin $B$ observes twin $A$’s time to be dilated so his clock runs faster than twin $A$’s clock. Each twin thinks their clock is running faster. How can this be? Isn’t this a paradox? How do I replace the *first instance* of a string in .NET? Check if user is using IE How can I disable the "Welcome, Name!" banner notification for Game Center? Non-interleaved vertex buffers DirectX11 Calibre No Longer Working Calibre stopped working recently. Tried uninstalling and reinstalling. When I open a terminal and call $calibre I get the following error message: Traceback (most recent call last):   File &quot;/usr/bin/calibre&quot;, line 20, in &lt;module&gt;     sys.exit(calibre())   File &quot;/usr/lib/calibre/calibre/gui_launch.py&quot;, line 73, in calibre     main(args)   File &quot;/usr/lib/calibre/calibre/gui2/main.py&quot;, line 543, in main     listener = create_listener()   File &quot;/usr/lib/calibre/calibre/gui2/main.py&quot;, line 514, in create_listener     return Listener(address=gui_socket_address())   File &quot;/usr/lib/calibre/calibre/utils/ipc/server.py&quot;, line 110, in __init__     self._listener._unlink.cancel() AttributeError: 'NoneType' object has no attribute 'cancel'  Do I need to install some dependency that the sudo apt install didn't get? I tried installing via command line and the software installer/updater GUI. I'm on version Ubuntu 20.04.1 LTS. I made sure that my software in general is up to date also. I am trying to copy specific attribute table rows from one shapefile to a new shapefile that will contain only the specified rows in its attribute table, that is, if a field GDP_TOP10 = 1.   I am using this at the moment:  #Grab rows using Search Cursor with WHERE clause rows = arcpy.SearchCursor("C:/CGDP.shp", "GDP_TOP10 = 1") for row in rows:     arcpy.CopyFeatures_management("CGDP.shp", "New.shp") del row del rows   However, this just copies the entire shapefile, not the rows I want to select with the cursor. If I execute a print statement under my for loop instead, it works and prints only the relevant fields, but the copy to new shapefile command does not seem to copy specifically.   Is there a different ArcPy command to only copy rows that match a query? Which variables should I transform, center, and/or standardize in my data for Principal Component Analysis? I have multiple datasets that I am attempting to use principal component analysis (PCA) on in order to infer the underlying structure of the data.  I'm attempting to predict growth increment of conifers (the response which apparently needed to be cube root transformed for my original linear models).  My predictor variables include: continuous (some log-transformed like size-related variables, some compositional, and some traditionally scaled); binary; ordinal; and nominal (of which I'm pretty sure I can order with respect to site productivity).  I'm mostly attempting to make inferences because some of my previous models from linear regression have not produced significant terms corresponding to categories like health or climate and I feel that they should have.  I've read in Joliffe (2002) that I might want to leave my compositional data uncentered.  Forgive my naiveté, but I'm wondering if I should also do this for my binary data.  It also seems that I might not want to scale these since they're already on a 0 - 1 scale but I'm not sure.  After only using my continuous predictors (including compositional variables), it appears that a size principal component makes up the largest portion of the variability and Joliffe also mentions that one might want to attempt a "double-centering" of the log-transformed data if this is the case.  I don't know if this is even possible since the log-transformed data only makes up a small portion of my datasets.  Finally, I should ask if having such a combination of data types will complicate analysis so much that it will make inferences unreliable.  I am currently using R for this analysis.  Thank you. What changes when an Alienist becomes an outsider? How to correctly enable Desktop Cube in Unity 3D? Invalid characters for XML parsing in posthistory.xml from Stack Overflow We got the April 2011 data dump from Stack Overflow and, while trying to put everything into a database, we got      SyntaxError: reference to invalid character number: line 1105549, column 5445   in the middle of the posthistory.xml file. It seems that there are sets of characters (&amp;#xC I believe) within a few posthistory entries which break our XML parser.  Is there a solution besides manually changing all the C's? Sudo is also user login password - why? how to change? When installing Ubuntu I can create a user with an encrypted folder, what I always do. Now after the installation the login password for this user is also the sudo password. Is that how it should be? I actually would like to have a different user login and sudo password, does is make sense? Should I do that, how? Apply function conditionally Series of books featuring dragons, enchanted forests, magic swords, etc When I was younger I read a series of 3 or 4 books involving largely traditional mages/dragons/enchanted forests.  I'd like to buy it for some of my younger extended family members but I can't remember the name of it.  Here's what I remember from this:   I read the last book first, in which the son of a woman turned out to be largely one of the main characters in the other books takes an enchanted sword into a forest on a quest to do something. At one point in one of the earlier books the group was travelling through a swamp like area to get to a tower where a witch(or mage lived). There was a king (possibly the son's father) who was frozen in a castle during the last book. The evil mages/witches would melt (but not die) when some spell was cast on them. I read the series circa 1992 (estimated), comparable to the DragonLance series it was geared towards a younger audience (I don't remember any sex/brutal slayings of anyone). I believe in the first book the mother of the son in the last book runs away from home and becomes something of a maid to a dragon. I can't remember if there were evil dragons or not, I seem to remember that at least some of them were good (or at least largely neutral).   If I think of any other details I'll add them. Seeking Python script for creating .mxd files? I want to block Facebook (the Facebook site) for concentrating purposes, but I don't want to break the share or like buttons on other pages (like a good article or a nice clip on YouTube I want to like or share). How can I do that? I use both Windows and Linux. Whole disk unallocated? Gparted error or partition overlapping?