Let $G$ be a finite Abelian group such that it contains a subgroup $H_0 \neq (e)$ which lies in every subgroup $H \neq (e)$. Prove that $G$ must be cyclic. What can you say about order of $G$ ?   I have examples to illustrate this result, but i can no idea about the general proof.  Here's my examples:  1) Take $G=\Bbb{Z}_8$ and $H_0=\{0,4\}$. Clearly $H_0$ lies every other subgroups $(\neq(e))$, namely lies in $\{0,2,4,6\}$ and lies in $\Bbb{Z}_8$   Here, $o(G)=8=2^3$  Edit:  2) Similarly, take $G=\Bbb{Z}_{16}$ and $H_0=\{0,8\}$. Clearly $H_0$ lies every other subgroups $(\neq(e))$, namely lies in $\{0,4,8,12\}$, $\{0,2,4,6,8,10,12,14\}$ and lies in $\Bbb{Z}_{16}$   Here, $o(G)=16=2^4$  3) Likewise, $G=\Bbb{Z}_{4}$ and $H_0=\{0,2\}$. Clearly $H_0$ lies every other subgroups $(\neq(e))$, namely lies in  $\Bbb{Z}_{4}$   Here, $o(G)=4=2^2$  From this examples, can i conclude $o(G)=p^n$, for a prime $p$ ?  Because, if $o(G)=p^n$, then there are unique subgroups of order $1,p,p^2,p^3,..,p^{n-1}$ and in this case we take $H_0$ as a subgroup of order $p$, then it is immediately lies in other subgroups.  Is this correct ? What is the general proof?
Let $G$ be a finite abelian group s.t. it contains a subgroup $H_{0} \neq (e)$ which lies in every subgroup $H \neq (e) $. Prove that $G$ must be cyclic.   Also what can be said about $o(G)$ ?    I'm clueless about this problem from Herstein. I tried to think $G$ might be a solvable group and its subgroups are all normal (as $G$ is abelian) . But I'm unable to get further.
Interaction of Invoke Duplicity and Misty Step How do I remove an element from a list by index in Python?  I found the list.remove method, but say I want to remove the last element, how do I do this? It seems like the default remove searches the list, but I don't want any search to be performed. Limit- almost finished, however hint needed. How do I show the current node title in page.html.twig OR in node.html.twig? If I want to show the current node title in page.html.twig (by default it seems to only be available in node.html.twig) I have to create a preprocess function like so:  function theme_preprocess_page(&amp;$variables, $hook) {   // Set our page title so we can access it in page.html.twig   if ($node = \Drupal::routeMatch()-&gt;getParameter('node')) {     $variables['my_node_title'] = $node-&gt;title-&gt;value;   } }   And then I have to overwrite page-title.html.twig by copying it into my current theme folder and commenting out everything.  This doesn't seem "right".  Is there a better way? How do I monitor the computer's CPU, memory, and disk usage in Java? Using R, I have fitted a linear model for a single response variable from a mix of continuous and discrete predictors. This is uber-basic, but I'm having trouble grasping how a coefficient for a discrete factor works.   Concept: Obviously, the coefficient of the continuous variable 'x' is applied in the form y = coefx(varx) + intercept but how does that work for a factor z if the factor is non-numeric? y = coefx(varx) + coefz(factorz???) + intercept   Specific: I have fitted a model in R as lm(log(c) ~ log(d) + h + a + f + h:a) where h and f are discrete, non-numeric factors. The coefficients are:  Coefficients:               Estimate  (Intercept)  -0.679695  log(d)        1.791294  h1            0.870735   h2           -0.447570   h3            0.542033    a             0.037362   f1           -0.588362   f2            0.816825  f3            0.534440 h1:a         -0.085658 h2:a         -0.034970  h3:a         -0.040637   How do I use these to create the predictive equation:  log(c) =  1.791294(log(d)) + 0.037362(a) + h??? + f???? + h:a???? + -0.679695   Or am I doing it wrong?   I THINK that that concept is if the subject falls in category h1 and f2, the equation becomes:   log(c) =  1.791294(log(d)) + 0.037362(a) +  0.870735  + 0.816825  + h:a???? + -0.679695   But I'm really not clear on how the h:a interactive term gets parsed. Thanks for going easy on me. Is this a valid partial fraction decomposition? Write $\dfrac{4x+1}{x^2 - x - 2}$ using partial fractions. $$ \frac{4x+1}{x^2 - x - 2} = \frac{4x+1}{(x+1)(x+2)} = \frac{A}{x+1} + \frac{B}{x-2} = \frac{A(x-2)+B(x+1)}{(x+1)(x-2)}$$ $$4x+1 = A(x-2)+B(x+1)$$ $$x=2 \Rightarrow 4 \cdot2 + 1 = A(0) + B(3) \Rightarrow B = 3$$ $$x = -1 \Rightarrow 4(-1) +1 = A(-3)+ B(0) \Rightarrow A = 1$$ Thus, $$\frac{4x+1}{x^2-x-2} = \frac{1}{x+1} + \frac{3}{x-2}\textrm{.}$$  The substitution of $x$ ($x = 2, -1$) is a common method to find out the coefficient of the partial fractions. However, the equation on the third line is obtained by multiplying $(x+1)(x-2)$, which is assumed to be nonzero. Here we have a contradiction. Furthermore, the original function is not defined at $x=-1,2$. How can we substitute these value for $x$? So is this method valid and rigorous? How to modify it so that it is rigorous? How do I get the CPU temperature? How do I get the CPU temperature? QGIS ->ArcGIS Server configuration possible? Is it possible to connect a QGIS Desktop 1.8.0 to an ArcGIS Server 10.0 ? and of course publish services on it?.   The configuration would be like this Quantum GIS-> ArcGIS Server 10.0 on Amazon EC2+ POstgreSQL (+PostGIS)-> web app.   Is it possible to create a configuration like this? Do you think I could use it for networking? Maybe I'm just hearing the wrong comments, but it seems that Slytherin is considered a dark house by almost everyone.  I know when both reading the books and watching the movies, I always had a feeling of foreboding when there were a number of Slytherin house members around or the time when Harry and friends went into Slytherin and felt this feeling was emphasized by the writing and film directing.  We hear very little positive about Slytherin (other than the repeated line that Harry would be a great wizard if he went in Slytherin) and there really isn't a list of great wizards who were in Slytherin House.  Even in the movie Deathly Hallows (I can't check the book - I loaned it to my sister 3 weeks after it was published and haven't seen it since!), McGonagall specifically asks Filtch to lead all the Slytherin students into the dungeon while Hogwarts is under siege by Voldemort.  She doesn't specify that any Slytherin students remain behind to help fight.  Is this intentional?  Did JKR intentionally create a bad reputation and not present any counter-examples so we'd believe all of Slytherin was bad?  Most characters in the Potter-verse are complex.  While we can admire James and Lily Potter, we also see how James teased and taunted Snape, showing us a harsh side of James, and we see that there is good in Snape.  We even see complexity in Voldemort as a villain who is mixed-blood, but hates those of mixed blood and wants purity for all witches and wizards.  With all the complex characters, it seems odd that there is essentially nothing to redeem the House of Slytherin - which is not just one character, but is a group represented by a number of characters.  But yet it seems as if it's presented, in both books and movies, as dark and full of schemers and bad people.  Are there indications, and especially more than a few or just tiny ones, that there are a fair number of good witches and wizards in Slytherin? Winforms Textbox - Using Ctrl-Backspace to Delete Whole Word Is it theoretically possible that every person in this village has a different jar of coins? A village has about $1$ million residents. Suppose each resident has a jar with $100$ coins in it.Two jars are considered to be “equivalent” if they have the same number of 10c, 20c, 50c, $\$$1 and $\$$2 coins in them. How many different (i.e. nonequivalent) jars of coins exist? Is it theoretically possible that every person in this village has a different jar of coins?  I understood that if we have $k_1$ 10 coins in the jar where $k_1$ is between $0$ and $100$ inclusive, then there are $100-k_1$ coins of other denominations in the jar. Likewise, we also have $k_2$ 20c coins in the jar where $k_2$ is between $0$ and $100$ inclusive, so the combined number of 50c, $\$$1, and $\$$2 coins in the jar is $100-k_1-k_2$. And we always have some $k_1, k_2, k_3, k_4, k_5$, let's say, so that the sum of those $k$'s is $100$.  I got stuck since I don't know how to 'compare' the number of coins in jars for different residents. After you get the Reindeer Stables, you can add decorations through a text form to increase production speed. I've added a few decorations myself, but I was wondering what all of them are. I know that some entries seem to be "synonyms". For example, if you enter "wreath" and then "garland", you'll only add the "a Christmas wreath" and trying to put up the garland will return "We already have a Christmas wreath".  These are the decorations I've already added along with discovered synonyms.   Nutcracker (Nutcracker) a Christmas tree (tree) baubles (bauble) glass ornaments (ornament) a star (star) Christmas lights (lights) a Christmas wreath (wreath, garland) mistletoe (mistletoe) a Christmas stocking (stocking) candy canes (candy cane) a chimney decoration (chimney, fireplace) a Santa decoration (santa) an elf decoration (elf) a reindeer decoration (reindeer) a nativity scene (stable) toys (toy) wrapping paper (wrap) candles (candles) tinsel (tinsel) a star (star) an angle (angle) holly (holly) paper chains (chains) Christmas crackers (cracker) Christmas cards (card) a figure of Mary (mary) a figure of Joseph (joseph) a figure of Jesus (jesus) wise men figures (wise men) a donkey decoration (donkey) a sleigh decoration (sleigh) a snowman (snowman) snow decoration (snow) Santa hats (hat) bells (bell) ribbons (ribbon) a turkey decoration (turkey) ? What does [param: NotNull] mean in C#? Turning off "Update page numbers only" in Word When I update the fields document-wide (Ctrl + A, then F9) in a Word document that has a table of contents, a dialog appears asking if I want to update page numbers only, or update the entire table. I never want to update page numbers only if the document headings have changed, nor will I ever.  (Apparently this is given as an option in case someone has manually changed an automatically generated ToC, which is almost always a really bad idea.)   Unfortunately, "Update page numbers only" is the default selection in the dialog, which means that I have to select the other option every time. I update the fields all the time when I'm working in a document that has figures and cross-references, so this is an annoying extra step I'd really like to get rid of.   Is there any way to tell Word to stop asking me this stupid unnecessary question? Bluetooth Broadcom 43142 isn't working Test condition in if command Any idea why echo statement is being executed in both scenarios?  Outputs it worked:  #!/bin/bash  rizwan='' if [ -n $rizwan ]; then   echo "it worked" fi   Replace -n with -z, and it outputs it worked when expecting no output:  #!/bin/bash  rizwan='' if [ -z $rizwan ]; then   echo "it worked" fi New three-tiered badge idea: Explainer →Refiner → Illuminator The difference between a poor or meh question and a stellar question can often simply be someone understanding it and providing it a great answer. I can't begin to count the number of times I've justified re-opening a question as a moderator by saying:     Look at the answer it got, though. This isn't something we want deleted, this is something we want fixed, because it's obviously valuable. Someone with domain knowledge can easily edit that question based on the answer it received.   This doesn't only apply to borderline-poor or poor questions – sometimes a good edit means the difference between 100 and 10,000 people finding something through typing things into Google. Good titles are hard.   We think we've come up with a way to provide incentive for good, opportunistic edits to happen more often, ideally without questions needing to go through closure or collecting a centimeter of dust before they do. What if we simply rewarded folks that answered and edited questions with a special, relatively difficult to earn badge?  Let's say that you answer something, and:   You edit the question 12 hours before or after answering it. This allows you to edit now, answer later &mdash; or answer now and edit later, when you have the time. Your edit isn't rolled back, or outright rejected if it was a suggested edit The question is not closed for any reason, even simply being a duplicate Your answer has a score of 1 or higher   &hellip; then you've done something that we probably want to recognize. You understood something, you provided the knowledge that you have, and then you provided an edit to make sure that more people in need of this knowledge can find it, while raising the overall quality of the site.  Here's the proposed tiers for the badge (within the above stated criteria):   Explainer (Bronze): Answered &amp; edited 10  questions Refiner (Silver): Answered &amp; edited 50 questions Illuminator (Gold): Answered &amp; edited 500 questions   We think this is going to be pretty hard to game. You can't go leaving streaks of terse snark all over the place on questions you know are going to be closed anyway, and junk edits aren't going to fly.  We're talking about rewarding folks that take a tiny bit more time out of their day just to write a more descriptive title so that folks can find the awesome answer that they wrote. Just doing that alone can make a big difference. Sure, it's a tad hard to get at the gold level, but it's supposed to be. If you routinely edit questions that you answer and have done so for some time, you'd probably get it the day we roll it out.  While this is technically something that we came up with on the quality initiative ( | ) &mdash; this badge could be earned on all of our sites.  What do you think? What did we miss? What could be better about the idea? Regular expression to match non-ASCII characters? What is the easiest way to match non-ASCII characters in a regex? I would like to match all words individually in an input string, but the language may not be English, so I will need to match things like ü, ö, ß, and ñ. Also, this is in Javascript/jQuery, so any solution will need to apply to that. In C, what is the difference between using ++i and i++, and which should be used in the incrementation block of a for loop?