Because order of element divide order of the group,the possiblility of order of element is 1,2,5,10. Frin this ,it must contains element of order 5 and 2. But I can't prove that it must gaurantee five elements of order 2 .Anyone can give a hint ?
This is almost a duplicate of the following questions (but, read further):      But, I wasn't able to apply the proof to my case of a group of order 10 and an element of order 5.  In my case, if I try a proof by contradiction, I'm left with the fact that every element in $G$ is of order 2, but if I divide them into subgroups, every subgroup is of order 2, and has one unique element.  I also tried other methods to prove the theorem, but couldn't come up with anything meaningful.  I'm a beginner in the field, and am not familiar with theorems such as Cauchy's or Sylow's theorem (I'm aware of Lagrange's theorem). Therefore, please try to provide a simple answer, thanks!
Assume   $$\lim_{n\to \infty}{x_ny_n}=0$$  then  $$\lim_{n\to \infty}{x_n}=0$$ or $$\lim_{n\to \infty}{y_n}=0$$  I couldnt find two series who disprove that.. just hint please thanks What’s the point in having “www” in a URL? Other than for historical reasons, is there is reason to have “www” in a URL?  Should I create a permanent redirect from www.xyz.com to xyz.com, or from xyz.com to www.xyz.com? Which one would you suggest and why? OAuth redirect URL is not honored The  of March 01, 2013 put it's distance at 14 hrs 04 mins 23 secs of light-travel time from Earth. A  (earlier today) tweet says it is 14 hrs 04 mins 22 secs of light-travel time from Earth. An  tweet put it still further out at 14 hrs 04 mins 25 secs of light-travel time.  The Voyager 2 tweets on Voyager 1's proximity to Earth too has reduced similarly to  earlier today from  reported on Feb 28 2013  It could quite simply be a typo ... If not a typo then my curiosity is piqued; I was under the impression the Voyagers are outbound from the Solar System now without any major in-system activities now.   How is it the Voyagers are closer to Earth than before? An Open Interval and a Half-open Interval are not Homeomorphic I tried Office 2013, but now wish to uninstall it. I tried to uninstall it but it always hangs at step  2 out of 4. It uses no CPU and no disk activity. So basically it's just idle.   Does anyone have a solution to this problem? I'm studying for a midterm and need some help with proving summation     $$\sum\limits_{k=0}^n\binom{n}{k}2^k = 3^n$$   using the binomial theorem.  This is what I've been thinking so far:   In the binomial theorem, we set $x = 0$ and $y = 2$, so:  $3^n = (x+y)^n = \sum\limits_{k=0}^n\binom{n}{k}y^k$  $ = \sum\limits_{k=0}^n\binom{n}{k}2^k$  Am I getting this correct so far or completely wrong? Any help would be appreciated. Thanks. I am looking at  (dual low-voltage power amplifier). On the fourth page, there is a sample circuit; right at the V+ input there is a symbol that looks like a capacitor. However, it looks like it is shaded in. What is that? Geometric interpretation of the multiplication of complex numbers? I've always been taught that one way to look at complex numbers is as a Cartesian space, where the real part is the $x$ component and the imaginary part is the $y$ component.  In this sense, these complex numbers are like vectors, and they can be added geometrically like normal vectors can.  However, is there a geometric interpretation for the multiplication of two complex numbers?  I tried out two test ones, $3+i$ and $-2+3i$, which multiply to $-9+7i$.  But no geometrical significance seems to be found.  Is there a geometric significance for the multiplication of complex numbers? Difference between static class and singleton pattern? What real (i.e. practical) difference exists between a static class and a singleton pattern?  Both can be invoked without instantiation, both provide only one "Instance" and neither of them is thread-safe. Is there any other difference? Do virtual particles actually physically exist? I have heard  pop in and out of existence all the time, most notable being the pairs that pop out beside black holes and while one gets pulled away. But wouldn't this actually violate the conservation of energy? There has been quite a few times that I wished I could send a message to another user on SO - not ask a question for everyone to see, but just a short message informing them of something or requesting them to do something.  Are there any plans to allow this to happen in the future?  Related: How can I add a "C" wire to my thermostat? So, the spiffy new wi-fi thermostats want power on the C wire from thermostat, but the old thermostat doesn't have a C wire at it.  What's the easiest way to add the C wire/24 volt to the thermostat location? TeX capacity exceeded with glossary package My document contains many thousands of glossary items. When compiling with XeLaTeX, about 1/8 of the way through the document, it reports:  ! TeX capacity exceeded, sorry [main memory size=3000000]. \@glolist@ ...},{13586},{9406},{4539},{12811},{179                                                   51},{5831},{1047},{14948},    The numbers you see above are ones placed in #1 of \newglossaryentry{#1}{#2}. These unique numbers are assigned to each entry for convenience. I have successfully compiled short tests of the document, which only used 1% of the full content. The resulting .glo file is about 1 MB in size.   Could this be the result of some error on my part or am I giving TeX too much work to do?  Update:  I have followed the instructions below (edited texmf.cnf and running  fmtutil-sys --all. This has allowed me to compile an additional portion of the book, but no matter the settings I add, I can't go beyond 5/8 of the way through.  A compile with extra_mem_top.xelatex = 30000000 and extra_mem_top.xelatex = 50000000 both result in:  TeX capacity exceeded, sorry [pool size=3186676].   Update:  I have done the following:  $ kpsewhich texmf.cnf /usr/local/texlive/2011/texmf.cnf $ nano /usr/local/texlive/2011/texmf.cnf   To the texmf.cnf file, following instructions at , I added:  main_memory = 230000000 extra_mem_top = 10000000 extra_mem_bot = 10000000 save_size = 150000 stack_size = 150000  main_memory.xelatex = 30000000 extra_mem_top.xelatex = 30000000   After this, I tried: - fmtutil-sys --all, then compiling xelatex file.tex. - fmtutil --all, then compiling xelatex file.tex. - fmtutil-sys --byfmt xelatex, then compiling with xelatex file.tex. - texhash, then compiling with xelatex file.tex.  All of these resulted in no changes in results. I still get:  TeX capacity exceeded, sorry [pool size=3186676]. Access denied for user 'root@localhost' (using password:NO) Where did the practice of using apostrophes for possessive nouns but not pronouns originate? Where did the practice of using apostrophes for possessive nouns but not pronouns originate?  For example, possessive nouns (both proper and common) are written with a apostrophe before the final s:        Bill’s, not Bills   Sarah’s, not Sarahs   the bear’s, not the bears   the lizard’s, not the lizards      On the other hand, several possessive pronouns do not have apostrophes before the s:        his, not hi’s   hers, not her’s   its, not it’s   theirs, not their’s   ours, not our’s   yours, not your’s      When and where did this usage originate? Was there any technical advantage of this usage? What is the expected value of the maximum of 500 IID random variables   with uniform distribution between 0 and 1?   I'm not quite sure of the technique to go about solving something like this. Could anyone point me the right direction?  Thanks I want to count the total number of files in particular directory that ends with ".mp4" extension.  I am getting following command:  ls -F |grep -v / | wc -l   It count all the files in particular directory, but I want the count of files that end with .mp4 extension.  Is there any Ubuntu command for that? How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$ Lemma vs. Theorem I've been using Spivak's book for a while now and I'd like to know what is the formal difference between a Theorem and a Lemma in mathematics, since he uses the names in his book. I'd like to know a little about the ethymology but mainly about why we choose Lemma for some findings, and Theorem for others (not personally, but mathematically, i.e. why should one classify a finding as lemma and not as theorem). It seems that Lemmas are rather minor findings that serve as a keystone to proving a Theorem, by that is as far as I can go.  NOTE: This  doesn't address my concern, so please avoid citing it as a duplicate.