Let $G$ be  a group of order 21 .Question is whether it is abelian or not?  Number of Sylow 3 subgroups =1+3k divides 7 hence k=0 or k=2.Number of Sylow 7 subgroups=1+7k divides 3 hence k=0.In both cases if we get a unique sylow 3 and a unique sylow 7 subgroup we are done that the group is abelian using Internal Direct Property of groups .  But what if  number of Sylow 3 subgroups is 7??It does not contradict anything here we have 2*7+7=21 elements i.e works fine  Can I extend this arguement to conclude my proposition or I need something else.Please help
solution :-  $21= 3 \times 7$  there is only one Sylow $3$ and Sylow $7$ subgroup  so, Sylow $3$ and Sylow $7$ subgroup are normal in group $G$  so $G$ is cyclic group of order $21$.  Am I right ?  somebody told me that group of order $21$ is not cyclic.  he gave me this .  If group of order 21 is not cyclic, then can we understand it by Sylow method ?
I'd like to remove my phone number from my iMessage account, but still have my phone enabled to receive messages sent to my email address (also used on iPad, Mac). Is there a way of doing this? A significant amount of zombies were detected in your city Can anyone please help and tell me if this sentence is correct?  &quot;A significant amount of zombies were detected in your city.&quot;  In my opinion it should be &quot;A significant amount of zombies was detected in your city.&quot; because &quot;amount&quot; should be in agreement with &quot;was&quot; even though amount usually refers to something plural. Are both (was and were) acceptable in this case? Edit1: Yes, &quot;number&quot; would be the correct one in this case. So allow me to ask one more thing.  &quot;A significant amount of purchases were detected...&quot;  Does the use of &quot;number&quot; apply in this case also? As purchase is not exactly a thing/person. And if no, is it correct to say &quot;A significant amount of purchases were detected...&quot;? Edit2: So, please tell me what is the correct phrasing?  &quot;A significant amount of purchases were detected...&quot; &quot;A significant amount of purchases was detected...&quot;  Or, since purchases are countable, is the use of &quot;number&quot; instead of &quot;amount&quot; correct here, as in:  &quot;A significant number of purchases were detected...&quot; &quot;A significant number of purchases was detected...&quot;  Or would amount be correct since a purchase involves money and it refers to the amount of money? Would greatly appreciate an answer, and if possible a detailed explanation. On August 5, 41 days ago, I submitted an application for a UK visitor visa. This is the third time I have applied. The only notice I have received so far says this: "Unfortunately, the processing of your application has not been straightforward. This means that we will be unable to make a decision within our published customer service standards." What should I do? Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$ Here is an exercise, on analysis which i am stuck.    How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on $\mathbb{R}$? Similarity of real matrices over $\mathbb{C}$ $A  \underset{\mathbb{C}}{\sim} B \overset{\text{def}}{\iff} A=C^{-1}BC, \space C\in M_{n}(\mathbb{C})$ and similarly for $\underset{\mathbb{R}}{\sim}$.     I want to prove that $ A \underset{\mathbb{C}}{\sim} B$ for $A,B \in M_{n}(\mathbb{R})$ therefore $A \underset{\mathbb{R}}{\sim} B$.   My idea is that elementary divisors of $A,B$ over $\mathbb{C}$ are the same, and if $(x-z)^k$ is elementary divisor than $(x-\overline{z})^k$ is also elementary divisor $\implies$ $A,B$ have same elementary divisors over $\mathbb{R}$. But i think it's not clear. Why on the first page Google says there are thousands of results but on the last page there are less than a hundred? Cannot truncate table because it is being referenced by a FOREIGN KEY constraint? Using MSSQL2005, can I truncate a table with a foreign key constraint if I first truncate the child table (the table with the primary key of the FK relationship)?  I know that I can either   Use a DELETE without a where clause and then RESEED the identity (or) Remove the FK, truncate the table, and recreate the FK.   I thought that as long as I truncated the child table before the parent, I'd be okay without doing either of the options above, but I'm getting this error:     Cannot truncate table 'TableName' because it is being referenced by a FOREIGN KEY constraint. Speed $v$ of a wave on a tightrope Should reputation from SO carry over to Meta? Huge close votes review queue on Stack Overflow Today, the size of the close-votes queue is 55k questions and growing. Is that a problem?   Which action could be taken to clean it up? It seems to be growing faster than Stack&nbsp;Overflow members are capable to handle it. By "hand cipher", I mean a symmetric cipher for which encryption and decryption can can both be performed with a pencil on graph paper, consuming about 10-20 seconds per character by a proficient user. Additional simple tools, e.g. a deck of cards or a multiplication table, may be used to expedite the process. No mathematical proficiency is required other than mental addition, subtraction, and maybe multiplication.  Basically, it should be a cipher that someone of average mathematical abilities can master, and it should require no digital technology or cryptographic tools that cannot be constructed out of paper.  For example, four-square in CBC mode seems secure to me because CBC is a well established mode of operation and four-square seems to be a good block cipher as it accommodates large keys and seems to produce a random-looking output. However, the fact that four-square in CBC was never widely adopted before digital encryption became a consumer technology leads me to believe that someone broke the cipher and thus discouraged people from using it.  So is there any hand cipher that the best supercomputers would take years to cryptanalyze? If not, then for which hand cipher is the best known attack the most complex? I am working on a SQL query that reads from a SQLServer database to produce an extract file. One of the  requirements to remove the leading zeroes from a particular field, which is a simple VARCHAR(10) field. So, for example, if the field contains '00001A', the SELECT statement needs to return the data as '1A'.  Is there a way in SQL to easily remove the leading zeroes in this way? I know there is an RTRIM function, but this seems only to remove spaces. It seems as if a shift occurred and the descriptive "Oriental" was replaced by "Asian" as the accepted term in polite society — what caused this shift? how to prove $\mathbb Z_n$ has divisors of zero if and only if $n$ is not prime A non-zero number $a \in \mathbb Z_n$ is called a divisor of zero if there is a non-zero number $b \in \mathbb Z_n$ such that $ab\equiv 0\pmod n.$  How can I prove $\mathbb Z_n$ has divisors of zero if and only if $n$ is not prime.  I filled out the addition and multiplication tables for modulo 6 and 7 and tried to find out the relation, and it's definitely true. I know I need to prove it in both directions since it is an 'iff' question. But I still don't get it. Is there a simple reason why the expected number of coin flips till getting $m$ more heads than tails or $n$ more tails than heads should be $mn$? I flip a coin until I get $m$ more heads than tails, or $n$ more tails than heads. Let the expected number of flips of the coin before stopping be $f(m,n)$.  I obtained $f(m,n)=mn$ from the recursion $f(m,n)=1+\frac{f(m-1,n+1)+f(m+1,n-1)}2$ with $f(k,0)=f(0,k)=0$ for all $k$.   Other than going through this recursion (and either solving by inspection or by writing as linear recurrence in single variable and solving brute force), is there an intuitive reason you should expect this process to take $mn$ flips? I was thinking about the more general problem with probability $p$ of getting heads and was struck by how simple the formula became when handling, what turned out to be a special case (general formula broke down) of $p=\frac12$. How can the behavior of InterpolationOrder->0 be controlled? Best option to export more than 15 millions records from Salesforce? We have a requirement to export around 15 millions records.   Which option will be better Salesforce DataLoader or Jitterbit Cloud DataLoader or any other better option available.  Can we export around 15 millions or more records in one go ! is it possible ? will CSV file able to handle that many record if exported in one go.  Also can we do this using Report API wherein we can create a report and fetch reports data using API, is it advisable considering volume of data?  Please advise the best option to go with, for exporting large volume of records from Salesforce. Is there the easiest way to toggle (show/hide) navigational grids in TikZ? In PSTricks, we can toggle the grids easily by showgrid=[false/true]. How to do it in TikZ? Doing as follows is tedious.    \documentclass{minimal} \usepackage{tikz}  \begin{document}   \tikzpicture  \draw[gray,very thin] (-4,-2) grid (4,2); \foreach \x in {-4,...,4}   \draw[red] (\x,-1.9) -- (\x,-2.1) node [below] {\x};   \foreach \y in {-2,...,2}   \draw[red] (-3.9,\y) -- (-4.1,\y) node [right] {\y};  \draw (0,0) circle (2);   \endtikzpicture    \end{document}   Is there the easiest way to toggle (show/hide) navigational grids in TikZ?  Edit  The navigation needs not only the grids but also the labeling numbers. (A kind soul at  suggested I post here as well, sorry if out of bounds.)  I'm trying to programmatically model a damped harmonic spring for use in mobile UI animations (physics mathematics isn't my background, please pardon any misconceptions).  Having derived the parameters for the general case equation, I can iteratively calculate values until I reach a suitable threshold, though because this is bound to "simple" trigonometric and $e^{x}$ functions on the CPU, the 4000-some-odd steps can cause about 0.25 seconds lag on slow devices while it calculates. I'd like to speed this up using my platform's super-optimized vector and BLAS/LAPACK variants.  The requirement for doing this is precalculating the number of steps necessary to reach my threshold value.    In the underdamped case, where the roots of the characteristic function of the differential equation are non-real, I can use algebraic tricks to get my values:  $$x(t) = c_{1}e^{r_{1}}\cos(i_{1}t) + c_{2}e^{r_{2}}\sin(i_{2}t)$$  (Given $r_{1}$, $i_{1}$, $r_{2}$, and $i_{2}$ are the real and irrational components of my two roots, respectively.)  Knowing that $r_{1} = r_{2}$ and $i_{1} = -i_{2}$, I can simplify to:  $$x(t) = c_{1}e^{r_{1}}\cos(i_{1}t)$$  And get my desired value of $t$ for my threshold $a$:  $$t = \arccos(a / c_{1} / e^{r_{1}}) / i_{1}$$    When the roots are real, the equation looks a lot simpler:  $$x(t) = c_{1}e^{r_{1}} + c_{2}e^{r_{2}}$$  However, I don't have my trig functions floating around to help me solve it (even if I did, the irrational components being 0 would cause problems, of course).  Take the concrete example on pages 3-4 of  (my bible during this process), since they at least solve cleanly:  $$x(t) = 1.5e^{-t} - 0.5e^{-3t}$$  I know how I would solve for $t$ to get my for when $x(t) = a$ on paper, by setting $x=e^{t}$, solving, and back substituting, but I don't have that luxury here.  I can make a few assumptions: the roots and constants are all real. I'm always going to be looking for the smallest, first, positive value of $t$. Obviously, the iterative solution is the simplest for this case, but in the end that would involve more steps and therefore be slower no matter what my other optimizations would be. How, then, would I go about solving for my threshold value algorithmically in this (supposedly) simplified case?  Addendum  The underdamped solution presents an extra requirement. The motion curve will oscillate back and forth a few times across the endpoint. Therefore, "first and lowest" $t$ requirement is not necessarily true. In my current, iterative code, the threshold value is both checked against the distance from the current $x(t)$ to the endpoint, as well as to the distance from the previous $x(t)$ as well to allow for a number of oscillations. This might make a more efficient solution nearly impossible. Find a non-negative function on [0,1] such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable Problem:  Find a non-negative function $f$ on $[0,1]$ such that $$\lim_{t\to\infty} t\cdot m(\{x : f(x) \geq t\}) = 0,$$ but $f$ is not integrable, where $m$ is Lebesgue measure.  My Attempt:  Let $f(x) = \frac{\chi_{(0,1]}}{\sqrt{x}}$. Then,  \begin{align*} \lim_{t\to\infty} t\cdot m(\{x \in [0,1]: f(x) \geq t\}) &amp;= \lim_{t\to\infty} t \cdot m(\{x\in (0,1]: 1/\sqrt{x} \geq t\})\\ &amp;=\lim_{t\to\infty} t \cdot m(\{x\in (0,1]: x \leq (1/t^2)\})\\ &amp;= \lim_{t\to\infty} t \cdot m((0, (1/t^2)))\\ &amp;= \lim_{t\to\infty} \frac{t}{t^2} = 0. \end{align*}  However, $f(x)$ is integrable over this interval. I have tried functions looking like $f(x) = 1/x^p$ but I cannot find any that will work here. There is a hint that says there is a monotonic function that fits this description.    Also, does anyone know of a list of non-Lebesgue integrable functions on $[0,1]$? I feel as though I could use this for many counterexamples if one were to exist. Thanks!