I'll use a concrete definition of a dihedral group $D_{2n}$ which emphasizes its group structure:      $D_{2n}$ consists of distinct elements $r_0,...,r_{n-1},s_0,...,s_{n-1}$ so that for any $i \in \mathbb{Z}$ we have $r_{i + n} = r_i, s_{i + n} s_i$ and for any $i,j \in \mathbb{Z}$ we have $r_ir_j = r_{i + j}, r_is_j = s_{i + j}, s_ir_j = s_{i - j}, s_is_j = r_{i - j}$.   It's easy to see that all $r_i's$ commute in $D_{2n}$, hence the only elements for which it is possible not to commute are $s_i$ and $s_j$ or $r_i$ and $s_j$.  $s_is_j = s_js_i$ whenever $i - j \equiv j - i \bmod n$ and $r_is_j = s_jr_i$ whenever $i + j \equiv i - j \bmod n$, hence $D_{2n}$ is noncommutative if and only if either   there is $0 \leq j &lt; n$ so that $n\nmid 2j$, or there are $0 \leq i,j &lt; n$  so that $n\nmid 2(i - j)$.   Using this result, it's not hard to establish that $D_2$ and $D_4$ are abelian.  I wonder how we can use this to prove that $D_{2n}$ is not abelian for $n \geq 3$. I'm not interested in a possible geometric proof.
Is it true that a dihedral group is nonabelian?  I'm not sure if the result is true. I checked it for some lower order and I think the result may correct.   But I failed to prove/disprove the result.
Strange set notation (a set as a power of 2)? I am not very well versed on set theory or syntax, but I thought I knew the basics.  However, in a book about databases I am reading now, the author uses $2^x$ to signify "a set of $x$."  For example, $2^{\text{dogs}}$ is a set of $\text {dogs}$, etc.    The author never really explained this or why he does it, I just picked up the meaning from context.    I am not sure why the exponent operator is used, nor am I sure what the number $2$ has to do with it.  The sets being represented are NOT powers of $2$ (in size)... they come in all sizes.  Is this a valid notation?  I have not seen it anywhere before... $1,000 is pronounced as "one thousand dollars". Reading from left to right, it seems like it would make more sense to write the value as: 1,000$. This way the pronunciation of the value follows the way it's written. Every other numerical value I can think of places the unit after the value:   1 kg 1 L 1 m, etc...   I'm not sure if this question fits in with the rest of the questions on this site. But I'd like to know why currencies are so different from other quantities. Not to mention the need to separate groups of three digits with commas. When I restart httpd, I get the following error.  What am I missing?  [root@localhost ~]# service httpd restart Stopping httpd:                                            [  OK  ] Starting httpd: Syntax error on line 22 of /etc/httpd/conf.d/sites.conf: Invalid command 'SSLEngine', perhaps misspelled or defined by a module not included in the server configuration   I have installed mod_ssl using yum install mod_ssl openssh  Package 1:mod_ssl-2.2.15-15.el6.centos.x86_64 already installed and latest version Package openssh-5.3p1-70.el6_2.2.x86_64 already installed and latest version   My sites.conf looks like this  &lt;VirtualHost *:80&gt; #    ServerName shop.itmanx.com     ServerAdmin webmaster@itmanx.com      DocumentRoot /var/www/html/magento     &lt;Directory /var/www/html&gt;         Options -Indexes         AllowOverride All     &lt;/Directory&gt;      ErrorLog logs/shop-error.log     CustomLog logs/shop-access.log &lt;/VirtualHost&gt;  &lt;VirtualHost *:443&gt;     ServerName secure.itmanx.com     ServerAdmin webmaster@itmanx.com      SSLEngine on     SSLCertificateFile /etc/httpd/ssl/secure.itmanx.com/server.crt     SSLCertificateKeyFile /etc/httpd/ssl/secure.itmanx.com/server.key     SSLCertificateChainFile /etc/httpd/ssl/secure.itmanx.com/chain.crt      DocumentRoot /var/www/html/magento     &lt;Directory /var/www/html&gt;         Options -Indexes         AllowOverride All     &lt;/Directory&gt;      ErrorLog logs/shop-ssl-error.log     CustomLog logs/shop-ssl-access.log     &lt;/VirtualHost&gt; Windows Active Directory naming best practices? This is a  about Active Directory domain naming.   After experimenting with Windows domains and domain controllers in a virtual environment, I've realized that having an active directory domain named identically to a DNS domain is bad idea (Meaning that having example.com as an Active Directory name is no good when we have the example.com domain name registered for use as our website).   , but I'm still not sure about what other rules there are around naming Active Directory domains.  Are there any best practices on what an Active Directory name should or shouldn't be? $n$ divides a product of $n$ consecutive integers (by $n$ divides one of them) Why does my item name not show up in 1.13? I am in 1.13 Pre-Release 7 trying to make a one command creation and I need an item with a specific name. The command I use is /give @p diamond_sword{display:{Name:"Sword of Power"}} and it gives my a diamond sword, but it doesn't have the name.   This is me running the command.   This is the command not working. How do I make windows save their position on screen after re-attaching a display? How to create a database directory structure for MySQL? I was following this tutorial:    So far, I installed mysql-server mysql-client libmysqlclient-dev, and I was suppose to tell MySQL to create its database directory structure where it will store its information using sudo mysql_install_db.  $ sudo mysql_install_db  [WARNING] mysql_install_db is deprecated. Please consider switching to mysqld --initialize [ERROR]   The data directory needs to be specified.   I thought running mysqld --initialize would solve the issue:  $ mysqld --initialize   mysqld: Can't create directory '/var/lib/mysql/' (Errcode: 17 - File exists) [Warning] TIMESTAMP with implicit DEFAULT value is deprecated. Please use --explicit_defaults_for_timestamp server option (see documentation for more details). [ERROR] Aborting   How do I tell MySQL to create its database directory structure where it will store its information? Insert result of a parallellized select query into a table, in Postgresql According to ,      "Even when it is in general possible for parallel query plans to be   generated, the planner will not generate them for a given query if any   of the following are true:       The query writes any data or locks any database rows. If a query contains a data-modifying operation either at the top level or within   a CTE, no parallel plans for that query will be generated. This is a   limitation of the current implementation which could be lifted in a   future release."   Indeed, when I try to insert result of a parallel SELECT query into a table (either by SELECT.. INTO either by WITH..SELECT..INTO, the query is not executed as parallel query.  My question is: Is there any way to trick the Postgresql so that a SELECT query is executed as parallel query and then its result inserted into a table? Web applications that want to force a resource to be downloaded rather than directly rendered in a Web browser issue a Content-Disposition header in the HTTP response of the form:  Content-Disposition: attachment; filename=FILENAME  The filename parameter can be used to suggest a name for the file into which the resource is downloaded by the browser.  (Content-Disposition), however, states in  (The Filename Parameter) that the file name can only use US-ASCII characters:     Current [RFC 2045] grammar restricts   parameter values (and hence   Content-Disposition filenames) to   US-ASCII.  We recognize the great   desirability of allowing arbitrary   character sets in filenames, but it is   beyond the scope of this document to   define the necessary mechanisms.   There is empirical evidence, nevertheless, that most popular Web browsers today seem to permit non-US-ASCII characters yet (for the lack of a standard) disagree on the encoding scheme and character set specification of the file name. Question is then, what are the various schemes and encodings employed by the popular browsers if the file name “naïvefile” (without quotes and where the third letter is U+00EF) needed to be encoded into the Content-Disposition header?  For the purpose of this question, popular browsers being:   Firefox  Internet Explorer  Safari Google Chrome  Opera Assume I have the following list:  foo = ['a', 'b', 'c', 'd', 'e']   What is the simplest way to retrieve an item at random from this list? Error propagation with asymmetric uncertainties I red the Wikipedia  on error propagation. I get into some trouble when I want to calculate the error propagation in a specific case. I have real numbers $A$ and $B$ with related upper ($\Delta A_u$, $\Delta B_u$) and lower ($\Delta A_l$, $\Delta B_l$) uncertainties. I have two functions calculated as follows: $$F = \sqrt{A^2+B^2}$$ and  $$G = \tan^{-1}\frac{B}{A}$$ What are the related upper and lower errors of $F$ and $G$?  I found general formulae for Gaussian error distributions ($\Delta A_u = \Delta A_l$ and $\Delta B_u = \Delta B_l$), for instance the first formula of the first answer on . Is there a way to generalize that formula for arbitrary distributions with asymmetric errors as in my example? Cannot log in with Apple ID after performing password reset My wife forgot the password to her Apple ID account. Her Macbook Air (OS X Yosemite) restarted to install some updates. She has her user account set up through her Apple ID account.  Since she forgot the password she reset it from my laptop.  She can log in to iCloud and Apple ID with the new password, but cannot log in with the new password on the Air.  When she tries to log in she is greeted with this message:     "This account uses an iCloud password to log in. If you need to change   the password, use another device to change it at appleid.apple.com." I need to do an  request in JavaScript. What's the best way to do that?  I need to do this in a Mac OS X dashcode widget. Fundamentally, what does a friction force do for a rolling object? I am very confused by the way my textbook explains it. In my textbook , it says:  A wheel rolling on a horizontal flat surface at a constant velocity experiences no friction force. Why?   On the same surface, there is an acceleration of the wheel pointing to the right (probably caused by a force), so the ball is angularly accelerating in the clockwise direction. In this case, a friction force appears, and it is also pointing to the right. How come? What does the friction do for this wheel?  On an inclined plane, a ball freely rolls down the surface. The direction of friction is up the ramp, which confuses me because in the previous example the friction force was in the direction of the wheel's acceleration.   And there is a difference if a wheel is freely rolling and if there is a torque acting on the ball's center of mass. WHY???  What does friction do in these cases? How does it cause an object to roll? Why does the addition of a \left and \right introduces an extra space around the formula? Specifically, why do $\cos(\theta)$ and $\cos\left(\theta\right)$ render differently?  Is there a way to use resizable delimiters without introducing the spurious spaces?  Why did Knuth design it like this?  Edit: This occurs too, if \operatorname{MYOP} or some other operator, like \lim, \min etc. is used. (This edit is basically to make it easier to find this answer.) Order of an element in a finite cyclic group Let $G$ be a cyclic group of order $m$ generated by an element $a$.  I want to show that the order of $a^k$ is $m/d$, where $d:=\gcd(k,m)$.  I have a simple proof, but want to make sure I haven't overlooked anything since the other proofs I've seen (such as in Dummit and Foote) look more complicated.    The order of $a^k$ is the cardinality of the set $\{a^{ks}: s \in \mathbb{Z}\}$, which equals the cardinality of the set $\{a^{ks} a^{mt}: s,t \in \mathbb{Z} \}$ since $a^m$ is the identity.  But the set $\{ks+mt: s,t \in \mathbb{Z} \}$ is the set $d \mathbb{Z}$ by the basic properties of integers. So we have that the cardinality of the set is $| \langle a^k \rangle| = |\{1, a^d, a^{2d}, \ldots, a^{m-d} \}| = m/d$. I'm trying to answer the following question from the book :  Let $X_1,X_2,\dots$ be a sequence of sub-gaussian random variables, which are not necessarily independent. Show that   $E\bigg[ \max_i \frac{|X_i|}{\sqrt{1 + \log i}} \bigg] \le CK$,  where $K = \max_i \|X_i\|_{\psi_2}$. Deduce that for ever $N \ge 2$ we have   $E\bigg[ \max_{i \le N} |X_i| \bigg] \le CK \sqrt{\log N}$.  I've tried to figure out what is the distribution of the maximum of Gaussians, but I'm reaching only inequalities that that don't help me answer the question.   I've also seen a similar question .  Does anyone have a clue or something to start with in order to answer this question?  Thanks! How do I install libtiff.so.3? When I try to run Maya from the terminal, I get:  error while loading shared libraries: libtiff.so.3: cannot open shared object file: No such file or directory.   How do I solve this?  Note: Maya is a third-party software. How to show a group is cyclic? One question asking if $\mathbb{Z}^*_{21}$ is cyclic.  I know that the cyclic group must have a generator which can generate all of the elements within the group.  But does this kind of question requires me to exhaustively find out a generator? Or is there any more efficient method to quickly determine if a group is a cyclic group?