Consider the dihedral group $D_8$ of order $16$.  Consider $D_8$ with the presentation $D_8=\{r^i s^j : i=0,...,7; j=0,1; r^8=s^2=e; sr=r^7s=r^{-1}s\}$, where $\{e\}, \{rs, r^3s, r^5 s, r^7s\}$ and $\{s, r^2s , r^4s, r^6s\}$ are three of its conjugacy classes (here $r$ is a rotation and $s$ is a reflection).   1) Find the remaining conjugacy classes.  2) Find the centre and class equation for $D_8$.  3) Find the subgroups of order $4$ and explain which is normal and why.
List all the conjugate classes in the dihedral group of order $2n$ and verify the class equation.  The dihedral group is generated by two elements $r$ and $s$.   The order of $r$ is two since $r^2=e$ and $s$ is $n$ since $s^n = e$. And I know all elements can be produced as either $s^k$ or $rs^k$. How can I list all the conjugacy class?
Thailand visa exemption: any rule limiting number of entries per year? Java. Splitting a multiple word string into two word strings every space How do I add permissions via command line for "everyone" on external HDD I have an external HDD and I kind of messed up the file permissions but when fixing it I thought it is ok bc with my username I can access the files perfectly fine. Now that I use this with two PC (actually ATM I don't have access to my other PC) I can't access these files.  The problem is this directory has hundreds of folders with no permission for "everyone". I would like to give it the default permissions including have all access for the user "everyone". How do I do that via command line for these hundreds of folders? Why would Voldemort spare Lily? In one of Harry's dementor-induced flashbacks, he presumably hears his mother protesting against Voldemort:  Harry was falling again through thick white fog, and his mother’s voice was louder than ever, echoing inside his head – ‘Not Harry! Not Harry! Please – I’ll do anything –’ ‘Stand aside – stand aside, girl –’ ‘Harry!’  Voldemort tells her to stand aside - why did he not just kill her? Export data as CSV in back end with proper HTTP headers I wrote a plugin that displays all products in a woocommerce store in admin settings option, now I want to add a link to download the products as CSV file.  The problem is, when I click the link I get a permission error saying I don't have permission to view this page.  Here's my code:  function extra_tablenav($which) {     if ($which == "top") {         echo '&lt;h3 style="display:inline"&gt;'         . __('These products are currently in the database:')         . '&lt;/h3&gt;' .         '&amp;nbsp;&amp;nbsp;&amp;nbsp;' .         '&lt;a href="' . admin_url('admin.php?page=download_csv.php') . '"&gt;' . __('Export to CSV') . '&lt;/a&gt;';     } }   How can I fix those permissions? Driver locked memory on a non-virtual machine Does Reincarnation allows skills to be passed into other classes I'm mainly asking this for Disgaea: Afternoon of Darkness, given I have enough Mana to reincarnate a character into 4 different types of Mages (Green, Blue, Red, Star), a Cleric and then a Warrior, would the Warrior have all the spells learned from the previous classes? How to revert to GNOME Classic Desktop? Making a string out of a string and an integer in Python Variable Sized Arrays vs calloc in C On the discussion of dynamic memory here:   The author states:     A memory block like this can effectively be used as a more flexible array. This approach is actually much more common in real-world C programs. It's also more predictable and flexible than a "variable size array"   The type of memory block he is talking about is as such:  const int size = 5; int * array = calloc(size, sizeof(int));   and then using another pointer to iterate over the array:  int * index = array; for (i = 0; i &lt; size; i++) {     *index = 1; // or whatever value     index++; }   My question is how is this method better than a standard variable sized array like this?:  int array[variable];   or dynamic:  char name[] = "Nick";   The author doesn't really shed much light as to why I should prefer the former method to the latter. Or more specifically: How is it more "predictable and flexible"? There is a statement that maximizing the likelihood is equivalent to minimizing the cross-entropy.  Are there any proof for this statement? SEO (or other) problems with a 404? 50 question per month limit? I get a message:     Sorry, you are only allowed to ask 50 questions in a 30 day period   Why is there a question limit? 50/month is just not enough :) Finding a basis for a certain vector space of periodic polynomials I am having a little bit of trouble solving an homework question.  I found that $S={ p(x) \in R_4[x]} \big| p(x)=p(x-1) $ is a vector space.  Now I need to find some set, K that holds ${span(k)=S}$  My idea was to substitute: $ a+bx+cx^{2}+dx^{3}+ex^{4}=a+b(x-1)+c(x-1)^{2}+d(x-1)^{3}+e(x-1)^{4}$  But here I get something that seems a little bit weird, I'd like to look if I have any mistakes that I did not notice. and if I had mistakes, how to make my answer right. I calculated the right side of the equation and I got to  :  $a+b(x-1)+c(x-1)^{2}+d(x-1)^{3}+e(x-1)^{4}=a+bx-b+cx^{2}-2cx+c+dx^{3}-3dx^{2}+3dx-d+ex^{4}-4ex^{3}+6ex^{2}-4ex+e= (a-b+c-d+e)+(b-2c+3d-4e)x+(c-3d+6e)x^{2}+(d-4e)x^{3}+ex^{4}  $  From here, $ (a-b+c-d+e)+(b-2c+3d-4e)x+(c-3d+6e)x^{2}+(d-4e)x^{3}+ex^{4}=a+b(x-1)+c(x-1)^{2}+d(x-1)^{3}+e(x-1)^{4}\Rightarrow $  $(-b+c-d+e)+(-2c+3d-4e)x+(-3d+6e)x^{2}+(d-4e)x^{3}=0+0x+0x^{2}+0x^{3}+0x^{4}$  And the solution to this equation is only whan $a=b=c=d=e=0$.  I think that I am maybe not in the right direction. Do you have any Idea how to find a set $K\Rightarrow span(K)=S$ ?  Thanks in advance. As per the following example, and as in  In bash, where has the trailing newline char gone?, I want to know "why" it happens    x="$(echo -ne "a\nb\n")" ; echo -n "$x" | xxd -p  # Output is: 610a62  # The trailing newline from the 'echo' command #   has been "deleted" by Command Substitution   I assume there must be some very significant reason for a shell action, namely Command Substitution, to actually delete some data from the command output it is substituting... but I can't get my head around this one, as it seems to be the antithesis of what it is supposed to do.. ie. to pass the output of a command back into the script process... Holding back one character seems weird to me, but I suppose there is a sensible reason for it...  I'm keen to find out what that reason is... Is there an AmpScript function that can be used in an equivalent manner to sql left, right and mid functions?   Thanks, Mark Prove that for all sets A and B, A ∩ B = ∅ implies ( A ∪ B ) - B = A In the next proof we avail ourselves of the next lemma: For all sets A and B, ( A ∪ B ) - B = A. Proof: Let A and B be arbitrary sets and let x ∈ ( A ∪ B ) - B. x ∈ ( A ∪ B ) - B ⇔ x ∈ ( A ∪ B ) ∧ x ∉ B ⇔ ( x ∈ A ∨ x ∈ B ) ∧ x ∉ B ⇔ ( x ∈ A ∧ x ∉ B ) ∨ ( x ∈ B ∨ x ∉ B) For p ∧ ~ p ≡ F and p ∧ q ⇒ p where p and q are prepositions, ( x ∈ A ∧ x ∉ B ) ∨ ( x ∈ B ∨ x ∉ B ) ⇔ x ∈ A ∨ F By modus tollendo ponens, x ∈ A ∨ F ⇔ x ∈ A So then, x ∈ ( A ∪ B ) - B ⇔ x ∈ A And ( A ∪ B ) - B = A Therefore, the statement for all sets A and B, A ∩ B = ∅  implies ( A ∪ B ) - B = A is true trivially. Is this proof right? Examples of provably${}^n$ unprovable statements Given any statement $A$ and a classical theory $T$ which we assume is at least as strong as Peano Arithmetic ($\sf PA$), we have that $T\vdash A$ implies $T\vdash T\vdash A$ (that is, if a statement is provable, then it is provably provable). Let $T_1=T$ and $T_{n+1}=T_n+{\sf Con}(T_n)$. (We need stronger theories if we want to prove unprovability.) Given this, simply by considering the truth values of the relevant statements, we get the following classification of statements by unprovability:   $T\vdash A$ ($A$ is provable) $T_2\vdash T\not\vdash A$ ($A$ is provably unprovable) $T_3\vdash T_2\not\vdash T\not\vdash A$ ($A$ is provably unprovably unprovable)   &nbsp;&nbsp;&nbsp;&nbsp;...  &nbsp;&nbsp;&nbsp;&nbsp;$\omega$. For all $n$, $T_n\not\vdash\dots \not\vdash T_2\not\vdash T\not\vdash A$ (we can't say anything about $A$'s provability)  We can refine this hierarchy by also considering the status of $\lnot A$. Let's call a class $m,n$ statement one such that $A$ is at level $m$ above and $\lnot A$ is level $n$ in the list. Since $T\vdash\lnot A$ implies $T_2\vdash T\not\vdash A$, the only nonempty classes at level $1$ are the class $1,2$ statements (provable statements) and the class $2,1$ statements (refutable statements). Independent statements are in class $2,2$. The class $1,1$ is nonempty iff the theory is inconsistent.  My question is: Are there any (preferably "natural") statements that fall further into this hierarchy? I know many examples of independent statements, like the continuum hypothesis or the axiom of choice relative to $\sf ZF$, or the Paris–Harrington theorem relative to $\sf PA$. But I know no examples of even any class $2,3$ statements. I expect there may also be natural examples of $\omega,\omega$ statements. My guess is that all the classes $m,n$ with $m,n\ne1$ are nonempty, but examples elude me. Explicit example of a compact manifold of dimension $>2$ with strictly negative sectional curvature I am looking for examples of compact manifold of dimension $&gt;2$ with strictly negative sectional curvature (for dimension 2 it is well-known). Can anybody please help? Reading the reviews of the  on Google Play,  said that since Chrome supports HTML5, he had managed to persuade the BBC iPlayer website to play him streams meant for iPad/iPhone devices by spoofing an IOS user agent string on his Nexus 7.  How might I go about getting the same thing to work myself?  I have looked at  and  but both seem to concentrate on the Dolphin browser. Also, please mention if a solution will require me to root my Nexus-7.  Incidentally, I also tried using  with Firefox, but although Firefox supports HTML5 and the iPlayer website shows pages without either install flash or use external app messages, it still won't play the streams (it just sits swirling its progress wheel), hence looking for a solution for Chrome.