A consequence of Lagrange's theorem in finite group theory is that $g^{|G|}=e$ for all $g\in G$.  I wonder whether this be generalized along these lines:     If $H$ is a subgroup of $G$ of finite index $n$, then $g^n\in H$ for all $g\in G$. $\qquad(\star)$   This is true when $H$ is normal, but not in general, as this counter-example shows:     $G=S_4$, $H=S_3$, $g=(234)$: $[G:H]=4$ but $g^4=g\notin H$.   A  says that, when $\gcd(|H|,n)=1$, $(\star)$ happens iff $H$ is normal.  Can $(\star)$ be true when $H$ is not normal? What other conditions on $G$, $H$, and $n$ ensure $(\star)$ ?
Let $G$ be a group and $H$ a subgroup with $[G:H]=n$. Is it true that $x^n\in H$ for all $x\in G$?   Remarks. The answer is positive whenever $H$ is normal, e.g., for $n=2$. In general, by using the normal core of $H$, one can find an $m\ge 1$ such that $x^m\in H$ for all $x\in G$.
Correct interpretation of a "double-negative" in American Pop Culture (i.e. "I ain't afraid of no ghost!") Create a virtual file that is actually a command What are the colored/highlighted edges in edit mode? Does either pill actually do anything? When Morpheus offers the pill choice to Neo, they are still in the Matrix. That leads me to belive that none of the pills are real thus having no particular effect in the real world. Was it some sort of placebo effect that freed Neo from the Matrix or do the pills have an actual effect? How can I include custom post types on a tag page? I'm using custom post types, but they don't appear on tag pages along with regular posts and i'd like to include both standard and custom post types on tag or category pages. What would be required before the loop to include all posts? (including custom type). Updated: To add, I found this code , which worked perfectly for the first part: add_filter('pre_get_posts', 'query_post_type'); function query_post_type($query) {   if ( is_category() || is_tag() &amp;&amp; empty( $query-&gt;query_vars['suppress_filters'] ) ) {     $post_type = get_query_var('post_type');     if($post_type)         $post_type = $post_type;     else         $post_type = array('post','cpt'); // replace cpt to your custom post type     $query-&gt;set('post_type',$post_type);     return $query;     } }  Second question still remains though: Is there a way to allow readers to filter by post type? So either of the following would work..  example.com/tag/blue?post_type=custom example.com/tag/blue?post_type=post I need to hide field 'B' when the value in field 'A' is NOT equal to 'Other'.  This is a custom list form.  Is jquery the way to go? Name for the center amplitude of a sine wave I am trying to find the name for the value at the center of a sine wave $$ y(t) = A\sin (kx \pm \omega t+\varphi) + D $$ i.e. the offset $D$ of the wave in the $y$-direction.  calls it Center Amplitude, but further googling this name is difficult and does not provide any further sources. I also found the expression , but I'm not sure that it refers to the same thing. Is there a commonly used name for the $y$-direction-offset of a sine wave? Complete beginner here.  I'm trying to follow  tutorial on Anisotropic shaders, but my property panel looks nothing like the property panel in the tutorial.  For example, when creating a material, this is what his property panel looks like:    But this is what mine looks like, if I follow the same path:    I'm sure I'm missing something obvious.  How can I see the property panel shown in the tutorial? Errors "*ERROR* UVD not responding, trying to reset the VCPU!!!" I've setup a HTTPS and FTP site in IIS but how can I add users to use the FTP site?  Does they necessarily need an account with password in the computer to be able to use it? How to cancel a credit card and verify it was closed Plotting a Continuous signal Why can't I run this C program? I'm trying to run my first "process" program, but I get the following error :  ./fork.c: line 4: syntax error near unexpected token `(' ./fork.c: line 4: `int main()'   I'm pretty sure that the code is correct:   #include &lt;sys/types.h&gt;  #include &lt;stdio.h&gt;  int main() {      pid_t pid;      printf("Hello fork()\n");      switch(pid=fork())  {          case -1: printf("Error by fork().....\n"); exit(0);          case 0: printf("I'm the child process \n"); break;          default: printf("I'm the dad \n"); break;      }   exit(0); }   What is wrong? I'm an avid book collector, and I was wondering what house has the most bookshelves in Skyrim. I currently only own the Whiterun house, which has 2 bookshelves of 2 shelves each.  Is there a house I could buy that has more bookshelves?  Edit  Since not all bookshelves hold the same number of books, I'm looking for the house that has the most spaces on bookshelves for books, not necessarily the most bookshelves What factors influence the spring constant? For certain springs we have been taught that Hookes law, $F=-kx$ works. For such springs, what factors determine k. For instance, k is proportional to length. Can k be expressed in terms of dimensions, mass and other fundemental quantities, or is k a fundemental property of the spring itself? Dimensionality of null space when Trace is Zero This is the fourth part of a four-part problem in Charles W. Curtis's book entitled Linear Algebra, An Introductory Approach (p. 216).  I've succeeded in proving the first three parts, but the most interesting part of the problem eludes me.  Part (a) requires the reader to prove that $\operatorname{Tr}{(AB)} = \operatorname{Tr}{(BA)}$, which I was able to show by writing out each side of the equation using sigma notation.  Part (b) asks the reader to use part (a) to show that similar matrices have the same trace.  If $A$ and $B$ are similar, then  $\operatorname{Tr}{(A)} = \operatorname{Tr}{(S^{-1}BS)}$  $= \operatorname{Tr}(BSS^{-1})$  $= \operatorname{Tr}(B)$,  which completes part (b).  Part (c) asks the reader to show that the vector subspace of matrices with trace equal to zero have dimension $n^2 - 1$.  Curtis provides the hint that the map from $M_n(F)$ to $F$ is a linear transformation.  From this, I used the theorem that $\dim T(V) + \dim n(T) = \dim V$ to obtain the dimension of the null space.  Part (d), however, I'm stuck on.  It asks the reader to show that subspace described in part (c) is generated by matrices of the form $AB - BA$, where $A$ and $B$ are arbitrary $n \times n$ matrices.  I tried to form a basis for the subspace, but wasn't really sure what it would look like since an $n \times n$ matrix has $n^2$ entries in it, but the basis would need $n^2 - 1$ matrixes.  I also tried to think of a linear transformation whose image would have the form of $AB - BA$, but this also didn't help me.  I'm kind of stuck...  Many thanks in advance! Prove that $2^{1/2}$ is irrational I'm trying to understand each part of this completed proof that my professor did, here is my interpretation in parentheses, please advise as necessary.  Proof: Assume that $2^{1/2}$ is rational, then $2^{1/2}$ = $p/q$ for some integer $p$ and $q$ with $q \neq 0$ and $\gcd(p,q) = 1$  $$(p/q)^2 = 2 = p^2 / q^2 $$ $$\text{ (What was the purpose of squaring √2 and p/q?)}$$  $$p^2 = 2q^2$$ $$ \text{ (Is this basically just canceling out the q^2 on left side, and multiplying the right side?)}$$  2 | p^2 => (2 divides p^2 since it is a factor)  2 | p => (thus 2 divides p)  p = 2k for some integer k => (definition of even)  (2k)^2 = 4k^2 = 2q^2 => (Why are we squaring 2k, and how does 4k^2 = 2q^2?)   q^2 = 2k^2 => (Canceled out 2 on one side, and dividing on other)  2 | q^2 => (2 divides q^2)  2 | q => (thus 2 divides q)  2 | gcd(p,q) => (2 divides gcd of p and q)  2 | 1 Which is a Contradiction (no clue how I got here) How do I make my program watch for file modification in C++? Make wifi preferable over wired connection in Ubuntu 14.04? How do I make Ubuntu choose wifi connection over wired connection when both are connected to internet?  I edited the Ethernet connection to route using the preference "Use this connection only for resources on this network", which other people have reported having success with as a means of getting WiFi preferred, but it doesn't work for me. The issue persists. Ethernet is still being preferred for internet, and WiFi gets ignored.   another thing, is it possible to use wired connection as fail over connection to wireless? Edge won't connect to internet, Skype for business won't screen-share, Windows Store won't update apps after Windows 10 version 1809 installed