If $G$ is a group and $H<G$ and $|G:H|=2$ prove that for each $a\in H$, $aH=Ha$ Question:    If $G$ is a group and $H&lt;G$ and $|G/H|=2$ prove that for each $a\in H$, $aH=Ha$      Can someone please provide some imagination of $G/H$? ( And also provide at least a hint on the question? )
Any group of index 2 is normal I'm trying to understand the proof that any group of index $2$ is normal.   Let $G$ be a group and $H$ a subgroup of index $2$.  I understand that if $x\in H$ then $xH=H=Hx$ since $H$ is a subgroup. However, I don't understand why if $x\in G\setminus H$ then $xH=G\setminus H.$  I know there are other proofs of this on this site but none of them specifically address this question.
Egyptian fractions with prime power denominators summing to 1? Inspired by        Can we solve the equation $$1=\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots +\frac{1}{a_n}$$ where $a_1,a_2,a_3,\cdots ,a_n$ are distinct prime powers (primes are allowed) ?   I tried a modified greedy-algorithm with different starting vectors and brute force with $6$ entris with limit $200$, but did not find a representation. The difference between two periodic functions converges to zero, is this two functions identical? If $f(x)$ and $g(x)$ are two periodic functions, that is, $f(x+T_1)=f(x)$ and $g(x+T_2)=g(x)$ for every $x \in \Bbb R$. Now that $\displaystyle\lim_{x\to\infty}(f(x)-g(x))=0$.  Conjecture: $f(x) \equiv g(x)$. Let $q \in \mathbb C$, $\|q\|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n: n \in \mathbb N\}$ is dense in $S^1$ Let $q \in \mathbb C$, $|q|=1$ and $q^n \neq 1, \forall n \in \mathbb N$. Show that $\{q^n:  n \in \mathbb N\}$ is dense in $S^1$.  My attempt: As $(q^n)$ is limited, there is subsequence $(q^{n_j})$ convergent. So given $\epsilon &gt; 0, \exists j_0 \in \mathbb N, \quad |q^{n_j} - q^{n_k}| &lt; \epsilon, \quad \forall j,k \geq j_0$. Supose that $n_j &gt; n_k$, then  $|q^{n_j} - q^{n_k}| = |q^{n_k}||q^{n_j - n_k} - 1| = |q^{n_j - n_k} - 1| &lt; \epsilon, \quad \forall j, k \geq j_0$. So, there is a subsequence of $(q^n)$ that converges to $1$.  Using this fact, how can I ensure that for all $w \in S^1$ there is a subsequence of $(q^n)$ that converges to  $w$? Prove or give a counterexample:  $$(A ∪ B = A ∪ C) \land (A ∩ B = A ∩ C) \implies B = C.$$  I think this is true, but I am not sure how to show it. I don't know if there are any manipulations with unions and intersections that allows me to move things around. Thanks in advance for any hints. How to make sense out of the $\epsilon-\delta$ definition of a limit? The informal intuition for the limit of a function is this:     What is the value of the function $f$ as $x$ gets infinitely close to $c$?   How on earth does this monster  $$ \lim_{x \to c} f(x) = L  \iff  (\forall \varepsilon &gt; 0)(\exists \ \delta &gt; 0) (\forall x \in D)(0 &lt; |x - c | &lt; \delta \ \Rightarrow \ |f(x) - L| &lt; \varepsilon)$$ capture our intuition?  I want the answer to explain and motivate every bit of this formal statement. Remember: I am able to parse the statement, I just don't see how it captures our informal intuition. Reputation displaying incorrectly on Careers (for some accounts) When I go to , go to the "Stack Exchange" section, and click on "edit" under "Accounts", I see the following at the bottom of the account list (partially truncated for reasons of space):    Now, as you can see from my reputation here on MSO, I should have the association bonus on all sites where I have an account. Yet, for some reason, my accounts on Android, Italian, Sound Design, and Windows Phone are listed as having just 1 reputation. However, if you look at my account on each of these sites (, , , and ), you will see that I do, indeed, have 101 reputation from the association bonus.   At first, I thought the commonality among those four sites was that I hadn't performed any actions on them (not even voting), but it turns out that I have actually cast a vote on Android, so I don't really know what's up.   Note also that all of these accounts are at least a few weeks old (and my association bonus is even older than that), so this doesn't seem like it would be a caching issue or anything. Can I publish an article or thesis about my research without supervisor? How do we solve (i.e. get the closed form of) $\int \sqrt{1 + t^2} dt$ ?   shows the closed form of it but not the steps in solving it.  I think I need some algebraic trick..   I thought of some hyperbolic functions and their derivatives and equalities:  If I make $ t = \sinh x$, then I can reduce the integral to  $$ \int \cosh ^2 x dx $$ though I'm not sure if this is valid. But from here, I cannot go on evaluating the integral... Why am I constantly running out of space on my Galaxy S2? Which movie/TV show has an evil cyborg harvesting eyes from androids in a hidden town? I watched it maybe more than 8 to 12 years ago, but as far as I remember there was this place where androids (or maybe they were cyborgs) lived. (I think it was more like a hidden town). This guy finds them by accident (or at least I remember it that way) and tries to help them. I don't remember seeing any aliens or spaceships or anything like that. It was live-action, and I think US-based.  So, what I remember:   The bad guy is a human who has one red flashy robotic eye (not like terminators, it looks like a square eye patch above one of his eyes that functions like a second eye), he is collecting eyes from these androids/cyborgs. He also kills some of these androids/cyborgs but not all of them, some of them he just collects their eyes and doesn't kill (or get to kill). There was this woman who was a sexbot or something, she tried to give pleasure to the bad guy at some point, yet the bad guy added one of her eyes to his collection instead, but didn't/wasn't able to kill her. Towards the end she says something like "I used to give pleasure, now I only want to give pain" The human guy who found them tried to help them defend themselves from the bad guys, but they were in bad shape, many of them needed repair. To repair some of them, they needed spare parts they didn't seem to have, but one of the android/cyborg dudes said something like "we don't really bury our dead" and he brought a box of spare parts (or maybe android/cyborg body parts), they used these parts to repair some of the androids/cyborgs. There was this android/cyborg guy who was an assassin or something, but lost his fingers or hands, they couldn't find any hands compatible to his, but he said something like "I can still feel my trigger finger, just attach the weapons to my hands".   I remember it as a great movie/tv show, but I can't find anything about it. Can someone help me?  Please and thank you in advance... Proof by induction that $ 169 \mid 3^{3n+3}-26n-27$ $ 169$ | $3^{3n+3}-26n-27$ ?  Fulfilled for $n=0$. Induction to $n+1$: An integer $x$ exists so that  $ 169x= 3^{3n+6}-26n-27-26$ $ 169x= 27*3^{3n+3}-26n-27-26$ $ 169x= 26*3^{3n+3}+3^{3n+3}-26n-27-26$  An integer $m$ exists so that   $ 169x= 26*3^{3n+3}+169m-26$ ($ 13x= 2*(3^{3n+3}-1)+13m$)  Now I'm stuck although it looks simple. Thanks for any input in advance. Is it possible to use the enumerate/itemize environment within the proof/remark environment in a nice way? Couldn't find hvm kernel for Ubuntu tree This command:  virt-install   \    --name bla  \    --ram=1024  \    --disk path=/home/me/libvirt/images/bla.qcow2,bus=virtio,size=10  \    --location /home/me/Downloads/ubuntu-18.10-desktop-amd64.iso   Reports:  ERROR    Couldn't find hvm kernel for Ubuntu tree.   How can I fix this?  (Since there are many duplicate posts and answers with regard to this issue, I am answering it, too.) Maybe I'm searching for the wrong thing, but I can't see how I can do this.  I have two servers:   Server A (has SQL Server 2012 installed) Server B (has SQL Server 2014 installed)   Server A provides an API via IIS. The code both inserts and gets data from the database depending on the API call.  Server B does exactly the same for its local database.  I would like to have it so that the two databases are consistent copies of each other. I do believe this is what HA accomplishes.   So a load balancer, for example, might direct API calls to A sometimes, and B sometimes, but the data needs to be consistent either way.  I found some documentation on HA for SQL Server 2012, but not on how to do this for a mixed environment of 2012 and 2014.  Any help or pointers to documentation would be great! Thanks. How can I reduce the delay before window previews are shown in the Windows 7 taskbar? I really love the preview feature of the Windows 7 taskbar. There's only one reason I don't use it more often -- it takes a relatively long time to display the actual window.  There is a registry trick () to reduce the delay used by general menus, and make the submenus open instantly when hovered upon. I'm wondering if there's a similar trick that can be used for the Windows 7 taskbar's previewing. Power Supply Capacity Formula Given that I am building a home computer from scratch and I have several components (MB, CPU, HDD, Video, Sound, etc) which have specific power requirements, is there a formula for me to calculate the size/capacity of the Power Supply I will need? I want to be able to expand the setup later on also... Duplicate objects transformation applies to both objects I'm facing this frustrating "feature" that I can't seem to remedy; when I duplicate an object (and now, even when I create two completely new objects), their transformations/scaling is synchronized. This is extremely annoying when trying to use any sort of transformation keyboard shortcut.  It should be noted by going to the properties menu (N) and manually setting the properties works just fine.  For instance, here is just the camera selected (trust me, I made sure):   And when I try to transform by clicking+dragging on the X axis OR by hitting G->X, this happens:   They are not grouped (verified by going to the Groups list up in the Scene panel) and they do not have parents (verified by going to Object->Relations).  I have tried the unlink command as well, to no avail.  I have seriously run out of options; what did I change to have my G->X shortcuts transform multiple objects? It seems completely at random, too; certain objects seemed to be "linked" to other random objects, and this "linking" is guaranteed when duplicating even when I do a non-linking duplicate (Shift+D).  What gives? If $A$ and $B$ are positive definite, then is $B^{-1} - A^{-1}$ positive semidefinite? New rule preventing multiple @notifies causing Convert to Comment to fail Apparently, an image is not worth a thousand words, as without at least a few words this won't meet quality standards. Can you not sell/destroy buildings? So I've started playing Gathering Storm, built a bunch of coal plants and now Global Warming is out of control mainly because of me. I've been trying to remove these power plants, but I haven't found any option to sell/destroy or even replace these buildings.  Is there seriously no way to do this?