This is my question : Let G be a group without subgroups with index 2 (i.ee. there is no subgroup M s.t $[G:M]=2$).prove that every group $H&lt;G$ s.t $[G:H]=3$ is normal in G.  What I thought to do: Let N be a subgroup s.t $[G:H]=3$ so $G/H={N,xN,yN}$ so every $g \in G$ is of the form : $g=n,g=xn,g=yn$ when $n \in \mathbb N$. I tried to prove that for every $g \in G$ : $gNg^{-1} =N$  but without a success.
Generalizing the case $p=2$ we would like to know if the statement below is true.  Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.
Can't SSH in as root I'm using the default installation of openssh-server for my Ubuntu 14.04 server. I can connect through the console as root like normal, with my username and password. When I try to SSH with the same username/password, however, I repeatedly get Access Denied errors. /var/log/auth.log reports Failed password for root from &lt;ip address&gt; port &lt;port&gt; ssh2, but I'm entering the correct password.  Why can't I connect to the server via SSH, even though the username and password ARE correct? In my book, it proves that an infinite subset of a coutnable set is countable. But not all the details are filled in, and I've tried to fill in all the details below. Could someone tell me if what I wrote below is valid?  Let $S$ be an infinite subset of a countable set $T$. Since $T$ is countable, there exists a bijection $f: \mathbb{N} \rightarrow T$. And $S \subseteq T = \{ f(n) \ | \ n \in \mathbb{N} \}$. Let $n_{1}$ be the smallest positive integer such that $f(n_{1}) \in S$. And continue where $n_{k}$ is the smallest positive integer greater than $n_{k-1}$ such that $f(n_{k}) \in S$. And because $S$ is infinite, we continue forever. Now consider the function $\beta : \mathbb{N} \rightarrow S$ which sends $k \rightarrow f(n_{k})$.   So in order for $S$ to be countable, $\beta$ would have to be a bijection. $\beta$ is injective since if $f(n_{k}) = f(n_{j})$, then $n_{k} = n_{j}$ because $f$ is injective. We also can conclude $k = j$ because the various $n_{i}$ chosen were strictly increasing.   Edit: $\beta$ also needs to be surjective. So for every $r \in S$ there exists a $q \in \mathbb{N}$ such that $\beta(q) = f(n_{q}) = r$. We know that $f^{-1}(r) \in \{n_{1}, n_{2}, ..., n_{k} ... \}$, so we can let $f^{-1}(r) = n_{d}$. Thus $\beta(d) = f(n_{d}) = r$. default permissions for /var/www I was trying to open a file and write to it with PHP at /var/wwwfolder but it wasn't working so I did  sudo chmod 777 /var/www   now I want to set the permissions for /var/www to the default. what are the default permissions for /var/www? Let $A,B$ be real symmetric matrices, and for any $n\in \Bbb N^{+}$, and for all $x,y\in \Bbb R$, we have $$tr(xA+yB)^n=x^ntr(A^n)+y^ntr(B^n).$$  Show that $AB=0$.  My try:since $$(xA+yB)^n=x^nA^n+\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}+y^nB^n$$ so $$tr\left(x^nA^n+\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}+y^nB^n\right)=x^ntr(A^n)+y^ntr(B^n)$$  then  $$tr\left(\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}\right)=0$$ so $$\binom{n}{1}x^{n-1}ytr(A^{n-1}B)+\cdots+\binom{n}{n-1}xy^{n-1}tr(AB^{n-1})=0$$ then I can't  I found this is a nice result, maybe this is an odd problem, and I can't solve it. Thank you. What books are recommended for learning calculus on my own? I recently graduated with a degree in bachelor of science with a focus interactive and multimedia design. I had to opportunity to take 1 C++ course and 1 HTML course. I was also only required to take one math class, introduction to calculus (and it was hardly calculus, it was more of a math class for designers)  Since graduating I have pursued a career in web development. I'm considering going back for a MS in computer science or MS in web technologies.   I feel my math has suffered dramatically as I have always been behind most others on my math studies (middle school through college). I want to learn calculus as I feel it's an integral part of any CS degree.   Does anyone have any recommendations on any great books that I can read through and teach my self a bit of calculus or other advanced math topics?  Thanks for any advice How to concatenate text from multiple rows into a single text string in SQL server? Consider a database table holding names, with three rows:  Peter Paul Mary   Is there an easy way to turn this into a single string of Peter, Paul, Mary? remove isp-provided dns My isp provides a dns service; but unfortunately they do block certain sites. I would like to use opendns or google; but it seems like it doesn't work.    I have set up a wired connection; given a manual ip and entered opendns-ip's in the nameserver field (all through the gui). Still url's are being resolved through my ISP. I have tried to configure the same on my girlfriends laptop (win 7) and I also can't seem to do the setup. My previous computer also was a win 7 and had no problem. (so I believe the problem is 'me') If it were only one site, i could edit the hosts file; but I prefere to circumvent their whole dns server.  Is there something an isp can do to block the use of other dns-servers than theirs? Is it possible to re-route the ISP dns to the opendns? The Stack Exchange beta iOS app distributed over TestFlight will expire in 2 days has expired. After the 100 day test period, the app won't start any more so all beta testers can't use the app then.  Can we please get an updated release so we don't have to remove the app and install the current release from the App Store instead? My phone was stolen last night. I made sure it was in Lost Mode in iCloud and it was immediately in Offline status when I open iCloud. I am positive the thief switched it off. Now I am 100% certain the Find my iPhone option was turned on on my iPhone, but will the "Lost Mode" be of any help if it was not turned on on the iPhone? Would it be showing "Offline" if I did not turn on the option? What is the purpose of LongLength for .Net Arrays? Typesetting 144...4 with "n times" under the 4's is easy, but what about \sqrt{144...4}? Inspired by  on the math StackExchange, suppose you want to typeset the square root of the number 144...4, that is, 1 followed by n 4's. If you write $\sqrt{1\underbrace{44\ldots4}_{n\ \text{times}}}$, as in the current version of the question, you get a huge radical sign that reaches down to match the "n times" text below the number. Is there an elegant way to typeset this so that the radical sign looks like it would with $\sqrt{144\ldots4}$, but also keeping the underset text in the right place? How do trees lift water higher than 10 meters? The atmosphere pressure is 10 meters of water (approx). This means that it is impossible to lift water higher than 10 meters with vacuum or сapillary action (on Earth, under normal conditions).  There are trees higher than 10 meters.  How do they lift water to their tops?  UPDATE  In other words: how cohesion-tension theory can be true if it apparently contradicts the laws of physics?   UPDATE 2  Atmospheric pressure helps to rise the water, not resists rising. What is resisting is water weight. When water column is 10 meters high, then atmospheric pressure can't help anymore.   Any adhesion/cohesion mechanism can't help here too, because it acts only in thin molecular layer. To transfer action force further the pressure is required, which is insufficient at 10 meters.  UPDATE 3  If we had capillary small enough to rise water to 10 meters and then we will build smaller capillary which we expect will rise water higher, we will fail. Water column will break and does not climb higher than 10 meters.     Menisci acts like small piston and can't help rising water higher than 10 meters.  UPDATE 4  Common pressure distribution in capillary is follows:    $P_0$ is atmospheric pressure. As you see, right under menisci, the pressure is lowered by $2 \sigma / R$ where $R$ is the radius of menisci and $\sigma$ is surface tension. The entire term is called "Laplace pressure". As you see, it can't supersede atmospheric pressure, because water continuity will be broken in the case.  I.e. no any menisci can rise water higher than 10 meters.  The existence of higher trees PROVES that there are some other significant mechanisms, not adhesion/cohesion, not capillary.  UPDATE 5  Current version, as I understood it, is based on a declaration, that a water, if put into thin capillary, can behave like a solid body. Particularly, it can resist tension up to minus 15 atmospheres.   This is a tensile strength of concrete, so I don't believe that without additional proofs.  I think it is just not hard to make thin tube, put water into it and check, how high it can climb.  Was it done ever? Why is notepad++ writing to my new document? I updated notepad++ and it started writing to the screen.  I quickly closed it not knowing what it was, but now I'm curious do I have a 'hacked' copy of notepad++? Where's the Strangedupe blog post? Previously, in order to explain the benefits of (certain limited kinds of) duplication on Stack Exchange sites, I've seen a blog.stackoverflow blogpost with the words "Dr. Strangedupe" in the title.  The post comes up in a search on the new blog, but       Can we please bring this post back? I'd love to link to it when , to reassure the questioner that there is value in not deleting their question because of search terms, but the blog post explained it a lot better than I could in a comment. Why are our planets in the solar system all on the same disc/plane/layer? Why does Li-Po battery voltage drop under load? I have very low background in electronics.  I have a Li-Po battery connected to an electric motor. The question is:   Why does battery voltage drop under load? Why is each cell of a Li-Po battery limited to 3.0 V (discharging)? Why is each cell of a Li-Po battery limited to 4.2 V (charging)? After turning census block data into a raster, how do I redistribute the density to reflect land cover types I plan on rasterizing (terminology?) census block data relating to density to a regular 30 m by 30 m grid for use with NLCD data.  I then hope to distribute the density of the original census block among each result set of grid cells within the original polygon according to the land cover type dictated by the NLCD dataset.  For example, let's imagine a string of city blocks with a park, all found within one census block.  If within this census block the average density is 1000 people/km^2, we ultimately would want the resulting grid cell containing the park to have a lower population density than the grid cells containing the surrounding apartments.  This information may be captured by the NLCD dataset, with the park being classified as '21' (open but developed land) versus 24 for apartment buildings which could be classified as '24' (high density development).  Is there a method, or set of methods, within ArcMap that would allow me to:   Redistribute the density based on the NLCD value Perform this redistribution only within the original census block polygon   I don't know if this is possible, but I figured I would ask.  Edit  Here's what I have been trying:  I converted the census block data to a raster, snapping it to the NLCD land cover data and ensuring the output grid cells were the same size.  Each cell contains the same info as the original census block (housing umbers, population, etc). I have been trying to figure out a way to edit the new grid so that if the NLCD grid cell contains a land cover type that doesn't support a business or housing (park, forest, farm, etc.) that it's housing and population values area set to 0.  My next move was going to be figuring out how to subtract the non-residential/non-business grid cells area from the area of the original census block, and force a recalculation of the density.  Converted the altered census block layer to grid cells again, with each having the new density value, and once again set the cells' housing, population, and density values that don't support a business or housing (park, forest, farm, etc.) to 0.  Kind of circular, and I'm not sure of all the steps (raster calculator for editing individual grid cells based on another raster's grid cell value?), but it seems to logically work out if I can get the parts to work.  On the other hand, @dof1985's answer seems to simplify the first part of my workflow (by directing me to the correct toolset), but I will still have to nullify the grid cells where parks, etc., do exits after the density has been recalculated for the census data. Long transit in Cairo (Egypt) I came to this conclusion: the primes that can't be written as the sum of two squares (lets use$p\prime$ to denote) can be written as ${p}\prime \equiv 3 \;(\bmod\; 4)$.  I need a simple proof to show this and am stuck. I would appreciate guidance! How to place a float at the top of a floats-only page? I just got a short algorithm on my page, and it keeps getting aligned at the center of the page. But I want to have it aligned at the top, what do I have to do?