Normalizer of some subgroup of a group is necessarily normal subgroup of the group.
Counterexample for the normalizer being a normal subgroup
Is normality testing 'essentially useless'? A former colleague once argued to me as follows:      We usually apply normality tests to the results of processes that,   under the null, generate random variables that are only   asymptotically or nearly normal (with the 'asymptotically' part dependent on some quantity which we cannot make large); In the era of   cheap memory, big data, and fast processors, normality tests should   always reject the null of normal distribution for large (though not insanely large) samples. And so, perversely, normality tests should   only be used for small samples, when they presumably have lower power   and less control over type I rate.   Is this a valid argument? Is this a well-known argument? Are there well known tests for a 'fuzzier' null hypothesis than normality? Get the title instead of the number of a referenced chapter/section? Python: Invalid Token Beryllium Vacuum Sphere Boat/Aircraft Is it possible to make a solid rigid evacuated "balloon" out of Beryllium or other elements or alloys?  The critical buckling pressure at which an evacuated sphere is given as $$ P_1=\frac{2E\cdot\left(h/R\right)^2}{\sqrt{3\left(1-\nu^2\right)}} $$ where $E$ is , $h$ the shell's thickness, $R$ the radius to the midsurface shell, and $\nu$ is .  Of course this is theoretical and experimental and literature values for submersibles hover around 1/4 of the theoretical pressure.  Beryllium has the following properties:   Density=1.85 g/cm$^3$ Young's Modulus=287*10$^9$ Pa Poisson Ratio=0.032   Calculating the upward force, buoyancy at 101.3 kPa atmospheric pressure and Radius of 10m gives 4.24*10$^8$ N. The weight of this sphere with shell thickness 1mm has 2.32*10$^7$ N downward force of Beryllium material, so it will theoretically float at ground level.   Calculating the critical buckling stress of this beryllium sphere is 331.57 kPa, which is 3.27 times the atmospheric pressure of 101.3 kPa, so theoretically it won't buckle. But in experimentation done in earlier experiments as stated before, due to construction/material non-ideal situations, the critical buckling formula predicts about 4 times stronger than actual, so it would need about 400 kPa, not 331 kPa as Beryllium sphere with this specification would hold. Increasing the strength capability also increases weight and therefore lowers it's ability to float, but Beryllium is the closest element with this situation I have calculated that could work if ideal (I have only been using periodic table elements and household materials)  Is it possible with an alloy of elements? Using the equations for buckling and buoyancy, the following is needed. What alloys/materials come to your mind?  Material considerations:   Material with larger Poisson ratio Material with larger Young's Modulus Less dense material Construction (is it possible)   Material considerations affect on lift   Thickness can be larger, increases strength by the squared. But also increases weight. If a material with lower density with similar structural properties as Beryllium is used, it would help. Radius can be lower for same reason, but also lowers buoyancy (volume)   My other designs (don't worry about these just yet, just if you are curious is a vacuum sphere with a pressurized "outer tube" with spokes in tension (carbon fiber nanotubes maybe) to keep the 2 spheres in position. Another is a spherical truss out of carbon fiber nanotubes with a light membrane around. The truss prevents the outer membrane from collapsing due to the atmospheric pressure  The "point" of this if you ask, as to why not use a traditional balloon is it can be used as a boat by filling the sphere with air, and then used as a aircraft at will by evacuating the sphere. An alternative mode of transportation for sea vessels etc. How can I make a program executable from everywhere What should I do if I want to be able to run a given program regardless of my current directory? Should I create a symbolic link to the program in the /bin folder? Can we divide two vector quantities? For eg., Pressure( a scalar) equals force (a vector) divided by area (a vector). Summability vs Unconditional Convergence I'm reading that summability and unconditional convergence are the same. (At least if the index set is countable, so unconditional convergence makes sense.)  Then unconditional convergence is nothing else than convergence of the net of all rearranged partial sums. Thus uniqueness of the limit becomes fairly easy as the limit of every net in a Hausdorff space is unique.  But the usual proof deals with separating functionals on Banach spaces, which is a much more serious technique. So I think there must be a hook in my reasoning. Can somebody resolve this issue? Thanks a lot!  Best Regards, Freeze_S Graph of Reputation Over Time for all linked accounts Execute bash commands over SSH while staying in interactive mode afterwards Probably it is a weird thing I would like to achieve but: I would like to SSH to a remote host and then execute some bash commands like "alias" automatically to use them afterwards in an interactive mode. The remote host does not allow the personal environment.    Is this possible? Would "expect" do it? Or is there any other way to achieve this? Basically I would like to have my bashrc without modifying any files on the remote host. What do the symbols on the main menu mean? If you swipe all the way to the left on the menu and then some, this sequence of symbols appears. (Do not pay any mind to the white circles: it's where I was holding my finger. The space between the first few and the last three symbols is also not present in game.)    What gives? I am using Selenium 2/WebDriver for automation. I have a table and am trying to select a row in it. When the test runs, I can see the row become highlighted as if it is clicked, but then immediately I get the "org.openqa.selenium.StaleElementReferenceException: Element is no longer attached to the DOM" error and the test fails.  The code is the following:  @Test public void rowSelection() throws Exception {     SeleniumHelper helper = new SeleniumHelper();     action = new SeleniumActionHelper(driver);      helper.login();      String testUrl = navigateToUrl("option/listOptions.xhtml");     driver.get(testUrl);      WebElement table = findElementById("tableSection:dataTableWrapped_data");     List&lt;WebElement&gt; allRows = table.findElements(By.tagName("tr"));     for (WebElement row : allRows)         {             List&lt;WebElement&gt; cells = row.findElements(By.tagName("td"));             for (WebElement cell : cells)             {                 WebElement listName = cell.findElement(By.xpath("./* [text()='body_build']"));                 listName.click();             }         } }   I have put a Thread.sleep(2000) in front and after the listName.click() action, but neither has helped. Any help would be greatly appreciated. How to make a particular app the default viewer / editor for ALL its supported file types at once? What's the difference between lists and tuples? What's the difference?  What are the advantages / disadvantages of tuples / lists? What does the Visual Studio "Any CPU" target mean? What is the best way to identify outliers in multivariate data? Suppose I have a large set of multivariate data with at least three variables. How can I find the outliers? Pairwise scatterplots won't work as it is possible for an outlier to exist in 3 dimensions that is not an outlier in any of the 2 dimensional subspaces.  I am not thinking of a regression problem, but of true multivariate data. So answers involving robust regression or computing leverage are not helpful.  One possibility would be to compute the principal component scores and look for an outlier in the bivariate scatterplot of the first two scores. Would that be guaranteed to work? Are there better approaches? Layout selector missing Counting overlapping polygons in GeoJSON to result in new set of polygons? The speed of light as it approaches a massive body No matter how fast you go, you will aways perceive the speed of light as constant. Taking that as a fact, the special relativity theory was formulated. Now, for what I understand about general relativity, just by standing here on the surface of the planet under the effect of gravity, my frame of reference is a little different from someone in geostationary orbit, and time goes by a little slower for me.  So, considering me and someone in geostationary orbit, does both of us would perceive light at the same speed? If so, does that means that the light is slower down here than up there? How do I install GNOME Shell extensions?  Is it possible to update, remove and manage the extensions? Do community and moderator deletes penalize in the same way that self-deletes do?