A common practice in "search bar with filtering result" is usage of &nbsp;&nbsp;▲&nbsp;&nbsp;&nbsp;&nbsp;▼&nbsp;&nbsp; arrows keys.  Can we have this feature in More Stack Exchange Communities in Stack Exchange dropdown in top bar?    picked those arrows keys from
When searching for the desired SE site, the down arrow key (down) should work in order to choose the site one is looking for and hit enter to visit it, instead of using the mouse point to choose it. Would be pretty convenient.
Showing group with $p^2$ elements is Abelian I have a group $G$ with $p^2$ elements, where $p$ is a prime number. Some (potentially) useful preliminary information I have is that there are exactly $p+1$ subgroups with $p$ elements, and with that I was able to show $G$ has a normal subgroup $N$ with $p$ elements.   My problem is showing that $G$ is abelian, and I would be glad if someone could show me how.   I had two potential approaches in mind and I would prefer if one of these were used (especially the second one).   First: The center $Z(G)$ is a normal subgroup of $G$ so by Langrange's theorem, if $Z(G)$ has anything other than the identity, it's size is either $p$ or $p^2$. If $p^2$ then $Z(G)=G$ and we are done. If $Z(G)=p$ then the quotient group of $G$ factored out by $Z(G)$ has $p$ elements, so it is cylic and I can prove from there that this implies $G$ is abelian. So can we show theres something other than the identity in the center of $G$?  Second: I list out the elements of some other subgroup $H$ with $p$ elements such that the intersection of $H$ and $N$ is only the identity (if any more, due to prime order the intersected elements would generate the entire subgroups). Let $N$ be generated by $a$ and $H$ be generated by $b$. We can show $NK= G$, i.e every element in G can be wrriten like $a^k b^l $. So for this method, we just need to show $ab=ba$ (remember, these are not general elements in the set, but the generators of $N$ and $H$).   Do any of these methods seem viable? I understand one can give very strong theorems using Sylow theorems and related facts, but I am looking for an elementary solution (no Sylow theorems, facts about p-groups, centrailzers) but definitions of centres and normalizers is fine. When is the past perfect exactly needed? I have never had this problem before. But after seeing some questions posted on ELU, I started to doubt. I have been through links1, pages of grammar books2, a lot of questions3 posted here and ELU. I now still can't seem to understand how the past perfect is exactly used. The references sometimes even contradict each other.  What I had understood so far was      The past perfect is not necessary if we are not 'going back' to the earlier past, and simply moving from one event to another.   but I then saw a lot of other 'rules' including      If the sequence is clear, the past perfect is not needed.   which is slightly different, less 'narrow' and can be applied in more situations than the first ruleA, and      The past perfect is optional only when talking about an action at a specific time.   This is confusing. Can anyone here clarify things out please?    A For example, for  Doing the survey [completed] -&gt; Showing the result      We showed the result of the survey we did.      If the second rule is applied, the sentence above is fine (the sequence is clear), but not if the first rule since we are not moving from any event to the next and possibly even going back.    1  and   2 Practical English Usage (2005) by Michael Swan, Collins Cobuild Student's Grammar (1991). Upper-Intermediate Cutting Edge.  3 including , , ,  etc. Strange result when removing item from a list while iterating over it Series about proxy space battles via implanted computers in kids The premise of the story is that there are 2-3 main coalitions on Earth and they fight over resources in the edges of the solar system with spaceships that are controlled from Earth by kids with computers in their heads.  If I recall correctly:   The main character's name is Tom, he starts out by conning people at some sort of gambling arcade before he is found and recruited by the "army" The Pentagon in the US has a super tall tower built in the courtyard called The Spire where most of the three books take place. The reason they need kids for this has something to do with brain elasticity and putting computers in adults causes them to go mad. All the cadets are super dismissive about learning how to program their computer aside from one girl who is super good at it. The best fighters from each side of this conflict were something of superstars, but everyone else was kept completely anonymous for their protection. There were at least 3 books last I knew, released circa 2011-2014 The third book ends with Tom being the first astronaut to Mars. The first book had a green cover, the second was yellow, and the third was red. Though I might have the order wrong. Difference between $:=$ and $=$ I am sorry if this is quite elementary question. But I always think, that why we use $:=$ at some places, instead of $=$. Is there any fundamental difference between these two? Before reading Terry Tao's blog (4 months ago), I had never seen a symbol like this (:=). Finding the limit of $(1-\cos x)/x^2$ Javascript and PHP functions I've just started playing with Guice, and a use-case I can think of is that in a test I just want to override a single binding.  I think I'd like to use the rest of the production level bindings to ensure everything is setup correctly and to avoid duplication.  So imagine I have the following Module  public class ProductionModule implements Module {     public void configure(Binder binder) {         binder.bind(InterfaceA.class).to(ConcreteA.class);         binder.bind(InterfaceB.class).to(ConcreteB.class);         binder.bind(InterfaceC.class).to(ConcreteC.class);     } }   And in my test I only want to override InterfaceC, while keeping InterfaceA and InterfaceB in tact, so I'd want something like:  Module testModule = new Module() {     public void configure(Binder binder) {         binder.bind(InterfaceC.class).to(MockC.class);     } }; Guice.createInjector(new ProductionModule(), testModule);   I've also tried the following, with no luck:  Module testModule = new ProductionModule() {     public void configure(Binder binder) {         super.configure(binder);         binder.bind(InterfaceC.class).to(MockC.class);     } }; Guice.createInjector(testModule);   Does anyone know if it's possible to do what I want or am I completely barking up the wrong tree??  --- Follow up: It would seem I can achieve what I want if I make use of the @ImplementedBy tag on the interface and then just provide a binding in the test case, which works nicely when there is a 1-1 mapping between the interface and implementation.  Also, after discussing this with a colleague it would seem we'd head down the road of overriding an entire module and ensuring we have our modules defined correctly.  This seems like it might cause a problem though where a binding is misplaced in a module and needs to be moved, thus possibly breaking a load of tests as bindings may no longer be available to be overriden. How to use the keyboard to drag and drop? I am not experienced in Number Theory but what I know is that some results of this filed are applicable in other areas, e.g. algebra. For sure FLT made (and makes) people be interested in Number Theory leading to the development of new methods which can be themselves applied not only for the proof of FLT (like financial problems motivated somehow development of stochastic analysis). I am interested - if there are applications or implications of FLT itself?  More precisely: if the fact "for each $n\geq3$ there are no integer solutions of $a^n+b^n=c^n$" leads to solutions of problems which are not in the field of Number Theory?   I would specify that I wonder about some problems which are already formulated: since FLT is known for more then 300 years I am pretty sure that there were formulated hypothesis which follow from FLT directly (if there are such hypothesis not in the field of Number Theory). If the plural of ‘man’ is ‘men,’ shouldn’t the plural of ‘German’ be ‘Germen’? What makes these two words so different that 'man' is changed to 'men', but 'German' is changed to 'Germans'? How do that charge gets transferred while rubbing? Is it possible to terminate a running thread without setting/checking any flags/semaphores/etc.? Prove there are no other solutions of functional equation $f(x+y) = \frac{f(x)+f(y)}{1-f(x)f(y)}$ How exactly does one “control for other variables”? Here is the article that motivated this question:   I liked this article, and it nicely demonstrates the concept of “controlling for other variables” (IQ, career, income, age, etc) in order to best isolate the true relationship between just the 2 variables in question.    Can you explain to me how you actually control for variables on a typical data set?      E.g., if you have 2 people with the same impatience level and BMI, but different incomes, how do you treat these data?  Do you categorize them into different subgroups that do have similar income, patience, and BMI?  But, eventually there are dozens of variables to control for (IQ, career, income, age, etc)  How do you then aggregate these (potentially) 100’s of subgroups?  In fact, I have a feeling this approach is barking up the wrong tree, now that I’ve verbalized it.  Thanks for shedding any light on something I've meant to get to the bottom of for a few years now...! Do matrices $ AB $ and $ BA $ have the same minimal and characteristic polynomials? Let $ A, B $ be two square matrices of order $n$. Do $ AB $ and $ BA $ have same minimal and characteristic polynomials?   I have a proof only if $ A$ or $ B $ is invertible. Is it true for all cases? I am a citizen of two different countries, and have two passports.  How should I use my passports when traveling? I have being using Skype for a long time on my Ubuntu 14.04 LTS. From this morning on, Skype simple cannot connect to my account anymore.  That is my Skype version:     Skype 4.3.0.37   © 2014 Skype and/or Microsoft   My Ubuntu system:  LSB Version:    core-2.0-amd64:core-2.0-noarch:core-3.0-amd64:core-3.0-noarch:core-3.1-amd64:core-3.1-noarch:core-3.2-amd64:core-3.2-noarch:core-4.0-amd64:core-4.0-noarch:core-4.1-amd64:core-4.1-noarch:cxx-3.0-amd64:cxx-3.0-noarch:cxx-3.1-amd64:cxx-3.1-noarch:cxx-3.2-amd64:cxx-3.2-noarch:cxx-4.0-amd64:cxx-4.0-noarch:cxx-4.1-amd64:cxx-4.1-noarch:desktop-3.1-amd64:desktop-3.1-noarch:desktop-3.2-amd64:desktop-3.2-noarch:desktop-4.0-amd64:desktop-4.0-noarch:desktop-4.1-amd64:desktop-4.1-noarch:graphics-2.0-amd64:graphics-2.0-noarch:graphics-3.0-amd64:graphics-3.0-noarch:graphics-3.1-amd64:graphics-3.1-noarch:graphics-3.2-amd64:graphics-3.2-noarch:graphics-4.0-amd64:graphics-4.0-noarch:graphics-4.1-amd64:graphics-4.1-noarch:languages-3.2-amd64:languages-3.2-noarch:languages-4.0-amd64:languages-4.0-noarch:languages-4.1-amd64:languages-4.1-noarch:multimedia-3.2-amd64:multimedia-3.2-noarch:multimedia-4.0-amd64:multimedia-4.0-noarch:multimedia-4.1-amd64:multimedia-4.1-noarch:printing-3.2-amd64:printing-3.2-noarch:printing-4.0-amd64:printing-4.0-noarch:printing-4.1-amd64:printing-4.1-noarch:qt4-3.1-amd64:qt4-3.1-noarch:security-4.0-amd64:security-4.0-noarch:security-4.1-amd64:security-4.1-noarch Distributor ID: Ubuntu Description:    Ubuntu 14.04.3 LTS Release:    14.04 Codename:   trusty   Any suggestion why or how to fix it?  Thanks in advance. Sending Email via gmail smtp server in JAVA Can the union axiom be weakened?