I wish to prove the exercise which states that for a set $A$ containing more than half elements of a group $G$, every element of $G$ is a product of two elements of $A$.  My attempt:  By Lagrange Theorem, subgroup generated by $A$ must coincide with $G$, so every element in $G$ is a product of elements from $A$. How can I prove that the product is of exactly two elements?
I'v found this proposition.  If $G$ is a finite  group , $ A \subset G $ a subset and $|A| &gt; \frac{|G|}{2} $ then $AA = G $.  Why this is true ?
Dual Canadian and British citizen - healthcare coverage in the UK? I have a dataset that has both continuous and categorical data. I am analyzing by using PCA and am wondering if it is fine to include the categorical variables as a part of the analysis. My understanding is that PCA can only be applied to continuous variables. Is that correct? If it cannot be used for categorical data, what alternatives exist for their analysis? Question regarding an inequality How to prove that $$ \frac{x_1}{1+x_1^2}+\frac{x_2}{1+x_1^2+x_2^2}+\cdots+\frac{x_n}{1+x_1^2+\cdots+x_n^2}&lt;\sqrt{n} $$ knowing that $(x_n)$ is a positive sequence ? I looked up all kinds of inequalities such AM-GM, Chebyshev, Cauchy-Schwarz, but I couldn't manage to obtain anything useful.. Can anyone help ? How to XZ a directory with TAR using maximum compression? I have some auto-generated T-SQL, which is probably valid, but I don't really understand.  ALTER TABLE [dbo].[MyTable] WITH CHECK CHECK CONSTRAINT [My_FORIEGN_KEY];   I know what a foreign key constraint is, but what's the CHECK CHECK? No error message if the URL is very long I entered a URL with a very long parameter on Stack Exchange.  I have type this url    It shows a blank screen , does not shown any messages.    We need to show a “Page Not Found” message, same as what you did on other Stack Exchange pages. In particular, I am referring to the Pearson product-moment correlation coefficient. Can the Hubble constant be measured locally? In other words, why do we need to connect the battery positive to the negative to get electron flow? As far as I know, voltage difference is what drives current flow.  From what I understand, there's a surplus of negative charge (electrons) in the positive end of a battery (weird I know, but I guess they do it for mathematical reasons). Between the positive and negative ends of the battery is some kind of wall that prevents the electrons from diffusing, so they have to go the long way (through a wire to the other end of the battery) to diffuse and reach the receiver atoms/molecules.  So why not just have the surplus of electrons on one end go to ground to equalize (e.g. 5 volts to 0 volts should produce current)? Why does it have to be the other end of the battery? I'm sure there's atoms/molecules in the ground that could accept the extra electrons (receivers atoms/molecules for the donors in the battery). What is the VC dimension of a decision tree? What is the  of a decision tree with k splits in two dimensions? Let us say the model is CART and the only allowed splits are parallel to the axes.  So for one split we can  and then for any labeling of the points we could get perfect prediction (i.e.: shattered points)  But what about 2 splits, or any general k? I am using a Collection (a HashMap used indirectly by the JPA, it so happens), but apparently randomly the code throws a ConcurrentModificationException. What is causing it and how do I fix this problem? By using some synchronization, perhaps?  Here is the full stack-trace:  Exception in thread "pool-1-thread-1" java.util.ConcurrentModificationException         at java.util.HashMap$HashIterator.nextEntry(Unknown Source)         at java.util.HashMap$ValueIterator.next(Unknown Source)         at org.hibernate.collection.AbstractPersistentCollection$IteratorProxy.next(AbstractPersistentCollection.java:555)         at org.hibernate.engine.Cascade.cascadeCollectionElements(Cascade.java:296)         at org.hibernate.engine.Cascade.cascadeCollection(Cascade.java:242)         at org.hibernate.engine.Cascade.cascadeAssociation(Cascade.java:219)         at org.hibernate.engine.Cascade.cascadeProperty(Cascade.java:169)         at org.hibernate.engine.Cascade.cascade(Cascade.java:130) Daily review limit exceeded I have just reached my limit of suggested edits for the day when I noticed someone else has 21, when the limit is 20, how can this be? How do I play Minecraft with people online, besides using a server? Make it clearer when looking at a question if I've already voted to close it Since "" we now have the possibility to retract close votes.   However, as mentioned in "" there is no indication when looking at a question to see if I have already cast such a close vote.  Therefore, could we have a UI improvement to show this? Adding some image with freehand circles to improve visibility. Kinematic Viscosity How would you define ? What does it physically represent? Around the Internet I've found it defined as just a ratio, and that's it.  I saw in an answer that I can think of it as "diffusivity of momentum". What does this mean? I am having trouble with the following problem:   $f: N\times N\rightarrow N$ and $f(i,j)=2^{i-1}(2j-1)$. Prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent.   Work: I am told that Euclid's Lemma will be useful to proving injection and also that for any positive integer $n$, $n$ is a product of prime numbers.   For injection, I set $f(a,b)=f(x,y)\rightarrow 2^{a-1}(2b-1)=2^{x-1}(2y-1)$.  Then I distributed across to get $2^ab-2^{a-1}=2^xy-2^{x-1} \rightarrow 2^ab-2^xy=2^{a-1}-2^{x-1} \rightarrow 2(2^{a-1}b-2^{x-1}y)=2^{a-1}-2^{x-1}$.   From here, I am unsure how to apply Euclid's Lemma to arrive at the injection result.  I am even more unsure how to prove surjection. TSA broke a lock for which they have a master key. How can I claim compensation? How to simulate from a Gaussian copula? Suppose that I have two univariate marginal distributions, say $F$ and $G$, which I can simulate from. Now, construct their joint distribution using a , denoted $C(F,G;\Sigma)$. All the parameters are known.  Is there a non-MCMC method for simulating from this copula? How do I restore the default repositories? Which equals operator (== vs ===) should be used in JavaScript comparisons?