Show that if $f : G → H$ is a homomorphism, $H$ is abelian and $N $ is a subgroup of $G$ containing $Ker f$, then $N$ is normal in $G$.      (First Isomorphism Theorem) If $f: G→ H$ is a homomorphism of groups, then $f$ induces an isomorphism $G/Ker f\cong Im f$.      Corollary 5.12. lf $N$ is a normal subgroup of a group $G$, then every subgroup of $G/N$   is of the form $K/N$, where$ K$ is a subgroup of $G$ that contains $N$. Furthermore, $K/N $ is normal in $G/N$ if and only if $K$ is normal in $G$.   Proof: Since $H$ is abelian, $f(H)$ is an abelian subgroup. (How?) By the First Isomorphism Theorem $f(H) \cong G/Ker f$ (How?), and so it too is abelian. (How?) Thus for every $g ∈ G, n ∈ N$, $$(gH)nH(g^{-1}H) = (gH)(g^{-1}H)nH = nH.$$  Therefore $(gH)(N/H)(g^{-1}H) = N/H $ and so $N/H$ is normal in $G/N$. From here Corollary-I.5.12 concludes $N$ must be normal in $G $ since $N/H$ is normal in $G/Ker f$.  Can anyone explain the (How?) parts ? Or is there any other proof suggested, since I understood so little from this.
Let $\varphi:G\rightarrow H$ be a group homomorphism from group $G$ to group $H$. Show that, if $H$ is abelian, all subgroups of $G$ that contain $\mathrm{ker} (\varphi)$ are normal in $G$.
Why enums require an explicit cast to int type? Reading shapefile using C#? Hi I would like to know whether the trace of the inverse of a symmetric positive definite matrix $\mathrm{trace}(S^{-1})$ is convex.  Actually I know that the trace of a symmetric positive definite matrix $S\in M_{m,m}$ is convex since we can find $B\in M_{n,m}$ such that $S=B^T\times B$ then we can write the trace as the sum of scalar quadratic forms, i.e. $\mathrm{trace}(S)=\mathrm{trace}(B^T\times B)=\sum_{j=1}^mb_j^T\times b_j$ where $b_j$ is the $j^{th}$ column of $B$.  for instance if we have  $trace([\begin{array}{cc} 1 &amp; 2  \\ 3 &amp; 4  \\ \end{array}] \times [\begin{array}{cc} 1 &amp; 3  \\ 2 &amp; 4  \\ \end{array}])= [\begin{array}{cc} 1 &amp; 2  \\ \end{array}]\times [\begin{array}{c} 1   \\ 2   \\ \end{array}]+ [\begin{array}{cc} 3 &amp; 4  \\ \end{array}]\times [\begin{array}{c} 3   \\ 4   \\ \end{array}]=30$  And so I wonder if $\mathrm{trace}(S^{-1})$ is convex too.. How can I make it so that each space has its own isolated dock?  Basically what I'm looking for is:   The dock in one space should not show the icon of an app that was open in another space. If I have the same program open in two different spaces, clicking on the icon in one space should not go to the app in the other space.   Is this possible natively, or with a third party program? A quick search for ESTA application returned many results. According to , the official site to apply for an ESTA is .  Are the others scams, legitimate third party resellers or "added value" services?    (defunct - domain is for sale circa 2020)     Can anyone confirm either way?  Updated 2020: The above have better "not affiliated with the U.S. Government" warnings now, but beware there are newer more sneaky sites starting all the time:        And fortunately more official guidance about this issue: Complete Statistic: Uniform distribution Take a random sample $X_1, X_2,\ldots X_n$ from the distribution $f(x;\theta)=1/\theta$ for $0\le x\le  \theta$.  I need to show that $Y=\max(X_1,X_2,...,X_n)$ is complete.  Now, I know I should multiply the sample distribution of $Y$ and multiply it with a function of $Y$, then integrate over the range of $\theta$ and equate them to zero. But how do I get the sampling distribution of $Y$? Appendix in LaTeX How can I programmatically force-stop scrolling in a UIScrollView? How to "freeze" a cloth animation at X frame? How would one freeze a simulated body at frame X and use it as a static mesh?  Following a , I've been able to do a cloth animation to my model, now I'd like to have that made static, without the use of the timeline.   The thing is, I'm modelling something that's not going to be an animation, but a static object (something like a statue/figurine). So I don't need the animation, only the shape at a particular frame that I find convenient. I'd like eventually once this is done, to also modify that deformed grid.   I've tried to follow what was said , but couldn't figure out what chipmasque was saying. I'm quite new to blender (been using it on and off, just enough to forget everything).  I'm using blender 2.63, if it is of any relevance. Jquery load() and PHP variables What does "javascript:void(0)" mean? How Many Genius Mixes Are There in iTunes? "To work for" vs. "to work in" vs. "to work at" Storing C++ template function definitions in a .CPP file Is there any way to view saved wifi passwords on the iPhone? Subset of a finite set is finite We define $A$ to be a finite set if there is a bijection between $A$ and a set of the form $\{0,\ldots,n-1\}$ for some $n\in\mathbb N$.  How can we prove that a subset of a finite set is finite? It is of course sufficient to show that for a subset of $\{0,\ldots,n-1\}$. But how do I do that? In Lieb and Loss's Analysis, I saw that they mentioned  $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$ (dense wrt the $L^2$ norm, I think). But I didn't find its proof in the book. So I wonder why it is?    Does this conclusion hold for $L^1(\Omega, \mathcal{F}, \mu) \cap L^2(\Omega, \mathcal{F}, \mu)$ for any measure space $(\Omega, \mathcal{F}, \mu)$?  Are there similar statements if replace $L^1$ with $L^p$, and  $ L^2$ with $L^q$ for $p \leq q \in (0, \infty ]$ or $\in [1, \infty]$?  Thanks and regards! Exploring ocean depths: is pressure an issue? Should it be? An upcoming adventure will have the party exploring the depths of the ocean. Breathing underwater shouldn't be an issue for them, as there are both items and rituals to take care of that, however, I was wondering what to do about about pressure, since none of these options mention it. I'll have to split my question into two, since they are closely related.   Is pressure an issue in 4e adventures? Is it mentioned in any of the published material? Has it been ruled anywhere that options that enable water breathing also grant immunity to pressure? Should it be an issue? This is somewhat subjective, but I really am wondering what other people have done. Would it be unresonable to assume that these options grant immunity to pressure? Should the players work to find another solution? Good textbooks on homological algebra How to show $(a^b)^c=a^{bc}$ for arbitrary cardinal numbers? One of the basic (and frequently used) properties of cardinal exponentiation is that $(a^b)^c=a^{bc}$.  What is the proof of this fact?  As Arturo pointed out in his comment, in computer science this is called .