Given a normal subgroup $N$ of a group $G$, do there exist sufficient conditions that allow us to conclude that we have an isomorphism $$ G\cong N\times G/N?$$
Suppose you have a group isomorphism given by the first isomorphism theorem: $$G/\ker(\phi) \simeq \operatorname{im}(\phi)$$ What can we say about the group $\ker(\phi)\times \operatorname{im}(\phi)$? In particular, when does the following hold: $$G\simeq \ker(\phi)\times \operatorname{im}(\phi)?$$ I ask this question because i want to prove that $GL_n^+(\mathbb{R}) \simeq SL_n(\mathbb{R}) \times \mathbb{R}^*_{&gt;0}$, with $GL_n^+(\mathbb{R})$ the group of matrices with positive determinant. I proved that $SL_n(\mathbb{R})$ is a normal subgroup and that $GL_n^+(\mathbb{R})/  SL_n(\mathbb{R}) \simeq \mathbb{R}^*_{&gt;0}$, using the surjective homomorphism $\det(M)$. I tried something with semidirect products but I got stuck.
Arch on WSL: libQt5Core.so.5 not found, despite being installed I was hit by some hardcore power-tripping forum mods on the Arch Linux forum, so here I am.  I'm trying to run GUI apps on WSL, such as dolphin or kate. However, none of them can find libQt5Core.so.5, despite it being installed in /usr/lib as you'd expect. Here's a terminal transcript:  /u/lib » DISPLAY=:0.0 dolphin dolphin: error while loading shared libraries: libQt5Core.so.5: cannot open shared object file: No such file or directory /u/lib » pwd /usr/lib /u/lib » ls -la | grep Qt5Core -rw-r--r-- 1 root root     1143 Aug  7 06:05 libQt5Core.prl lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so -&gt; libQt5Core.so.5.11.1* lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so.5 -&gt; libQt5Core.so.5.11.1* lrwxrwxrwx 1 root root       20 Aug  7 06:05 libQt5Core.so.5.11 -&gt; libQt5Core.so.5.11.1* -rwxr-xr-x 1 root root  5234312 Aug  7 06:05 libQt5Core.so.5.11.1* /u/lib » ldd `which dolphin`         linux-vdso.so.1 (0x00007fffdb426000) ...         libQt5Core.so.5 =&gt; not found /u/lib » pacman -Qk qt5-base qt5-base: 3546 total files, 0 missing files   Note that other GUI apps work fine - anything compiled against GTK, including Firefox, work perfectly. I just have to set the $DISPLAY variable. Not setting the var doesn't help, as you'd expect.  Any ideas? Thanks! Do tag edits count for Archaeologist? I know tag-edits do not count towards the 500 edits required for the Copy Editor badge. Do they count to the 100 for the Archaeologist badge? It seems they should because you are making a 6-month-inactive post active again. From a previous question, we have that $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[4]{3})$, and I am assuming that we would need to use this to justify the answer to the question.   Would it be right to use some idea of a splitting field here? Or is it something to do with $\mathbb{Q}(\sqrt[3]{3}, \sqrt[4]{3})$ not containing a primitive 3rd root of unity?   This is a past exam paper question worth 8 marks, so it doesn't seem like it's a few lined answer, but I am a little stuck on where to go. The batcave, in any version of Batman, is large.  Plus there's the batmobile, which required special construction, plus the batcomputer and numerous other batwidgets.  How did a good sized batcave get built without the land above it being torn up and without requiring a huge construction crew?  And, after the cave was built, how was the batmobile built in secret, without someone involved in it being able to make the connection between the car they built and the one that shows up in the news when Batman is first seen?  (While he's in Gotham, a masked vigilante is news that would end up nation wide.)  There would also be the need for secrecy so vilains can't find out how it is built and what the specs are.  And then there's the continual supply of batwidgets.  Just as with the batmobile, there's the need for these items and how they work to be kept secret.  It'd take several factories to produce all that Batman uses and major construction for any batcave I've seen (which is in movies and on TV).  How is all this done in secret?  (Or is there a trail of bodies?) Merge two data frames while keeping the original row order How to replace ${} placeholders in a text file? Why is sample standard deviation a biased estimator of $\sigma$? According to the Wikipedia article on  the sample SD   $$s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x})^2}$$  is a biased estimator of the SD of the population. It states that $E(\sqrt{s^2}) \neq \sqrt{E(s^2)}$.   NB. Random variables are independent and each $x_{i} \sim N(\mu,\sigma^{2})$   My question is two-fold:   What is the proof of the biasedness? How does one compute the expectation of the sample standard deviation    My knowledge of maths/stats is only intermediate. No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof? How to give a direct proof of the following result?     Let $A$ be a subset of $\mathbb{R}$ such that $A$ is both open and closed. Then $A$ is either empty or all of $\mathbb{R}$?   My work:   If $A$ is not empty, let $u \in A$. Then there is some open interval $(a,b)$ such that $$u \in (a,b) \subset A.$$  Now if $\mathbb{R} - A$ is not empty either, then let $v \in \mathbb{R} - A$. Then there is some open interval $(c,d)$ such that  $$v \in (c,d) \subset \mathbb{R} - A.$$  Suppose that $u &lt; v$.   Since $$\emptyset \subset (a,b) \cap (c,d) \subset A \cap (\mathbb{R} - A) = \emptyset,$$ we must have $$(a,b) \cap (c,d) = \emptyset.$$ So we can conlclude that $$b \leq c.$$ But $b \in A$ and $c \in \mathbb{R} - A$. So we must have $$b &lt; c.$$  What next? Can anybody here please help complete the proof from here on?  An edit based on a comment by Marc Paul:   Let us assume that the set $A$ is a non-trivial clopen set in $\mathbb{R}$. Let us define a function $f \colon \mathbb{R} \to \mathbb{R}$ as follows:  $$f(x) \colon= \begin{cases} 1 \ \mbox{ for } \ x \in A; \\ 0 \ \mbox{ for } \ x \in \mathbb{R} - A. \end{cases}$$   Let $V$ be an open set in the range space $\mathbb{R}$. We show that the inverse image set $f^{-1}(V)$ is open in the domain space $\mathbb{R}$.   The following cases arise:   If $0, 1 \in V$, then $f^{-1}(V) = \mathbb{R}$.   If $0 \in V$ but $1 \not\in V$, then $f^{-1}(V) = \mathbb{R} - A$.   If $1 \in V$ but $0 \not\in V$, then $f^{-1}(V) = A$.   And, if $0 \not\in V$ and $1 \not\in V$, then $f^{-1}(V) = \emptyset$.   Thus, $f^{-1}(V)$ is open in the domain space $\mathbb{R}$.   Hence the function $f$ is continuous.   What next? How does this lead to our desired conclusion?   [Yet another edit, again based on valuable comments from Marc Paul: ]  So if both $A$ and $\mathbb{R} - A$ were non-empty, then let's suppose $a \in A$ and $b \in \mathbb{R} - A$, and we can assume without any loss of generality that $a &lt; b$.   Then $f(a) = 1$ and $f(b) = 0$. So by the intermediate value theorem there is a real number $c \in (a,b)$ such that $f(c) = 1/2$, which is a contradiction because the image set of $f$ does not contain $1/2$. Generating random numbers in C Whenever I boot Ubuntu, I get a message that it cannot mount my windows partition, and I can choose to either wait, skip or manually mount.  When I try to enter my Windows partition through Nautilus I get a message saying that this partition is hibernated and that I need to enter the file system and properly close it, something I have done with no problem so I don't know why this happens.    Here's my partition table, if any more data is needed please let me know.     Device Boot      Start         End      Blocks   Id  System /dev/sda1            2048    20000767     9999360   83  Linux /dev/sda2        20002814   478001151   228999169    5  Extended /dev/sda3   *   478001152   622532607    72265728    7  HPFS/NTFS/exFAT /dev/sda4       622532608   625141759     1304576   82  Linux swap / Solaris /dev/sda5        20002816   478001151   228999168   83  Linux Can I use Stack Exchange logos on a t-shirt or other apparel? I happened to notice while on the Super User site that one user's picture shows him wearing a shirt with the Super User logo on it. Interested, I wanted to find out where he got it from. Looking through old questions on Meta, I found out that there used to be a shop.stackexchange.com site that sold Stack Exchange gear. Unfortunately, they no longer sell any merchandise.  Is there any site where I can purchase a Stack Exchange site shirt/hat/etc.? If not, could I create a shirt/hat/etc. myself with one of the sites' logos on it? If so, where can I locate the logos in their largest size? Suppose you do bad in undergraduate school in say computer science. But you do very well in a masters program in computer science. If you want to apply to a PhD program in computer science, will the masters degree grades offset the undergraduate degree grades? Error :: You must put some 'source' URIs in your sources.list I'm trying to install PIL a python module on ubuntu   I first need to install a package called python-imaging using the following command    sudo apt-get build-dep python-imaging   When I try and do this I get the error   E: You must put some 'source' URIs in your sources.list   I saw  question whihc shows how to fix via the GUI but I'm using the command line. Does anyone know how I can fix this error..?    UPDATE: Based on the below answers I've updated my package list but I now get this error. Not sure if it is related, if not I'll start a new question as I can't find reference to the error anywhere.   E: Build-Depends dependency for python-imaging cannot be satisfied because candidate version of package python-all-dev can't satisfy version requirements Purpose of @ Symbol Before Strings? What is the JVM thread scheduling algorithm? Do newer laptop batteries still need to be drained before recharging? 10 years ago when I was doing hardware support it was best practice to fully drain a laptop battery prior to recharging in order to maximize its life.  Is this still the case or can a battery now be recharged at any point without worrying about shortening the lifespan? Glucagon stimulates glycogenolysis and gluconeogenesis, thus increasing the plasma glucose concentration — so that tissues get enough glucose in the fasting state. However glucagon also inhibits glycolysis. In this case, what is the point of having high blood glucose if it cannot be catabolized? What is the difference between "speaking" and "talking"? It seems more politically correct to say that someone is speaking rather than talking.  What is the definitive difference between these terms? Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$ For an odd prime $p$, prove that any primitive root $r$ of $p^n$ is also a primitive root of  $p$   So I have assumed $r$ have order $k$ modulo $p$ , So $k|p-1$.Then if I am able to show that $p-1|k$ then I am done .But I haven't been able to show that.Can anybody help me this method?Any other type of prove is also welcomed. What does the "private" modifier do? Considering "private" is the default access modifier for class Members, why is the keyword even needed?