The basis of my question is as follows:  I am very aware that, within the realm of quantum gravity research, string theory is a far more popular formulation than loop quantum gravity (LQG). Why is this? Is there a reason that strings are more attractive, or perhaps is there something predicted by LQG which is considered unattractive? Personally, I could imagine a mindset where LQG is a far more attractive and natural option, so why is it considered an "alternative" quantum gravity description?  I am mostly looking for objective/physical/historical reasons.
Basically asking what were the motives behind each theory. What was it that led physicists toward these ideas?
$\mathbb Z_{p^n}$ is not the direct product of any family of its proper subgroups I'm trying to solve this question of Hungerford's Algebra book:     $S_3$ is not the direct product of any family of its proper subgroups.   The same is true of $\mathbb Z_{p^n}$.   The first claim is easy: we note that every subgroup of $S_3$ is of order $2$ or $3$ by Lagrange, since its subgroups are of prime order, they are cyclic, then abelian, but $S_3$ is not abelian, contradiction because $S_3$ is not abelian while the direct product of abelian groups has to be abelian.  My problem is with the second claim, I need help. Combinatorics identity question I encountered this particular combinatorial identity :$$\begin{align}&amp;1\dbinom{n-1}{r-1}+2\dbinom{n-2}{r-1}+\cdots+(n+1-r)\dbinom{r-1}{r-1}\\=&amp;\dbinom{n}{r}+\dbinom{n-1}{r}+\cdots+\dbinom{r}{r}\\=&amp; \dbinom{n+1}{r+1}\end{align}$$How does one figure it out? They knew where she was. Why didn't the Ministry of Magic ask Moaning Myrtle's ghost about her death? Moaning Myrtle said she made sure Olive Hornby was haunted by her memory of finding Myrtle's body until Olive's dying day. Olive had to seek help from the Ministry of Magic in an effort to be free of the spirit, so the Wizards there knew a witness (Myrtle's ghost) to the 1943 murder had become available. After the Ministry of Magic exorcised Myrtle, she was relegated to Hogwarts, generally haunting the same stall in the same lavatory she died in. The powers that be knew where she was  What does JKR say about why no one at the Ministry of Magic, or any of the Hogwarts Professor's, asked Moaning Myrtle about her death? How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$? Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$  Apart from induction, I tried with  to check the validity, but I can't think of an easy (manual) alternative.  Please suggest an intuitive/easy method. What to do with meta questions referring to currently non-existing features? I sometimes come across such questions like that:    The "Not a real question" close reason doesn't exist for long time. It doesn't make sense to answer, comment, or even up/downvote (since there's nothing more to vote for/against it).   Shouldn't such questions be locked as historical ones? Should I flag them for moderator attention or just ignore them? The question asks it all really. When referring to a master's degree, do you use an apostrophe or not? That is, is it "a master's" or "a masters"? How to prove this question? If a graph with $n$ vertices and $n$ edges it must contain a cycle? When a Java VM crashes with an EXCEPTION_ACCESS_VIOLATION and produces an hs_err_pidXXX.log file, what does that indicate? The error itself is basically a null pointer exception. Is it always caused by a bug in the JVM, or are there other causes like malfunctioning hardware or software conflicts?  Edit: there is a native component, this is an SWT application on win32. Weird line in my mirrored mesh was following a tutorial where I stumbled into this weird line in my mirrored mesh (sorry for the feet lol)      You can also download the blender file I'm trying to define \conjunction by means of \textbullet instead of \bullet, since the latter looks too big in logical formulas.  How can I make the spacing around \conjunction to behave like the spacing around \bullet when there is an opening parenthesis just after it?  Compare (2) and (2'):     \documentclass[12pt]{article} \usepackage[paperwidth=10cm,paperheight=7cm,margin=1em]{geometry} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ebgaramond-maths}  \DeclareMathOperator\conjunction{\raisebox{-0.25ex}{\textbullet}}  \begin{document}\begin{align} \intertext {Good spacing but bad size:} (\exists x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\bullet Gx)\supset\psi)\\ \exists  x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\bullet (Gx\supset\psi))\\ \intertext {Good size and spacing:} (\exists x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\conjunction Gx)\supset\psi)\\ \intertext {Good size but bad spacing:} \exists  x(\forall y(\phi(y)\equiv y\!=\!x)&amp;\conjunction (Gx\supset\psi)) \end{align}\end{document}   This won't do:  \DeclareMathOperator\conjunction{\raisebox{-0.25ex}{\textbullet\ }}   because it adds an unwanted extra space to (1'). I have the following code in my theme's archive.php:  &lt;?php the_archive_title( '&lt;h1 class="page-title"&gt;', '&lt;/h1&gt;' ); ?&gt;   This gives me titles like "Category: Russia", "Tag: America", "Author: John".  I would like to remove the "Category:", "Tag:" and "Author:" part and just display the category, tag and author names.  Does anyone know how to accomplish this?  Thank you. How do I get the Glare node to output transparent instead of Black background This is what I have so far:  The rendered image has a transparent background, but the glare filter creates an output with a black background, so after merging the two togheter, the background turns black but I need it to be transparent, what can I do? Splitting a string into words and punctuation Ejecting a USB 3.0 drive connected to USB 3.0 port reconnects immediately GeoReference tiff image into the Google map? I want to Geo-reference a tiff image contain a Map for a specific region, I need to add this region(image) and Geo-reference it on the Google map, SO I can see on the map for the region I select the image which I have inserted.  the image also need information such as the Latitude and Longitude of corners and size, and the pixel per meter measurement.  How to do the correct Geo-reference for that ? Why isn't Force Speed used more often? Field level permissions in Drupal 7 What is the best way to create a piece of content with multiple fields... allowing users to edit the content as a whole yet blocking the ability to edit.  The setup I have is a piece of content that has 3 fields: Body, PDF, SWF... and a "Selector" (List Text) that's tied via dependancy. So the user chooses BODY, PDF or SWF and that's what available to edit and shown on the page when live.  What I want to do is add a PHP field... that's either:     greyed out (unable to edit) if your not admin...   doesn't show up in the Drop Down list... Can't select it if you can't see it Shows up on the drop down, but the field doesn't show up in the editor without permissions.   I hope I'm making sense... I can provide an example if needed. Binomial Distribution Problem - Airline Overbooking How to do joins in LINQ on multiple fields in single join I need to do a LINQ2DataSet query that does a join on more than one field (as  var result = from x in entity join y in entity2         on x.field1 = y.field1  and            x.field2 = y.field2   I have yet found a suitable solution (I can add the extra constraints to a where clause, but this is far from a suitable solution, or use  solution, but that assumes an equijoin).  Is it possible in LINQ to join on multiple fields in a single join?  EDIT  var result = from x in entity              join y in entity2              on new { x.field1, x.field2 } equals new { y.field1, y.field2 }   is the solution I referenced as assuming an equijoin above.  Further EDIT  To answer criticism that my original example was an equijoin, I do acknowledge that, My current requirement is for an equijoin and I have already employed the solution I referenced above.  I am, however, trying to understand what possibilities and best practices I have / should employ with LINQ. I am going to need to do a Date range query join with a table ID soon, and was just pre-empting that issue, It looks like I shall have to add the date range in the where clause.  Thanks, as always, for all suggestions and comments given Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$. Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.   I do not understand how to go about completing this problem or even where to start.