An alternative formulation of this question is: why is the set of four axioms that define a group so fundamental, rather than, for example, the same four axioms except that associativity is interchanged with commutativity?  What makes this particular set of 4 axioms so special that abstract algebra is based on it?  The four axioms for a group $(G, +)$ are:   Closure: $\;\forall a, b \in G$, $\;(a+b)\in G$ Associativity: For all $a, b, c \in G,\,$ $\;a+(b+c)=(a+b)+c$. Identity: $\;\exists e \in G$ such that $\forall a \in G,\; a+e=e+a = a$. Invertibility: $\forall a \in G, \exists b = -a$ such that $b+a=a+b= e$.   If we add commutativity we have a commutative group. Why not for example make commutativity the key axiom and speak of an "associative group" as a special case? Or why not something else entirely?  What's so special about this combination of 4 axioms?  Edit: For example, is there a theory of "commutative non-associative magma's with an identity and inverses"?
I've only just begun studying group theory (up to Lagrange) following on from vector spaces and I am still finding them almost frustratingly  arbitrary. I'm not sure what exactly it is about the axioms that motivated them defining groups.  My textbook asks you to list 'common features' of vector spaces and then later defines a set of axioms for vector spaces under addition, scalar multiplication and both, noting that the axioms under addition form an Abelian group. So are groups just a generalisation of vector spaces under any binary operation?  My main problem is that the book notes that 'the axioms may seem rather arbitrary' and links groups with vector spaces but doesn't elaborate.   When introducing groups, you are tasked to complete Cayley tables for the symmetries of an equilateral triangle and square.  Then, similarly to the delivery of vector spaces, notes that the tables have common properties (Closure, Identity, inverse and association) and defines a group as a set of elements under a binary operation that has these features.  So what's so important about these 4 properties? For example, if 1 or 2 the properties were excluded form the  axioms, or we added an extra few properties as axioms how would that cripple the effectiveness of groups?  Are the group axioms ever difficult to work with or do they always work, forgive the crude Littlewood analogy, like a mathematical skeleton key?  What is it about these 4 properties that make groups such a powerful tool in mathematics and physics?  My best guess is that a group is the best way to express our sense of  symmetry and what is symmetric mathematically, but I would prefer some elaboration.
I have used Windows since childhood, and when I hear the phrase "Windows filesystem" I think of directories (folders) within directories, a folder called SYSTEM, a folder called PROGRAM FILES etc. Is this what the system is? Just the layout of the folders?  And then I recently started using Linux, and my reference book says in the linux filesystem everything starts at root and branches off from there. How is that really different from Windows? I mean, it seems the linux system and the windows system are just two ways of setting up a directory tree.  Is this what file system means? Vertical text (not in table) Why can we use sample variance of the treatment group to estimate population variance of the controls? Where can I ask about Software Licenses/Licensing? I'm trying to make a page where a user provides an ID of a attachment and the site will set the headers and deliver that file instead of a 404 page. so /download/1754 would lookup the attachment with that Id and output it for viewing/download depending on the file type.  I just can't think of a wordpress friendly way to do this from inside a plugin. I considered using add_rewrite_rule but couldn't see a good way around it, also considered checking $_SERVER['REQUEST_URI'] to deliver the content before any HTML but then I'd have to stop the page from rendering a 404 and I was not sure of a Wordpress friendly way to do this.  My current idea was to have my plugin make a download page, assign a template and have that template run the logic instead of my plugin class. while I have used this before for another situation it was very messy as I had to write a lot of code to force users to be unable to update or edit the file and recreate/delete it on activation.deactivation. so ideally I'd like to avoid that. Why does the SARS-CoV2 virus not remain infectious forever? Or does it? Given that the majority of biologists do not currently consider viruses to be alive, a virus can never die.  It can, however, get destroyed by long exposures to soapy water, alcohol, and apparently certain frequencies of UV light.  Why is an individual SARS-CoV2 virion not infectious forever?  Or is it? How can I pass a class member function as a callback? Is there a more complete way of proving this theorem (that I devised)? OR does this theorem already exist and have a name? Last night, I was working on re-solidifying my understanding of parametric equations, and came up with a neat trick for evaluating certain definite integrals. I'm not sure if it exists as a formal theorem, though I refuse to believe I'm the first to notice such a relation. The 'theorem' is a useful trick to evaluate the definite integral of a continuous function over an interval where the function is either strictly increasing or strictly decreasing, given that the inverse of the function is 'nicer' to work with than the function itself. My idea goes as follows:  If $y = f(x)$ is a continuous function over the interval $I$, and either:  $f'(x) &gt; 0$   OR   $f'(x) &lt; 0$   for all values of $x \epsilon I$, then:  $\int_{a}^{b}{f(x)dx} = bf(b) - af(a) - \int_{f(a)}^{f(b)}{f^{-1}(x)dx}$  where $a,b \epsilon I$  Quasi-proof:  Clearly, if a function is strictly increasing or decreasing, it is injective. This is clear because no two values in the domain can correspond to a single value in the range. I feel intuitively that it must also be surjective (though I'm not entirely sure how to prove this part). Assuming it is both injective and surjective over $I$, the function must have an inverse.   If we were to draw two rectangles out of our bounds, they will have areas:  $af(a)$ and $bf(b)$, so that   $bf(b) - af(a)$ will be the area of the region we are observing.  This area is equivalent to the sum of the areas connecting the curve $f(x)$ to the $x$-axis and to the $y$-axis. Thus:  $bf(b) - af(a) = \int_{a}^{b}{f(x)dx} + \int_{f(a)}^{f(b)}{f^{-1}(x)dx}$  Solving for $\int_{a}^{b}{f(x)dx}$, we get:  $\int_{a}^{b}{f(x)dx} = bf(b) - af(a) - \int_{f(a)}^{f(b)}{f^{-1}(x)dx}$  A particularly illuminating example is evaluating:  $\int_{a}^{b}{ln(x)dx}$  $ln(x)$ is strictly increasing over any interval, so the 'theorem' applies.  $\int_{a}^{b}{ln(x)dx} = bln(b) - aln(a) - \int_{ln(a)}^{ln(b)}{e^xdx}$  $\int_{a}^{b}{ln(x)dx} = ln(b^b/a^a) - (e^{ln(b)} - e^{ln(a)})$  $\int_{a}^{b}{ln(x)dx} = ln(b^b/a^a) + a - b$  and so we've evaluated the integral without the need of integrating the function itself (by method of integration by parts), but rather by integrating its inverse, which is trivial to evaluate.!  Response to Jason DeVito's answer: [This image was taken from   () Storing the original document class page layout with geometry Is it possible, using the geometry package, to store (and restore) the original layout specified by the document class (in my case, the class is report)?    Here is some context.  I have been editing a long document in the default page layout for the report document class.  Now I found out that I need a title page.  There is a particular title page layout, and in fact, it works perfectly if I just load the geometry package with its default and change just two settings.  But then I want the rest of the document back in the original page layout!  I can't do something like \restoregeometry, since I think geometry just overrides the class options... i.e. there doesn't seem to be anything to restore.  Similarly, it seems I can't use \savegeometry or \loadgeometry.  So how can I get back to my original layout for the rest of the document without painstakingly hard-coding in all of the specific page layout variables?    Here is a minimum working example:   \documentclass[a4paper,10pt]{report}  \RequirePackage[a4paper,dvips]{geometry} % g1 \geometry{top=2.5cm,bottom=2.3cm}        % g2  \begin{document}  \input{the_title_page}  \SomeGeometryCommandsHere  % g3  \input{the_rest}  \end{document}   If I run the above code with the lines labeled "g" commented out, then the title page does not have the correct page layout, but the rest of the document is formatted perfectly.    If I leave lines g1 and g2 uncommented, then the title page has exactly the correct layout, but no matter what I put for g3 (e.g. \loadgeometry, \restoregeometry, etc.) I can't recover the original layout that I had when g1 and g2 were commented.  Of course, I could completely specify the original layout exhaustively in line g3, and this should in principle work.  However, I spent almost one hour on this last night and when I still couldn't get it formated correctly I decided I really needed a smarter solution. 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). Do I need a visa for Georgia? I am a Nigerian, holding a valid schengen visa issued by the French embassy, and a multiple entry visa for the United States. Will I be allowed to enter Tbilisi in Georgia? I have a DisplayPort based monitor, and a laptop with a DP out.  I recently got a second laptop with HDMI out, and went looking for a HDMI -> DP converter.  I was very surprised to find that such a thing effectively doesn't seem to exist.   and the  seem to imply that it's just not possible to go HDMI -> DP, without a very expensive converter box (not just an adaptor).  However, I can't find anything that explains why this is the case.  Is DP designed to be backwards compatible but can't interpret HDMI?  Is it a case of HDMI-quality signal not being high enough quality to be convertible?  Some other reason?  Note that this question is not looking for an adaptor - I'm looking for the underlying cause of the lack of them. Possible to do a MySQL foreign key to one of two possible tables? Well here's my problem I have three tables; regions, countries, states.  Countries can be inside of regions, states can be inside of regions.  Regions are the top of the food chain.  Now I'm adding a popular_areas table with two columns; region_id and popular_place_id.  Is it possible to make popular_place_id be a foreign key to either countries OR states.  I'm probably going to have to add a popular_place_type column to determine whether the id is describing a country or state either way. Consider a particle in a potential well. Let’s assume it’s a simple harmonic oscillator potential and the particle is in its ground state with energy E0 = (1/2) &#8463;&omega;0. We measure its position (measurement-1) with a high degree of accuracy which localises the particle, corresponding to a superposition of momentum (and therefore energy) states.  Now we measure the particle’s energy (measurement-2) and happen to find that it’s E10 = (21/2) &#8463;&omega;0. Where did the extra energy come from?  In the textbooks it’s claimed that the extra energy comes from the act of observation but I wonder how that could work. Measurement-1 which probed the position of the particle can’t have delivered to it a precise amount of energy, while measurement-2 might just have been passive. No doubt there is entanglement here between the particle state and the measuring device but where, and which measurement? What are the difference between:    the action $\int_{t_{1}}^{t_{2}}(L+H) dt$ that we use in the principle of least action, and  the action integral $\int_{t_{1}}^{t_{2}}L dt$ that we use in Hamilton's variational principle?   References:   H. Goldstein, Classical Mechanics, Section 8.6. H. Goldstein, Classical Mechanics, Chapter 2. Variance of sample variance? What is the variance of the sample variance? In other words I am looking for $\mathrm{Var}(S^2)$.  I have started by expanding out $\mathrm{Var}(S^2)$ into $E(S^4) - [E(S^2)]^2$  I know that $[E(S^2)]^2$ is $\sigma$ to the power of 4. And that is as far as I got. In some scenes in Avengers: Age of Ultron Iron man, War Machine, and Captain America try to lift Thor’s hammer but they end up failing. At some point Vision lifts it easily. How can Vision lift it? Background reflected in glossy glass, but I didn't set it I've made a roughness 0 glass pane, when I'm in lookdev mode it's reflecting a background that I didn't put there. I know I could just ignore it, but I'd rather be able to see what it'll mostly look like when I'm in the later stages. I also just want to know how on Earth I managed to set this background without realising it  Reflection completely ignores any objects I put in front of it. If I placed this glass inside my house, it would still just reflect trees (This is the point where I felt I couldn't ignore it - reflections are going to play a big part in my project and I don't want to be in render mode all the time)  The nodes are the default glass BSDF, except I set roughness to 0  The glass is set to 20mm thick, if that makes a difference  The background moves with my perspective and moves when I relocate the glass  I have a few HDRI's on my PC, but none look remotely like this   Also, I'm using cycles as I find Evee buggy as hell For every twice differentiable function $f : \bf R \rightarrow [–2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$, which of the following statements are TRUE? For every twice differentiable function $f : \mathbf R \rightarrow [–2, 2]$ with $(f(0))^2 + (f'(0))^2 = 85$, which of the following statement(s) is (are) TRUE ?  (A) There exist $r, s\in \bf R$, where $r &lt; s$, such that $f$ is one-one on the open interval $(r, s)$  (B) There exists $x_0 \in (–4, 0)$ such that $|f'(x_0)| &lt; 1$   (C) $\lim\limits_{x\to\infty}f(x) = 1$   (D) There exists $\alpha\in(–4, 4)$ such that $f(\alpha) + f''(\alpha) = 0$ and $f '(\alpha)\ne0$  My attempt is as follows:-  $A)$ If function is continuous, then definitely in some part of the interval it will be increasing or decreasing hence will be one-one on the open interval $(r,s)$   $f(x)$ cannot be a constant function as in that case $f'(x)=0$ which means $f(0)=\sqrt{85}$, but it doesn't belong to co-domain of $f$  B) By mean value theorem we can say that there exists some $c\in(-4,0)$ for which $4f'(c)=f(0)-f(-4)$  Suppose if $f(0)=2$ and $f(-4)=-2$, then we can only say that there exists $c$ such that $f'(c)=1$. It will not be necessary that there exists a $c$ for which $|f'(c)|&lt;1$  C) $\lim\limits_{x\to\infty}f(x) = 1$, this is not at all necessary.  D) If function is not constant in $(-4,4)$, then we can safely say that at some point $\alpha$, $f(\alpha)\ne0$  But how to know $f(\alpha)+f''(\alpha)=0$  Actual answer is $A,B,D$ but I am able to ascertain only option $A)$. Please help me in this. So in the first 5 or 6 books, it was very clear that the wand chooses the wizard and the power/level of the spells the wizard produces is based on the knowledge and skill of the wizard.  This is referenced by people constantly being telling Harry that he wasn't skilled enough to perform certain spells, or Bellatrix telling Harry that he needed to mean the Cruciatus curse for it to be effective.  However, in the 7th book, this entire idea is changed to the wand being the thing that's powerful and regardless of the wizard's/witch's skill or knowledge, he/she could still perform a very advanced spell if it's something the wand could handle.  Goyle accidentally creating a fire serpent is something that instantly comes to mind from the movies(I don't remember if that happened in the book).  So, my question is, what actually holds the power, the wizard/witch, or the wand?