I know that $A_5$ is simple. But how do I tell which even conjugacy classes of $S_5$ split into two conjugacy classes of alternating permutations. When we know that $A_5$ is normal and normal subgroup is a disjoint union of conjugacy classes.
Suppose $n$ is a non negative integer $\geq 4$ and $\sigma\in S_n$ a permutation. Conjugacy classes in $S_n$ are completley determined by the cycle structure of $\sigma$. If we let the alternating group $A_n$.act on $S_n$ by conjugation, the orbits coincide with the conjugacy classes in $S_n$ except for those (even) permutations whose cycle structure is composed of even cycles (which corresponds to cycles of uneven length) of distinct order (fixed points are treated as cycles of length $1$.) See   For instance a $3$-cycle in $A_4$ has cycle structure $[3,1]$ which fits the bill, and a $5$-cycle in $A_5$ or $A_6$ works too, since the cycle structure is $[5]$ and $[5,1]$ respectively. On the other hand, a $3$-cycle in $A_5$ will does not satisfy the criteria as it has cycle structure $[3,1,1]$ with two fixed points.  In this case, the conjugacy class of $\sigma$ inside $S_n$ splits in $2$ orbits of the conjugation action of $A_n$, and if $\tau$ is any odd permutation, for instance $(12)$, then we have $$\mathrm{Conj}_{S_n}(\sigma)=\mathcal{O}(\sigma)\coprod\mathcal{O}(\tau\sigma\tau^{-1})$$  My question is How can we differentiate between the two orbits? Does it have geometric meaning?    I can see that this information can be useful when looking at representations of the alternating groups, since there are as many irreducible representations as conjugacy classes. As for some geometric interpretation, I ask because there is one for the three cycles in $A_4$. If $S_4$ is understood as the group of symmetries of a regular tetrahedron, then $A_4$ is the subgroup of direct symmetries, and a $3$-cycle corresponds to a rotation of angle $\frac{2\pi}{3}$ or $\frac{4\pi}{3}$ with axis passing through one of the vertices. Conjugacy classes inside $A_n$ preserve the angle, and that's how we can tell them apart. In this case the computations are so simple we don't really need this picture, but I think it is a nice way to understand the fact $3$-cycles in are not all conjugate inside $A_4$...
Is it true that $aH = bH$ iff $ab^{-1} \in H$ Let $H$ be a subgroup of a group $G$. The I know that for $a,b\in G$ we have $aH = bH$ if and only if $a^{-1}b \in H$.  My question is if it is also true that $aH = bH$ if and only if $ab^{-1} \in H$? Does it matter where the $-1$ goes?  I am guessing that this is not true since we keep putting the $-1$ on the left element, but I am also thinking that it might be true because they look like the same.  If $G$ is abelian, then it doesn't matter, so a possible counter example would have to involve a non-abelian group. If $H$ is a normal subgroup, then it should also be true because then $a^{-1}bH = Ha^{-1}b$. How does an isolated body in deep space 'know' it's rotating? We can imagine an object floating in the known universe, maximally distant from any other large mass. Maybe it has been there since coalescing after the big bang.  What physical phenomena tell it whether it is rotating relative to the rest of the universe and therefore experiencing a centrifugal (?) force.  Is it the combined gravity of all other matter? Is it 'spooky action at a distance'?  Is it because it is rotating relative to 'empty space'? How can I get Stack Exchange to send me to a conference? I read , which states,  Depending on the circumstances and location, we can also sponsor community leaders to attend an event on behalf of their site. We will subsidize your costs to attend, within reason, and provide you with a bunch of swag to use as an ice-breaker when introducing yourself.  This sounds great. But the blog is sparse on the details. What do I actually have to do to get in on this? The limit of a sum $\sum_{k=1}^n \frac{n}{n^2+k^2}$ as $n\rightarrow\infty$ The more I read about this, the more confused I become. Illuminance, as I understand it,  measured in lux, is the perceived brightness of light on a surface. It's what you'd measure with an incident light sensor. Luminance, then, is how much reflected light you'd perceive looking at that surface. This is what the light meter in a camera measures. Am I correct so far?  In both cases, the word "perceived" is important, because the scales are weighted using a  to map wavelengths to . However, in the case of illuminance, there's only actual perception if your eye happens to be the surface in question.  I can basically grasp this, but then I come across charts saying things like "family living room: 50 lux". Wait, hold on! Does that really mean that the typical lights in a house are that bright, or is it just confused and wrong, or am I confused and wrong?  If you're not taking pictures of light sources directly, why would an incident light meter reading ever be useful in photography? Reflected light recorded on film or by a sensor is usually what makes a photograph. So, if I have an incident light meter, how does that reading relate meaningfully to my camera settings?  Since incident light meters are sold and used, this implies that there must be some useful conversion. But this is where my brain explodes. Google tells me that 1 lumen equals 1 candela, so therefore 1 lumen/m² (i.e. 1 lux) must equal 1 candela/m2 (i.e. 1 nit). But clearly something is missing from this. There's something called a "steradian". Cones are involved. I'd never heard of this before; how does it fit in? I can see how it might be determined when calculating the usefulness of LED home lighting, but for a photo, I'm at a loss.  Some part of my (exploded, from the last paragraph) brain is trying to relate this to the difference between flash metering with TTL and with an incident meter. But how can the incident meter work without knowing the reflectance properties of the objects in the scene? Is this what the "C" in the  is all about? Is it just an average-this-probably-will-work value, or is there more to it than that? And if it is just an average, what knowledge is necessary for compensating for off-averages scenes? (As with the , where the photographer simply judges if the scene should be rendered brighter or darker than the average given by the meter.)  So yeah, so much confusion. In short:   What's the difference? Can one meaningfully convert between the two? When and how are illuminance / incident light measurements useful for photography?     Update: I appreciate Stan's answer, which covers the third point of when and how pretty well. And I think I've basically got the first point figured out, as described above. But I'd appreciate some answers covering the issue of conversion as well, both in the mathematical abstract and as the practical for photography. And I wouldn't mind more why and how, either. How to deserialize a JSON String to Apex How can I deserialize this json object:      {     "response": {         "count": 1,         "benchmark": 0.22567009925842,         "requests": [             {                 "request": {                     "id": 537481,                     "image_thumbnail": "",                     "title": "Request for new bin(s) - residential",                     "description": "Propmain ref  3234-1114",                     "status": "submitted",                     "address": "36 Pine Tree Close",                     "location": "Peterborough, England",                     "zipcode": "PE1 1EJ",                     "user": "",                     "date_created": 1417173208,                     "count_comments": 0,                     "count_followers": 0,                     "count_supporters": 0,                     "lat": 52.599967,                     "lon": -0.233482,                     "user_follows": 0,                     "user_comments": 0,                     "user_request": 1,                     "rank": "0"                 }             }         ],         "status": {             "type": "success",             "message": "Success",             "code": 200,             "code_message": "Ok"         }     } }   What i've tried:   Map&lt;String,Object&gt; rawObj = (Map&lt;String,Object&gt;) JSON.deserializeUntyped(jsonString);     Map&lt;String,Object&gt; responseObj = (Map&lt;String,Object&gt;)rawObj.get('response');     List&lt;Object&gt; reqs = (List&lt;Object&gt;) responseObj.get('requests');     System.debug('Map Size = ' + reqs.size());     Map&lt;String, Object&gt; i = new Map&lt;String, Object&gt; ();     for (Object x : reqs) {          i = (Map&lt;String, Object&gt;)x; }     Map&lt;String,Object&gt; requests = (Map&lt;String,Object&gt;)i.get('request');     System.debug('Map Size = ' + i.size());     for (String field : i.keySet()){          Object id = i.get(field);         Object title = i.get('title');         System.debug('Id : ' + id);         System.debug('title : ' + title);          //System.debug('Title : ' + title);       } Does non-mass-energy generate a gravitational field? At a very basic level I know that gravity isn't generated by mass but rather the  and when I wave my hands a lot it seems like that implies that energy in $E^2 = (pc)^2 + (mc^2)^2$ is the source of gravity.  If the total energy of a particle contributes to the generation of a gravitational field, that seems to imply some strange things.  For example, you could accelerate a particle like an electron to such a high velocity that it has enough momentum / kinetic energy to collapse into a black hole.  An observer traveling with the electron would see an electron and stationary observers would see an arbitrarily massive black hole flying by.  These observations seem to be contradictory.  The same would be true of an extremely high energy photon.  Another consequence is that a stationary particle could appear to have so much mass that it could attract / deflect the trajectory of a rapidly moving observer.  It seems odd that at low velocities you wouldn't even notice a stationary electron as you pass it by but at high velocities the gravitational field you observe could be significant enough to alter your path.  Am I correct that non-mass-energy generates a gravitational field?  If it does, how are these strange observations like a particle looking like a black hole in one reference frame but not another reconciled? I have a laptop that has Windows 8 which I use for playing Minecraft. However when using the inventory, I have to hold my mouse to drag and pick the items and I don't like that.   I have tried switching mouses but that didn't work, is this possible to fix? (exercise from Tao's analysis book) Proof of a lemma relating to power set of X I'm trying to remove some characters from file(UTF-8). I'm using tr for this purpose:   tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat    File contains some foreign characters (like "Латвийская" or "àé"). tr doesn't seem to understand them: it treats them as non-alpha and removes too.  I've tried changing some of my locale settings:  LC_CTYPE=C LC_COLLATE=C tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat LC_CTYPE=ru_RU.UTF-8 LC_COLLATE=C tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat LC_CTYPE=ru_RU.UTF-8 LC_COLLATE=ru_RU.UTF-8 tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat   Unfortunately, none of these worked.  How can I make tr understand Unicode? Using getopts to process long and short command line options You should choose your Display Name when signing up I am using the full pager in views module. I am displaying 40 items per page. The full pager display links like this  1 2 3 4 5 next last   I want to change it to this  1-40 41-80 81-120 next   How will I do this? Any idea about this? How do I emphasize role-playing during combat? Right now, the group that I GM for is very, very tactical-combat focused. To some extent, we may as well be playing a dungeon crawl board game for all the role-play we manage.  Now, in and of itself that's not bad, and I enjoy progressing the story that we're telling, but I would like more "theatre of the mind" than we're achieving right now.  The one bit of advice that I always seem to come across is to have the villains and other NPCs in the fight invite role-play by taunting and otherwise engaging in banter with the PCs, but unfortunately I've tried and failed at that - largely because quick witted banter is not my forte, and so it comes across as strained and less than engaging.  What I would like are other play-proven tactics that I as a GM can employ to lead my players towards more role-play during combat, without taking them away from their (and my) roll-play comfort zone. Repairing corrupt MP4 I took a video from my Android phone (Sony Xperia P, if you need to know), and before I could stop recording it, the phone memory got full and the phone got hanged too. I rebooted the phone, the video was there, but I couldn't play it from any media player on Android. It's in mp4 format.    So I tried to open it on PC and Mac with different video player, like VLC, quick Time, Windows Media Player etc., but none of them can play it. I know that the video is corrupted, but I believe it's repairable, since it got more than a GB in size. Is there anything that I can try  to repair the video file? The video is really important. It is a win7 ultimate x64 machine. The machine was in a domain where it got those group policy settings. Now it has left the domain but it still receives the settings from the group policy. For example, the power options. I set a certain power option but soon it will be reset to another power option which is endorsed by the domain.  Is there a way to remove the settings? What is the inverse of CoefficientList? How can I do case insensitive string comparison in Python?  I would like to encapsulate comparison of a regular strings to a repository string using in a very simple and Pythonic way. I also would like to have ability to look up values in a dict hashed by strings using regular python strings. Let $A$ be a nonempty and bounded below, and define $B= \{b\in \mathbb{R}: b$ is a lower bound for $A\}$. Show sup$B$ = inf$A$.   So far I have: Let $A$ be nonempty and bounded below. This implies $\exists l \in A, \forall a \in A, l\leq a$ and $l =$ inf$A$. Let $B = \{b\in \mathbb{R}: b$ is a lower bound for $A\}$ and let $M=$ sup$B$. This implies $\exists b\in B, \forall b \in B, b\leq M$. By definition of $B$, we know inf$A \in B$. Since $b\leq$ sup$B$, with sup$B$ the largest element in $B$, we have inf$A \leq M =$ sup$B$, so inf$A \leq$ sup$B$.  I realize I have to prove that sup$B \leq$ inf$A$ to get that sup$B$=inf$A$. I am having trouble doing so.   Part B: Use (a) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. On what criteria should I evaluate domain registrars? Though I've been a web developer for a fair amount of time, I am going for the first time to buy a few domain names.  I have looked into the domains I'm going to buy and know that they're available, and I've been looking into which sellers to use.  After doing a lot of research, the main ones I'm considering are 1&amp;1, Namecheap, and Gandi.  The problem is, when continuing to research, I'm not really sure what makes one domain seller distinct from another.  I don't need much in the way of services--definitely not hosting, since I plan to use Heroku for that.  I mainly only need the domain itself and DNS management, as well as possibly SSL certificates and WHOIS protection.  Question:  What makes one domain seller different from another?  How can I go about evaluating which one is the best for me?  Note:  This question is not which domain seller is the best, but rather, what criteria can I use to evaluate them and rank one over another.  I'm trying to find out what makes one domain seller different from another, since they all seem to be pretty similar to me right now.