If $G$ is a finite group of order $n$ and $p$ is a prime divisor $|G|$, then according to one of Sylow's theorems $G$ has a subgroup of order $p^m$, where $p^m$ is the highest power of $p$ which is a factor of $|G|$.  Here is a rough outline of the proof I am trying to understand. Let $|G|=k\cdot p^m$, where $k \in \mathbb{Z^+}$.  The proof begins by defining the set $X=\{A \subseteq G:|A|=p^m\}$ and the action of $G$ on $X$ by left translation, $G \times X \rightarrow X:(g,A)\mapsto gA$. The size of $X$ is the binomial coefficient ${k \cdot p^m}\choose {p^m}$. Here is where I lose the line of argument:  (1) the size of $X$ is not divisible by $p$, and (as a result)...  (2) ... there must be an orbit $G(A)$ whose size is not a multiple of $p$.  Any clarification as to why this is the case?
It seems that there are many ways to prove the Sylow theorems. I'd like to see a collection of them. Please write down or share links to any you know.
I see more and more servers that have no domain defined in the /etc/resolv.conf file.  Only a search entry. In what cases is this setup used ? Are there any cases where it must not be used ?  nameserver xxx.xxx.xxx.xxx nameserver yyy.yyy.yyy.yyy nameserver zzz.zzz.zzz.zzz  search domain1 domain2 domain3 ... How can I convince hyperref and xindy to play together nicely? Consider the following code:   0.1 + 0.2 == 0.3  -&gt;  false   0.1 + 0.2         -&gt;  0.30000000000000004   Why do these inaccuracies happen? How could a statement be true without proof? Does the Mordenkainen's Magnificent Mansion spell's mansion exist on a separate plane of existence? In particular: the  spell (PHB, p. 281) states:  [...] The creature can be anywhere on the same plane of existence as you. The spell ends if you or the target are no longer on the same plane. [...]  Can telepathy connect two characters, one inside the extradimensional dwelling created by the  spell and the other in the same plane as the portal to the magnificent mansion? In PHP (>= 5.0), is passing by reference faster? In PHP, function parameters can be passed by reference by prepending an ampersand to the parameter in the function declaration, like so:  function foo(&amp;$bar) {     // ... }   Now, I am aware that this is not designed to improve performance, but to allow functions to change variables that are normally out of their scope.  Instead, PHP seems to use Copy On Write to avoid copying objects (and maybe also arrays) until they are changed. So, for functions that do not change their parameters, the effect should be the same as if you had passed them by reference.  However, I was wondering if the Copy On Write logic maybe is shortcircuited on pass-by-reference and whether that has any performance impact.  ETA: To be sure, I assume that it's not faster, and I am well aware that this is not what references are for. So I think my own guesses are quite good, I'm just looking for an answer from someone who really knows what's definitely happening under the hood. In five years of PHP development, I've always found it hard to get quality information on PHP internals short from reading the source. Will the silhouette formula change depending on the distance metric? I am using Silhouette width to compute the best value for k in k-means. As I am performing document clustering, I am calculating the values of a and b as follows in Python:  a = distance(data[index], centroids[clusters[index]], metric=metric, p=p) b = min([distance(data[index], c) for i,c in enumerate(centroids) if i != currentindex]) score = float(b - a) / max(a, b) if max(a, b) &gt; 0 else 0.0   The original formula from the  is the following:    I am using a cosine similarity measure to compute the distances and was wondering if this formula needs to be changed or it can be left as is. In the above snippet of code, the function distance computes the cosine similarity metric. How to prove wave nature of large object? What are useful websites for scouting GIS related jobs? Link to the recommended job is broken in the mobile apps The link to the recommended job in the app brings you to a page with the text: Not Found    Looking at the link it seems like after changing the URL structure in Stack Overflow Jobs they have not been updated in the apps.  For the job in the screenshot this is the used URL in the app:    The correct URL is: Do not allow giving minus one for new user questions I always see Stack Overflow users giving -1 to new users' questions. This is not how you welcome a new user to the community. A new user does not know the rules yet.    I suggest that if one wants to give -1 to a new user, you prevent that and ask the downvoter to write a comment instead. new user = 2 first questions    I see the credit rating on my question is now at -9?  My question, is she not well formed or formatted? If you don't agree with the content of a question do you down-rate it? If an Ubuntu user asks a question that might not be reasonable, will you also down-rate it?  No. You post an answer that will present your thinking and correct him? I think that is part of the problem. When you first go to a community like Facebook, and you don't know the faces, you don't feel like any of those people are your relations. Newbies exist here as well, especially in a professional community.    First, welcome the new user in more appropriate way. And most importantly, force them to see/read how to post a good question. The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function.  $F: \mathbb{R} \to \mathbb{R}$ and $F^{−1}:\mathbb{R} \to \mathbb{R}$ are defined by $F(x)=3x+2$ and $F^{−1}(y)=\dfrac{y−2}{3}$. for all y ∈ R  My attempt:     Inverse Function  For each particular but arbitrarily chosen $y \in \mathbb{R}$, according to the definition of $f^{-1}$, $f^{-1}(y) = \dfrac{y-2}{3}$ is a unique real number $x$ such that $f(x) = y$. \begin{align*} F(x) &amp; = y\\ 3x + 2 &amp; = y\\ x &amp; = \frac{y-2}{3} \end{align*} Therefore, $f^{-1}(y) = \frac{y-2}{3}$.  Compositions of Functions  The functions $g \circ f$ and $f \circ g$ are defined as follows: $$(g \circ f)(x) = g(f(x)) = g(3x + 2) = 3x + 2$$  for all $x \in \mathbb{Z}$. Javascript mechanism to autoscroll to the bottom of a growing page? How to list of all the tables defined for the database when using active record? How do I get a list of all the tables defined for the database when using active record? SELECT GETDATE()   Returns: 2008-09-22 15:24:13.790  I want that date part without the time part: 2008-09-22 00:00:00.000  How can I get that? Forcing decimal numbers to use comma in ArcGIS Field Calculator? Hint:  $a^2 -ab +b^2 = (a+b)^2 -3ab.$  I know we can say that there exists an $x,y$ such that $ax + by = 1$.  So in this case, $(a+b)x + ((a+b)^2 -3ab)y =1.$  I thought setting $x = (a+b)$ and $y = -1$ would help but that gives me $3ab =1.$  Any suggestions? Is it unreasonable to assign a MySQL database to each user on my site? I'm creating a user-based website. For each user, I'll need a few MySQL tables to store different types of information (that is, userInfo, quotesSubmitted, and ratesSubmitted).  Is it a better idea to:  a) Create one database for the site (that is, "mySite") and then hundreds or thousands of tables inside this (that is, "userInfo_bob", "quotessubmitted_bob", "userInfo_shelly", and"quotesSubmitted_shelly")  or  b) Create hundreds or thousands of databases (that is, "Bob", "Shelly", etc.) and only a couple tables per database (that is, Inside of "Bob": userInfo, quotesSubmitted, ratesSubmitted, etc.)  Should I use one database, and many tables in that database, or many databases and few tables per database?    Edit:  The problem is that I need to keep track of who has rated what.  That means if a user has rated 300 quotes, I need to be able to know exactly which quotes the user has rated.  Maybe I should do this?  One table for quotes. One table to list users. One table to document ALL ratings that have been made (that is, Three columns: User, Quote, rating).  That seems reasonable. Is there any problem with that? Leave Convert to Edit button enabled all the time in the moderator flag list In the moderator flags list, there is a &quot;Convert to Edit&quot; button on answers which are posted by the person who posted the question. This button is disabled at times, and I am not sure that the algorithm used is useful. My proposal is that the Convert to Edit button in the flag list should always be enabled, as it seems fairly frequent that it's disabled when it would be useful. Justification When the button is disabled, it is required that we pop open the answer view in the flag list, and use the regular Mod menu and choose convert to edit there. When we do that, however, we experience this issue:  (that should also be fixed, by the way!) Here are two items currently in the queue:    Note that one has the button enabled, while the other does not. The only differences I can see in those two answers is the age and the number of answers. Neither question has an accepted answer, but the one with the button disabled only has that single answer. I've only just now noticed that difference with these two posts, so I'm not 100% sure if that's the determining factor. 1 It is perhaps coincidental, but these buttons being (dis)abled are reverse of the action needing to be taken. The one with the disabled button needed to be converted to an edit; the one with the enabled button was an attempt to answer their own question. If I'm right about the determining factor here (that the answer is the only one), I'm not sure that's useful. A good bit of the time, an asker will post an answer due to people requesting additional information. In such cases, it seems like it will be likely there are no other answers. 1  Note 1: I was looking to create a userscript to enable these buttons, and I found something that I think is related as to the reason. It looks like the ones which are disabled are on answers which have comments on them. I noticed the title attribute of those buttons says, &quot;There are comments on the post. Please review&quot;. That does seem to make some sense in theory. However, it still seems like I am often converting them, anyway. With the above-linked, separate issue regarding how the 'mod' link on the expanded view works, this is still a bit of an impediment to our workflow. It occurred to me to check if the button was reenabled once you expanded the view, but it does not. Maybe that's an alternate solution. My plan for my userscript is to enable the button, but change the text color (and leave the title attribute), to point out that there are comments. Is there a 'biblatex in a nutshell' guide out there? I'd like if someone explained to me the essentials on how to use biblatex (what lines I have to write in my document, which files I have to have, how many times and what I have to compile), so then I can go to  to customise it further.