Assume that $G$ is an abelian group, I read somewhere that it can be derived from Lagrange's theorem that it has a number of subgroups that is equal to the number of $G$'s divisors.  Why does it hold? I'm confused...
Let $G$ be a finite abelian group, then $G$ has a subgroup of order $n$ if and only if $n\mid G$.  Proof: by Lagrange if $H\leq G$ then $|H|$ divides $|G|$ so this proves one of the implications.  We prove the other implication by strong induction on the order of $G$, for primes this is just Cauchy, so we can take these like base cases. Now let $n$ be a proper divisor of $|G|$, take $p$ a prime in the factorization of $|G|$ (instead of $|G|$ is should be $n$, I had made a mistake when typing the first time), then by cauchy there is an element $x$ of $G$ of order $p$, since $G$ is abelian $\langle x\rangle$ is normal in $G$ so we can look at the quotient group $G/\langle x\rangle$, this group has order $\frac{|G|}{p}$, now take a prime $p_1$ in the factorization of $\frac{|G|}{n}$. By induction the quotient group has a subgroup $E$ of order $G/pp_1$ now look at the preimage of $E$ under the natural homomorphism from $G$ to $G/\langle x\rangle$, it is easy to see this is a subgroup of $G$, we call it $I$, consider the restriction of the natural projection to $I$, this is a surjective homomorphism from $I$ to $E$ with kernel $\langle x\rangle$ so by the first isomorphism theorem $I/x\cong E$ so $\frac{|I|}{|\langle  x\rangle|}=|E|\implies  |I|=\frac{|G|}{pp_1}{p}=\frac{|G|}{p_1}$ Notice $\frac{|G|}{p_1}$ is a multiple of $n$ since $p_1$ was in the fatorization of $\frac{|G|}{n}$. So by induction $I$ has a subgroup of order $n$ which is also a subgroup of $G$.
How to prove $n^5 - n$ is divisible by $30$ without reduction How can I prove that prove $n^5 - n$ is divisible by $30$? I took $n^5 - n$ and got $n(n-1)(n+1)(n^2+1)$ Now, $n(n-1)(n+1)$ is divisible by $6$. Next I need to show that $n(n-1)(n+1)(n^2+1)$ is divisible by $5$. My guess is using Fermat's little theorem but I don't know how. horizontal alignment of dot over hat over bold greek letter! For some strange reason, this:   \hat{\dot{\bm{\phi}}}   prints the dot and the hat slightly off the left of the phi letter. Any ideas if this can be fixed? thanks. Using bm package for bold symbols. pgfplots -> contour gnuplot with lualatex instead of pdflatex My sister, cousins, and other relatives speculate that my mom's grandmother was Jewish (for various reasons). I've heard people say at times things like, for example, "I'm 50% German" or "I'm 25% Swedish." This has to do with their ancestry. But, how exactly is this calculated? Say for example that my mother's paternal grandmother was indeed 100% Jewish, what would that make me and my siblings, and how is it calculated? Is posting a very long quote OK for an answer? I just found  to a question on SO. While it provides a source for the (very long) quote, I do not like the fact that the answerer just posted the quote, without adding anything. Maybe add highlights/emphasis or remove uninteresting passages. Or skip the quote altogether and write a short summary. I feel, in the current form, this should have been a comment that just links to the original article.  Am I too pedantic? Should I flag this as not an answer? Should I downvote it? What do you think about such answers? What is the amplitude of a light wave? Referring to this question   I'm curious about what is a amplitude of a light wave. Especially for light from a thermic source. Is it possible to exclude questions I've already answered from the search criteria? Recently I asked if it were possible to find questions with just one answer and as a result we got an .  That's great, but ...     Is there a way to restrict the search to questions where that one answer isn't mine? That would help. :-)   It's certainly possible to search for , but it's not possible to exclude your own posts as far as I can tell:   -user:me searches for posts that . user:1-67 doesn't search for posts from user #1 to user #67, but searches for .  (And I'm not sure that will help anyway since I don't think it would be possible to simultaneously exclude 69-max(userID).) -Ericson searches for posts that , but since my last name is in the the metadata and not the searchable portion of posts: no joy.   And of course none of this will actually work to eliminate the questions that that I've answered already and want to ignore.  In general, If I'm involved in a Q&amp;A (or ) the notification procedures do a good job of keeping me up-to-date with developments.  But when I search, it's often to find questions I've not noticed before.  So being able to eliminate questions I have already answered would be a generally useful feature for me. How to avoiding losing your current browsers websites when being redirected by wireless login pages Jump/Scroll to selected object in the outliner? In a  of Amstat News, the authors (Mark van der Laan and Sherri Rose) stated that "We know that for large enough sample sizes, every study—including ones in which the null hypothesis of no effect is true — will declare a statistically significant effect.".  Well, I for one didn't know that. Is this true? Does it mean that hypothesis testing is worthless for large data sets? Statistical significance (p-value) for comparing two classifiers with respect to (mean) ROC AUC, sensitivity and specificity I have a test set of 100 cases and two classifiers.  I generated predictions and computed ROC AUC, sensitivity and specificity for both classifiers.  Question 1: How can I compute p-value to check if one is significantly better than the other with respect to all scores (ROC AUC, sensitivity, specificity)?    Now, for the same test set of 100 cases, I have different and independent feature assignments for each case. This is because my features are fixed but subjective and provided by multiple (5) subjects.  So, I evaluated my two classifiers again for 5 "versions" of my test set and obtained 5 ROC AUCs, 5 sensitivities and 5 specificities for both classifiers. Then, I computed the mean of each performance measure for 5 subjects (mean ROC AUC, mean sensitivity and mean specificity) for both classifiers.  Question 2: How can I compute p-value to check if one is significantly better than the other with respect to mean scores (mean ROC AUC, mean sensitivity, mean specificity)?    Answers with some example python (preferably) or MatLab code are more than welcome. Add self-answer delay for all users TL;DR:  I believe there are still some issues with self-answers, particularly near-immediate self-answers, which are not addressed by the current system.  This could be remedied by imposing an automated restriction on how soon self-answers will be displayed in the thread.  Allow users to submit self-answers at any time, but do not display them until certain reasonable conditions (a relatively short timeout, or the posting of a competing answer) are met.    History  This thread covers a good deal of the concerns surrounding self-answers, and some mechanisms in place to prevent "bad" self-answers from new users and other abuse of the self-answer system.    As I understand it, the following restrictions apply:   Users with &lt;100 rep cannot self-answer within 8 hours of posting a question. Self-answers cannot be accepted within 48 hours of posting. Self-answers do not provide rep from being accepted. Accepted self-answers are not pinned to the top of the thread.   Problem  This does a good job of preventing users from posting answers which should probably be comments (or not posted at all), allows superior answers to rise above accepted self-answers, and prevents some other abuses of the system.  However, there is a less tangible element of self-answers which this does not address:    Self answers are a disincentive to other users who may otherwise be willing to post more accurate, detailed material stemming from broader or deeper knowledge and/or experience than the asker has.  When someone posts an answer to their own question, it presumably means that the problem which spawned the question is solved to their satisfaction.  This is fine for the purposes of sharing information with the community, but it deters others from providing solutions to the question which may be more complete or accurate since there is not so much likelihood that the self-answerer will accept a competing answer.  Worse, in the case of near-immediate self-answers, this can remove the incentive for other users to gain reputation via up-votes because the asker has an unfair head-start.  Unless everyone is required to view new answers to questions where they have already voted on an answer, most people probably vote on a good answer and never look at the question again.  This can put (potentially) better answers at a disadvantage even when the head-start is within an hour.  Suggested Solution  Add a delay to the posting of self-answers.  The delay should be the least of these:   Until a competing answer has been posted. A reasonable amount of time to allow competing answerers a chance to post, without preventing self-answers outright.  (Suggestion: 2 hours)   To better facilitate this, and avoid completely turning people off from appropriate self-answering, I suggest the delay be handled through automation more than user limitation.  That is, the user should be allowed to submit the self-answer at any time.  But the answer will not actually be displayed to other users until one of the above conditions have been met.  This could also be applied to the existing 8-hour delay for low-rep users.  Example:  T:0 - A user finishes resolving a tough problem, just before he would have gone to SE for assistance, and decides to share his new-found knowledge with the community.  He posts a question describing his problem.  T:5 minutes - The user then posts a self-answer.  There has not yet been another answer posted to the thread.  The user is given a notice describing the restrictions in place regarding self-answers, and the answer is submitted into the database but not displayed to other users.  (Similar to how suggested edits are only displayed to the user suggesting them, until they are approved.)  T:1 hour, 15 minutes - Another user sees the unanswered question, and posts their answer.  Their answer is displayed immediately, and the self-answer is now also displayed.  T:2 hours - Had the second answer not been posted, the self-answer would automatically be displayed after this point. Add review queues to Android app Are these adjectives interchangeable?  I always hear: 'An Arab man.' but never 'An Arabic man.' and I always hear 'Arabic coffee' but never hear 'Arab coffee.' Updating the SubscriberKey on the All Subscriber table through the API I'm trying to update an existing Subscriber in the All Subscriber Table through the API; i've set up the AppId in the App Center and have connected and created Subscribers, but i'm still having issues updating.  What call do I need to be making to do this? These are the ones i've been trying. If the order of all nontrivial elements in a group is 2, then the group is Abelian. I know of a proof that is just from calculations (see below). I'm wondering if there is any theory or motivation behind this fact. Perhaps to do with commutators?    Proof: $a \cdot b = (b \cdot b)  \cdot (a \cdot b) \cdot (a \cdot a) = b \cdot (b \cdot a) \cdot (b\cdot a) \cdot a = b \cdot a$. Serializing anonymous delegates in C# I am trying to determine what issues could be caused by using the following serialization surrogate to enable serialization of anonymous functions/delegate/lambdas.   // see http://msdn.microsoft.com/msdnmag/issues/02/09/net/#S3 class NonSerializableSurrogate : ISerializationSurrogate {     public void GetObjectData(object obj, SerializationInfo info, StreamingContext context)     {         foreach (FieldInfo f in obj.GetType().GetFields(BindingFlags.Instance | BindingFlags.Public | BindingFlags.NonPublic))             info.AddValue(f.Name, f.GetValue(obj));     }      public object SetObjectData(object obj, SerializationInfo info, StreamingContext context,                                 ISurrogateSelector selector)     {         foreach (FieldInfo f in obj.GetType().GetFields(BindingFlags.Instance | BindingFlags.Public | BindingFlags.NonPublic))             f.SetValue(obj, info.GetValue(f.Name, f.FieldType));         return obj;     } }     Listing 1 adapted from   The main issue I can think of that might be a problem is that the anonymous class is an internal compiler detail and it's structure is not guaranteed to remain constant between revisions to the .NET Framework. I'm fairly certain this is the case based on my research into the similar problem with iterators.  Background  I am investigating the serialization of anonymous functions. I was expecting this not to work, but found it did for some cases. As long as the lambda did *not&amp; force the compiler to generate an anonymous class everything works fine.   A SerializationException is thrown if the compiler requires a generated class to implement the anonymous function. This is because the compiler generated class is not marked as serializable.  Example  namespace Example {     [Serializable]     class Other     {         public int Value;     }      [Serializable]     class Program     {         static void Main(string[] args)         {             MemoryStream m = new MemoryStream();             BinaryFormatter f = new BinaryFormatter();              // Example 1             Func&lt;int&gt; succeeds = () =&gt; 5;             f.Serialize(m, succeeds);              // Example 2             Other o = new Other();             Func&lt;int&gt; fails = () =&gt; o.Value;             f.Serialize(m, fails); // throws SerializationException - Type 'Example.Program+&lt;&gt;c__DisplayClass3' in Assembly 'Example, Version=1.0.0.0, Culture=neutral, PublicKeyToken=null' is not marked as serializable.         }     }   Listing 2  This is similar to the issue of trying to serialize iterators and I had found the following code in a previous search (see ) Using the code from Listing 1 and an ISurrogateSelector I was able to successfully serialize and deserialize the second failing example.  Objective  I have a system that is exposed via a web service. The system has a complex but small state (many objects, not a lot of properties per object). The state is saved in the ASP.NET Cache, but is also serialized to a BLOB in SQL in case of cache expiration. Some objects need to execute arbitrary "events" upon reaching some condition. Hence they have properties accepting Action/Func objects. Contrived example:      class Command     {         public Command(Action action, Func&lt;bool&gt; condition);     }   Somewhere else      void DoSomethingWithThing(Thing thing)     {         state = Store.GetCurrentState();          Command cmd = new Command(() =&gt; thing.Foo(), () =&gt; thing.IsReady())         state.Add(cmd);          Store.Save(state);     } The category Set has the property that any morphism f can be decomposed into an epimorphism $e$ and a monomorphism $m$ with $f$ = $m\circ e$. (The intermediary set object is the image of $f$.)  Is there a name for this construction?  Is there a name for this property of categories?  In Set, the intermediary object is unique up to isomorphism. Is this the case for all categories with the property? What’s the best way to reload / refresh an iframe? I was wondering whether the time shown is the time I would be arriving in France so its time zone etc.  I know France is an hour ahead so not sure if I'm right in my assumption... It would be useful to know as I plan on traveling to the States and other European countries soon as well.