Let $G$ be a finite abelian group. Let $\phi:G\rightarrow G$ be a group homomrphism. Prove that there exists $n$ such that $G\cong ker(\phi^n)\bigoplus \phi^n(G)$.  Any ideas?
Let $G$ be a finite abelian group and let $\phi: G \rightarrow G$ be a group homomorphism. I am trying to show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$.  I know that since $G$ is abelian we have that $\ker(\phi^{k})$ and $\phi^{k}(G)$ are normal subgroups of $G$ for any $k$  I also suspect we have the following towers which stabilize at $m$ and $m'$  $$\phi(G) \unrhd \phi^{2}(G) \unrhd \cdots \unrhd \phi^{m}(G)=\phi^{m+1}(G)=\cdots$$  $$\ker(\phi) \unlhd \ker^{2}(\phi) \unlhd \cdots \unlhd \ker^{m'}(\phi)=\ker^{m'+1}(\phi)=\cdots$$  I know that if given a group of the form $HK$ where $H$ and $K$ are normal subgroups of $HK$ and $H\cap K=1$ then $H \times K \cong HK$. I want to apply this to this situation but I am unable to show how write $G$ as a product $HK$  Resolution One of the comments directed me to the following article:  The last part of which answers the question. Below is what is says:  Choose $n=\max(m,m')$, then we have for $x \in \ker^{n}(\phi) \cap \phi^{n}(x)$, this means that $x=\phi^{n}(y)$ for some $y \in G$. This gives:  $0=\phi^{n}(x)=\phi^{2n}(y)$ which means that $y \in \ker^{2n}\phi=\ker^{n}{\phi}$. and then we have that $0=x=\phi^{n}(y)$.  The answer below kindly points out that every element $x$ is contained in one of the cosets of $G/\ker^{n}(\phi)$, this means that $x=k+g$ for some $k \in \ker^{n}(\phi)$ and $g \in G$. We also have that $g=\phi^{n}(h)$ for some $h \in G$. Writing $x=k+\phi^{n}(h)$ essentially shows that $G=\ker(\phi^{n}) + \phi^{n}(G)$. Using the fact above, the claim follows.
Explain why $\mathbb{Z \times Z}$ and $\mathbb{R \times R}$ is not a field Watts P.M.P.O. and audio equipment power I am a beginner in electrical engineering, and to begin learning I decided to look into audio amplifiers. I have seen that the power of audio amplifiers is often described in the units of watts, watts RMS, and watts PMPO. So far, the only one of those units that I think I understand is watts. For instance, I just recently made . If I think: &quot;What is the power of my amplifier supposed to be?&quot;, then my thought process goes like this:  The transistors I used are the TIP120, which have a maximum rated power of 65W. Then, I decide that 60W is a safer maximum which would be reached if my amplifier runs 19V through one of my 6Ω speakers. Therefore, the power of my amplifier is 2x60W.  Then, I look at the datasheets of audio amplifier ICs, like the TPA3116 that says: &quot;50W into a 4Ω load at 21V&quot;, which makes me conclude that in that setting it'll output 14V maximum into the speaker, which I think makes sense. But then, I look at the datasheet of the TDA7294, and they talk about &quot;RMS Power&quot; and &quot;RMS Music Power&quot;, units which I couldn't find clear information about. Then, I look at home audio systems, like a Panasonic Stereo System I have which has &quot;9500 Watts P.M.P.O.&quot; written on the front. 9500W? That sounds stupidly high to me, but then I look again and it has 300W written on the back of it. If I'm correct, maybe a 1kW home stereo system could make sense, but a 2kW or greater one would not exist. So... Are my conclusions about  and the TPA3116 correct? Are my conclusions about audio equipment power correct? Are WattsPMPO, WattsRMS and RMSmusicpower, real useful units? My internet connection is very slow. I suspect someone in the network is using a BitTorrent client which takes huge amount of bandwidth.  Is there a way to detect who is using my bandwidth (network sniffing tools etc)?  I should have pointed out that I have no access to the router, what I am looking for is a way of detecting someone's MAC/IP if she uses Bittorrent. I’m making a little movie with using multiple computers. I can't use the network so I've divided the frames of the animation over the computers. Unfortunately now there are frames missing. What I'd like to do is using 1 machine to render all the missing frames. For example: 4-24-35-64-143 but ideally in such a way that I don't have to re-start blender multiple times. Is this possible somehow? Recently when skimming through my physics-text I encountered an interesting definition of Force $$F(x) = -\frac{\mathrm dU(x)}{\mathrm dx}$$  We were taught that some functions are continuous but not differentiable. So for the force to exist $U(x)$ has to be differentiable. So how can we prove that $U(x)$ is differentiable everywhere? Is there an optimal setting for the number of CPU cores exposed to a VM? VirtualBox, and possibly other VM software, allows the user to specify the number of CPUs exposed to the VM. In my particular case, I have a dual core CPU, with Windows 7 64-bit running natively, and Ubuntu 9.04 64-bit as the VM.  The help file suggests not setting this above the number of physical cores available. This makes sense. So, is it best allocating 1 or 2 CPU cores? Does this depend entirely on the native CPU load? What is a good word for “best example”? What is a good word for “best example”? I would like to describe a film as the greatest example of its features. The full prophecy by Trelawney, the part my question is about in bold:     "The one with the power to vanquish the Dark Lord approaches ... born to those who  have thrice defied him, born as the seventh month dies ... and the Dark Lord will mark him as his equal, but he will have power the Dark Lord knows not ... and either must die at the hand of the other for neither can live while the other survives ... the one with the power to vanquish the Dark Lord will be born as the seventh month dies ..."    – Sybill Trelawney (to Dumbledore)    I am interpreting      for neither can live while the other survives   as: Harry can't live while Voldemort survives and vice versa. But why? It clearly worked until the end of book 7. One could argue with Dumbledore's thery of the self-fulfilling prophecy. But clearly Voldemort was smart enough to see that there was no mechanism that would kill him just because Harry stayed alive.    and either must die at the hand of the other    Again: Why? If somebody else destroyed all Horcruxes and then killed Harry, Voldemort would be just as vulnerable and could be killed by anyone. On the other hand, Harry could have been killed by anyone - his mum's protection transferred to Voldemort by Harry's blood was only helping against Voldemort.  I guess that I am interpreting this part of the prophecy in the wrong way or maybe there is an alternate, less obvious and more tricky one. So my question is: What exactly does the highlighted part of Trelawney's prophecy state? What is the simplest way (least error-prone, least lines of code, however you want to interpret it) to open a file in C and read its contents into a string (char*, char[], whatever)? Declare a const array First post review queue: answers are questions, questions are answers I believe there is a typo in the  review queue. When reviewing a question, the information message asks the user to review the answer. Take a look at the image below.    Similarly, when an answer is posted in the queue, the description box calls it a question instead. Has anyone else noticed this? Require v8 only contrib module with v9 patch Given I have Drupal 9 composer installation with composer-patches plugin and given a contrib module with a stable v8 release, but no v9 release (not even dev branch) and given that contrib module has a working v9 patch in the issue queue  is there any method to install that module + patch in composer? Even if I manually add both, the package and patch, to my composer.json, I still can't require or update this module with composer due conflicting versions. I really want to avoid duplicating /contrib code into my project's /custom codebase. My current workaround is:  forking that module to my own, private git repo applying patch there creating a new composer.json in my private git, and changing the package vendor to my custom_private_vendor adding my private git as VCS repo in the D9 project's composer.json and then composer require custom_private_vendor/contrib_module  This fulfills my goal of not duplicating the contrib module in my project's custom codebase, but every time I do this I feel the urge to wash my dirty hands. Is there something more elegant like composer require drupal/contrib_module --apply-patch-first or can I somehow target drupal.org's git with a specific patch included? A term for words that change pronunciation with part of speech I'm talking about words like:   construct: CON-struct(n.), cun-STRUCT(v.) present: PRE-sent(n.), pre-SENT(v.) record: RE-cord(n.), ri-CORD(v.)   They are pronounced differently based on whether they are a noun or a verb.  Here are the terms I DON'T think would apply:   Homograph: same spelling, different meanings: for example, bark (the sound of a dog) and bark (the skin of a tree). Homophone: same pronunciation, different meaning: for example, to and two Homonym: same pronunciation and spelling, different meanings: for example: bank (river bank or savings bank) Heteronym: same spelling,  different pronunciations and meanings: for example bow (the front of a ship) and bow (a ranged weapon)   Bow is particularly interesting. It is a heteronym based on the two nouns I described. But in the sense of the body movement, the noun and the verb are pronounced alike (rhyming with wow), thus, not belonging to the category I've explained.  In fact none of the terms explained  satisfy my requirement.  I'm looking for the term for words that have same (or related at best, NOT different) meanings, same spelling and different pronunciation.  Specifically, different pronunciation depending on the part of speech. As against the words which have the same pronunciation in different parts of speech like bill, turn, case (or bigger ones like manoeuvre)  TL;DR:  Fill in the blank:     The noun and verb forms of construct are pronounced differently   because the word construct is a _____   or      The noun and verb forms of construct are pronounced differently   because they are _____s. What is the "Event Horizon" of a black hole Why don't we have an isomorphism between $R[x]$ and $ R[[x]]$? OLS is BLUE. But what if I don't care about unbiasedness and linearity? How do I format a number in Java? How do I format a number in Java? What are the &quot;Best Practices&quot;? Will I need to round a number before I format it?  32.302342342342343 =&gt; 32.30 .7323 =&gt; 0.73  etc. Orthogonal basis for infinite-dimensional vector spaces Does $1.0000000000\cdots 1$ with an infinite number of $0$ in it exist? Display error message if same record exist with combination of some fields? I have taken three fields from case object called AccountName, IntegrationTo__c and SfType__c (checkbox).  I have created one case record with a specific combination of three fields.  For the second time I create a record with that combination, then  display error message only if SfType__c is true in first record otherwise allow  for record creation.  trigger AvoidDuplicate on Case(before insert)  {          set&lt;string&gt; newNameSet = new set&lt;string&gt;();     newNameSet = new set&lt;string&gt;();      set&lt;string&gt; newEmailSet = new set&lt;string&gt;();      set&lt;string&gt; dbNameSet = new set&lt;string&gt;();      set&lt;string&gt; dbEmailSet = new set&lt;string&gt;();      for(Case newLead : trigger.new){          newNameSet.add(newLead.Account.Name);          newEmailSet.add(newLead.IntegrationTo__c);      }      for(Case dbLead : [select id, Case.Account.Name, IntegrationTo__c from Case where IntegrationTo__c IN: newEmailSet OR Case.Account.Name IN: newNameSet]){          dbNameSet.add(dbLead.Account.Name);          dbEmailSet.add(dbLead.IntegrationTo__c);      }      for(Case newLead : trigger.new){          if(dbNameSet.contains(newLead.Account.Name) &amp;&amp; dbEmailSet.Contains(newLead.IntegrationTo__c))              newLead.addError('You are inserting Duplicate Case');      }  }   i wrote this this not checking the account field name and integration__c values only with combination of account and integration__c shows error