Let $$f: G \to C^{×}$$ is homomorphism, whereas $C^{×}$ is the multiplicative group of non-zero complex numbers, |G|=n and G is abelian. Prove that \begin{align} \sum_{g \in G}f(g)&amp;=0,\quad \text{or}\\[1em] \sum_{g \in G}f(g)&amp;=n \end{align}  I know how to break abelian group into modulo $n$, but after that  I could not imagine a homomorphism.
Let $G$ be an abelian group of order n.   Let $\mathbb{C}^*$ be the multiplicative group of non-zero complex numbers, and suppose $f:G \rightarrow \mathbb{C}^*$ is a homomorphism. Prove that $\sum_{g \in G} f(g) = 0$ or $\sum_{g \in G} f(g) = n$.  I can prove this for cyclic groups but am unable to generalise to all abelian groups.
I have noticed there is a Stack Overflow in language differs from English which is global and mostly used by programmers.  I think there is no need for a site in other languages. Otherwise we will end up with 127 StackOverflow sites. I am afraid that we will lose the amount of people sharing data. Why didn't Harry Potter tell Aunt Petunia he was rich to improve how he was treated by Dursleys? This is a follow-up question to   One reason Aunt Petunia mistreated Harry was that she did not know he was rich. If she knew, she would naturally have treated this golden goose nicely. Her retirement would be secured if Harry financially supported her in old age.  Why didn't Harry tell Aunt Petunia he was rich? Surely every boy wants to be treated nicely by his family. Mixed DML operation error I am facing Mixed DML error in the following case:  static testmethod void method1() {     User u1 = DataFactory.createUser('testUser','test@user.com','permissionSetAPIname');        Obj1 rec1 = DataFactory.createrec1('Record1');     insert rec1;      Obj2 rec2 = DataFactory.createrec2('Rec2',rec1,system.today()-10, system.today()+10 );     insert rec2;       Obj3 rec3 = DataFactory.createrec3(rec2,'Tester',u1,'US');     insert Role;      test.starttest();         //testing code      test.stoptest();  }   However the following is working fine  static testmethod void method1() {      User u1 = new User(Alias = 'testRole',                   Email             = 'role@test.com',                   EmailEncodingKey  = 'UTF-8',                   LastName          = 'user',                   FirstName         = 'test',                   LanguageLocaleKey = 'en_US',                   LocaleSidKey      = 'en_US',                   ProfileId         = [select Id from Profile where Name = 'Basic User' limit 1].Id,                   TimeZoneSidKey    = 'America/Chicago',                   Username          = 'stakeholder@test' + Math.round(Math.random() * 10000) + '.com',                   UserRoleId        = null,                   Country           = 'Great Britain');     insert u1;         Obj1 rec1 = DataFactory.createrec1('Record1');     insert rec1;      Obj2 rec2 = DataFactory.createrec2('Rec2',rec1,system.today()-10, system.today()+10 );     insert rec2;       Obj3 rec3 = DataFactory.createrec3(rec2,'Tester',u1,'US');     insert Role;      test.starttest();         //testing code      test.stoptest();  }   Please explain the difference. As my understanding is that transaction remains the same in both the cases. Context to distinguish "we" inclusive versus exclusive What does character set and collation mean exactly? I can read the MySQL documentation and it's pretty clear. But, how does one decide which character set to use? On what data does collation have an effect?  I'm asking for an explanation of the two and how to choose them. Short story: rocket fuel which oxidizes iron destroys life on earth I'm looking for the title of a short science fiction story, probably published in a Hugo winner collection 30 years ago. In the story, scientists have developed an extremely efficient rocket fuel that has one dangerous side-effect: it oxidizes iron. A man takes off in a rocket powered by this fuel but returns to find the earth completely devoid of life. His subsequent investigation reveals that a single drop of the fuel began a domino effect, as the substance oxidized all iron it came into contact with. In the end, the man is dying.  He drags himself to the ocean, believing that if he can get his living cells into the water they will restart the primordial soup. If $\gcd(a,b)=1$ and $a$ and $b$ divide $c$, then so does $ab$ Edit: The previous title didn't really ask the same thing as the question (sorry about that), so I've changed it. To clarify, I understand that the action isn't always a minimum. My questions are in the points 1. and 2. below.  I understand that &quot;principle of least action&quot; is somewhat of a misnomer, since we find that to determine the path taken by a system, we need only impose the condition that the action be stationary, i.e., that $\delta S$ should vanish to first order for small variations of the path, and this leads to the Euler-Lagrange equations. In The Classical Theory of Fields, Landau &amp; Lifshitz discuss the relativistic action for a free particle:  So for a free particle the action must have the form $$S = -\alpha \int_a^b ds$$ (...) It is easy to see that $\alpha$ must be a positive quantity for all particles. In fact, as we saw [earlier], $\int_a^b ds$ has its maximum value along a straight world line; by integrating along a curved world line we can make the integral arbitrarily small. Thus the integral $\int_a^b ds$ with the positive sign cannot have a minimum; with the opposite sign it clearly has a minimum, along the straight world line.  There is also a footnote, addressed a couple of paragraphs earlier, but which is relevant:  Strictly speaking, the principle of least action asserts that the integral $S$ must be a minimum only for infinitesimal lengths of the path of integration. For paths of arbitrary length we can say only that $S$ must be an extremum, not necessarily a minimum.  I have two questions regarding this:  How is the condition &quot;the action must be a minimum for infinitesimal displacements&quot; formulated? I've never heard of that outside of Landau &amp; Lifshitz's books, and in Mechanics they mention it as well but doesn't go into detail. Is this discussed a bit more somewhere?  If for the whole path the action only needs to be stationary, how can we make the argument for the negative sign? If the action had to be a minimum then it would make sense, but surely the fact that $\delta S$ = 0 isn't affected by an overall sign? Show that any set of 7 distinct integers includes two integers $x$ and $y$, such that either $x-y$ or $x+y$ is divisible by 10 Show that any set of 7 distinct integers includes two integers $x$ and $y$, such that either $x-y$ or $x+y$ is divisible by 10. I'm trying to apply the pigeonhole principle, but haven't been able to come up with a convincing argument.   Though similar to , I think the difference between the two is enough to warrant a second post. Set horizontal width on custom listings environment I want to create a custom listings environment and want to set its width as something like 90% of textwidth, centred.  Per an  for algorithms, the first thing I tried was to embed the listing in a minipage environment. However, . I found a way to still make standard environment work, but it requires to use the escape command every time before ending the environment.  The second thing I tried was this latter command, but as far as I can see, it doesn't offer a feature to set the width of the listing or centre it.  \documentclass{article} \usepackage{listings} \usepackage{lipsum}   \lstnewenvironment{queryl} %% APPROACH 1  {\lstset{frame=shadowbox,escapechar=`}}  {}  \newenvironment{query} %% APPROACH 2  {\begin{minipage}{4cm}\centering\begin{queryl}}  {\end{queryl}\end{minipage}}   \begin{document} %% APPROACH 1 Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. \begin{queryl} begin { do nothing } end ; \end{queryl} Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis.  %% APPROACH 2 Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis.  \begin{query} begin { do nothing } end ;`%&lt;- needs escape-to-LaTeX character to work \end{query}  Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. \end{document}     One can note that for the first environment, the frame extends beyond the width of the text. Also, puzzlingly the minipage environment doesn't seem to want to centre and swallows space (though I guess I could perhaps at least get that to work with tinkering).  In any case, does anyone have any ideas on how to set a width for a custom listings environment and centre it without requiring use of the escape character each time to end the environment? Number Theory: Complete set of residues modulo $n$ I have this problem assigned for homework and I'm struggling with the proof of it:  If $a_1,a_2,\dotsc,a_n$ is a complete set of residues modulo $n$ and $\gcd(a,n)=1$, prove that $aa_1,aa_2,\dotsc,aa_n$ is also a complete set of residues modulo $n$.  (Hint: It suffices to show that the numbers in question are incongruent modulo $n$.)  I'm in elementary number theory so I'm not allowed to use an high-level theorems to prove this, I pretty much have to use the basics of modulo.  Thanks! Tools for tracking NPC/PC relationships and attitudes I have always had difficulty keeping track of NPCs, their attitudes and their relationships with each other and the PCs. I find relationship webs/diagrams are helpful, but find them difficult to keep up to date and accurate from session to session.  Are there any tools that would help with this task? I am aware that some general mapping software exists, but am more interested in tools specifically designed for RPGs. Also, I would prefer a graphical interface if software is suggested, and to not have to learn a text based language to define the diagram. However, if it turns out that the only tools available are more generalised, and/or complex, I would still like to hear about them as answers. Generalized Pell equation $x^2-2y^2=49$ How to remove session entries from LightDM? An example of a norm which can't be generated by an inner product I realize that every inner product defined on a vector space can give rise to a norm on that space.  The converse, apparently is not true.  I'd like an example of a norm which no inner product can generate. Global warming and air temperature The layman's question. Is there a maximum temperature that can be reached by air on Earth due to global warming? For example, one that would prevent living organisms from functioning? I read somewhere that with the increase of the sun's brightness by 10%, the average temperature on our planet will be 47 C. This, however, only for about 600 million years. Would global warming enable reaching such a high average temperature? Capitalizing "the" when mid-sentence in a company name? Consider a company called:     The Association of Examples Ltd.   When you refer to this company, should you have to capitalize "the" when it's mid-sentence and would have been used even if it wasn't in the name?   It looks odd to me:     In other news, The Association of Examples announced that...      Sarah is the founder of The Association of Examples and has...   But I imagine it's the correct way.  For the grammar-conscious, why or why not?    To give you context, I am asking also when branding ones own company like that. The Economist did it (as did ) and , but I can imagine it can cause confusion. "He acted strange(ly?)" Can't save my render as an image I seem to be having issues when I try and save the output file as a .png, or .jpeg for that matter. I've set up my user preferences, File, Render Output to my desktop as well as render output in the editor window. What am I missing here? I'm obviously very new to this program and your help would be appreciated. Thanks for your time How to prevent upper case in page header? I just came across this small problem which I can't seem to fix. I have a section which looks like this:  \section{Properties of $C_q$}   The problem is that the header of the page uses upper case and so it displays this section title as if I had written  \section{Properties of $C_Q$}   How can I make sure the q is always lower case?