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True or false. (Prove or give a counterexample.) Let $G$ be a group. Then $|g| = |\phi(g)|$, for all homomorphisms $\phi: G \to G$ and all $g \in G$.

Solution. False. $\phi: \mathbb Z_{10} \to \mathbb Z_{12}$ defined by $\phi(x)=0$ for all $x \in \mathbb Z_{10}$ is a counterexample. This function is a homomorphism because $\phi(x+y) = 0 = 0+0 = \phi(x) + \phi(y)$ for all $x, y \in \mathbb Z$ (this function is discussed in problem 3 in assignment 7 as the function sending $[1]$ to $[0]$). The order of $g=1$ is infinity. The order of $\phi(1)=0$ is one.

The problem is that he said $G \to G$ is equivalent to $\mathbb Z_{10} \to \mathbb Z_{12}$, which is think is not right. Explain please?

EDIT: Please look at this test and give me your honest opinion, http://zimmer.csufresno.edu/~ovega/teaching/151/Exam2Solutions.pdf

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    Did you ask your teacher? What happens if you consider the morphism $\phi:\mathbb Z_{10}\to\mathbb Z_{10}$ defined by the same formula?2011-10-29
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    I added the relevant question. The original source is [here](http://zimmer.csufresno.edu/~ovega/teaching/151/Exam2Solutions.pdf). (Note that the solution seems to have a lot of errors.)2011-10-29
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    Could you please point on those errors?2011-10-29
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    (1.) Your observation is correct. This is not a counterexample unless you change the morphism to $\mathbb Z_{10} \to \mathbb Z_{10}$, and define it appropriately. (2.) The sentence $\phi(x+y) = \phi(x) + \phi(y)$ for all $x,y \in \mathbb Z$ is meaningless. It should be "...for all $x,y \in \mathbb Z_{10}$". (3.) This is probably not an error, but I do not understand what "function sending $[1]$ to $[0]$" could possibly mean. (4.) It is true that the order of $\phi(1)=0$ is $1$. But the order of $1$ is $10$, not infinity. No element has infinite order in this group.2011-10-29
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    The other way out would be to ask for a morphism $\phi:G\to H$ with no condition that $G=H$ (as was done in a previous exam in same course), with same counterexample. // (Bis) Did you ask your teacher?2011-10-29
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    Probably two typos in the original: they mean $G\to H$, not $G\to G$, and $\mathbb{Z}$, not $\mathbb{Z}_{10}$.2011-10-29
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    Not necessarily, @anon. I second Gerry's guess that the question makes sense, but the answer doesn't. Perhaps the counterexample intended was $\phi: \mathbb Z \to \mathbb Z$.2011-10-29
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    No talk with teacher yet, Test was taken ~10 hours ago. I dont know whats the best way to approach him about it, because he usually is right. (with a few minor mistakes here and there) What the questions are in the link is exactly how they were on my test earlier, and I feel it is unfair to say G->G is the same as Z_10->Z_122011-10-29
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    @Sri: Ah, that seems even more likely.2011-10-29
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    @Eidbanger To be frank, this question is not that difficult. Also, even if the official solution is wrong, it still has a correct approach that you must try to understand. Since these solutions are anyway provided \*after\* the test, this has certainly not impacted your performance in the test. You should point out the mistake to the teacher and ask what correct solution was intended, but I do not see a reason to feel cheated by the mistake.2011-10-29
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    So, replace one $\mathbb Z_{12}$ and one $\mathbb Z$ by $\mathbb Z_{10}$ and everybody is happy. // Some remarks about wording and etiquette: (1) Typos are not the same as wrong answers. (2) *The best way to approach him about it* includes communicating the address of this webpage to him.2011-10-29
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    @Sri We had 50 minutes to answer these questions. Half of the class is failing. only 1 or 2 people get A's in the class (out of 25 that makes throough half a semester). I understand the material is difficult. But the results show the results. And little mistakes could be very misguiding. I appreciate all the help, indefinetely, but I feel that I am destined to fail. (without your help ofcourse)2011-10-29
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    I can sympathize with you, but how can we can help here? Even if I/we seconded your claim that these little mistakes are misguiding (hypothetically speaking), will that help your cause?2011-10-29
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    @Sri Who knows how many misguided directions we were lead to by our leader (prof)? Maybe that directly shows in our testing performance. It wont help my cause at all, that is correct, but this prof is the best math prof in the department and I feel like he is taking things carelessly sometimes. (Even though he is very into education which is weird) I guess I Just feel like its unfair that our department gives out such horrible grades when our education is actually tougher than some private's, in my honest opinion.2011-10-29
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    Just want to add, that my presence here is to learn in a right way, by myself, and not have to worry about silly teacher misguidance. (which is still minor, mind me) I love getting input and providing some myself. And this is the best place I can do it.2011-10-29

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The answer is right, but the solution is odd. The question asks for $G\to G$, the answer doesn't give that, and also the answer veers mid-course from the domain being ${\bf Z}_{10}$ to it being $\bf Z$.

Let $G$ be any group with more than one element, let $\phi$ map everything in $G$ to the identity element of $G$, end of story.

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    Great comment. I feel like so much freedom now with building function. But if it's not too much to ask, could some people please give me their honest opinion on how this test was structured. Solve 4 out of 6 with 1 for extra credit. Not only structure but emphasis on the questions and how well the solutions were written.2011-10-29
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    I suggest that if you have a problem with a test given at your institution that in the first place you discuss it with your instructor and, if, after due consideration of your instructor's replies, you are still dissatisfied, you speak to the head of the department (or the head of undergraduate teaching if the department has one, or whatever). I think that's much better than trying to air any grievances in public.2011-10-29
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    Well I would rather get some advice/ guidance if the test given was in fact not that good.2011-10-29