Timeline for alternative construction of the quotient group
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Feb 7, 2010 at 23:30 | comment | added | Bjorn Poonen | By the way, I agree with Martin that the construction of G/N given in this "answer" is not a handy one. I think its pedagogical value lies mainly in the fact that a similar argument can sometimes be used in other situations to prove the existence of universal objects. | |
| Feb 7, 2010 at 23:22 | comment | added | Bjorn Poonen | @VA,Tom: I define "normal subgroup" as "subgroup that is the kernel of some homomorphism"; then that homomorphism to its image gives rise to one of the (H,f) pairs, so N is the kernel of the big homomorphism! :) Well, that was mainly intended as a joke, but in any case, it'll be necessary to specify which definition of "normal" is being used. If the definition is "left cosets are the same as right cosets", then it's going to be hard to avoid cosets in any argument using this! If it's "gNg^{-1}=N for all g", then that's dangerously close to the coset definition too. | |
| Feb 7, 2010 at 21:33 | comment | added | Tom Church | To understand VA's point, consider the above construction applied to a non-normal subgroup of G, in which case the kernel is the normal closure of your subgroup. Without explicitly constructing the quotient, how do you know that for N normal in G and x not contained in N, there is some homomorphism G -> H vanishing on N but not on x? | |
| Feb 7, 2010 at 20:22 | comment | added | VA. | Surjectivity of $G\to G/N$ implies nothing. Please prove that the kernel is $N$, without using cosets. No, please. P.S. The joke is due to Arnold. | |
| Feb 7, 2010 at 20:19 | comment | added | Martin Brandenburg | @VA: the kernel of G -> G/N can be derived from the universal property, also that G -> G/N is surjective. | |
| Feb 7, 2010 at 20:18 | vote | accept | Martin Brandenburg | ||
| Feb 8, 2010 at 0:58 | |||||
| Feb 7, 2010 at 20:18 | comment | added | Martin Brandenburg | this is a little better than the construction I've given in the comments of the question. It's still rather "unhandy". anyway, thank you :) | |
| Feb 7, 2010 at 20:14 | comment | added | VA. | My goodness, this actually works. But from this definition you don't know what the kernel of $G\to "G/N"$ is. Reminds me of a joke. Q: What is 2+3? A: I don't know but it is the same as 3+2 because the addition is commutative. | |
| Feb 7, 2010 at 18:45 | comment | added | user2734 | @BP: Even if I had found the solution set, I would not think of this nice explicit construction. Thank you for this answer. | |
| Feb 7, 2010 at 18:00 | history | answered | Bjorn Poonen | CC BY-SA 2.5 |