Timeline for No injective groups with more than one element?
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jun 22, 2012 at 18:23 | comment | added | Michael Barr | See the comment added by Sergio Buschi added to #2. I don't know enough homotopy theory to see why that map is injective on $\pi_1$. Actually, I should have recalled that Eilenberg-Moore had it. I was very familiar with it back when it first came out. But it was kind of overtaken by Beck's thesis and other things and disappeared from my memory. | |
| Jun 21, 2012 at 22:17 | comment | added | Lee Mosher | @Michael: I'm not sure what inductive argument you are referring to here. Nogin's proof uses only that the inclusion of a subgraph into the full graph is $\pi_1$-injective, and that covering maps are $\pi_1$-injective. The subgraph and covering map to which these facts are applied are very concrete in her paper. | |
| Jun 21, 2012 at 20:41 | comment | added | Michael Barr | I just wanted to say that the proof attributed to Maria Nogin uses more about homotopy than I wanted my readers to have to know. And there is something wrong with the inductive argument that that map is 1-1 since clearly the image of the length 1 word $a^{-1}$ does not end in either $c$ or $d^{-1}$. I guess the argument of Eilenberg-Moore may be the easiest. Now I have to track down the reference to Baer. Thanks to all who answered. | |
| Jun 21, 2012 at 20:35 | vote | accept | Michael Barr | ||
| Jun 21, 2012 at 19:22 | comment | added | Theo Buehler | I forgot: Eilenberg-Moore refer to the paper by R. Baer, Die Kompositionsreihe der Gruppe aller eineindeutigen Abbildungen einer unendlichen Menge auf sich, Studia Math. 5 (1934), 15-17 which is available on the home page of the Polish virtual library of science: pldml.icm.edu.pl. Direct link to the paper: matwbn.icm.edu.pl/ksiazki/sm/sm5/sm512.pdf | |
| Jun 21, 2012 at 17:04 | history | edited | Theo Buehler | CC BY-SA 3.0 |
added screen shot of D.L. Johnson's exercise.
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| Jun 21, 2012 at 16:31 | history | edited | Theo Buehler | CC BY-SA 3.0 |
added 38 characters in body
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| Jun 21, 2012 at 16:21 | history | answered | Theo Buehler | CC BY-SA 3.0 |