Timeline for What is the automorphism group of the additive group of the p-adic integers?
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11 events
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| Oct 2, 2022 at 12:52 | comment | added | Yiftach Barnea | $\mathbb Z_p$ is cyclic in the category of pro-$p$ groups. | |
| Oct 2, 2022 at 0:00 | comment | added | Paul Fabel | But isn't $Z_{p}$ uncountable and consequently NOT cyclic? ncatlab.org/nlab/show/p-adic+integer. Treated as a group, aren't the units of its associated ring extension precisely the group automorphisms of $Z_{p}$? Please see Andreas Blass's answer below. | |
| Oct 23, 2018 at 6:57 | history | edited | Yiftach Barnea | CC BY-SA 4.0 |
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| Jul 5, 2010 at 9:47 | comment | added | Yiftach Barnea | fherzig, yes you are right the case of $\mathbb{Z}_p$ is triviel. I am not sure what is the origin of Serre's argument, I think it was in some letter, but I have a terrible memory so don't trust me too much. Luis Ribes knows the history of the problem, so if you are interested you can ask him. | |
| Jul 5, 2010 at 8:30 | comment | added | fherzig | Incidentally: did Serre not publish his result? E.g., in the announcement of their results in CRAS (2003), Nikolov-Segal didn't have any reference to a paper or book of Serre. | |
| Jul 5, 2010 at 8:16 | comment | added | fherzig | Surely the continuity of group automorphisms is obvious for $\mathbb Z_p$: the subgroups $p^n \mathbb Z_p$ are basic open neighbourhoods of the identity, and they are preserved by group homomorphisms. | |
| Jul 5, 2010 at 7:21 | comment | added | Yiftach Barnea | Indeed, Serre proved that for a finitely generated pro-$p$ group a subgroup of finite index is open. Thus, the topology is determined by the algebra. Hence, every automorphism must be continuous. The questions whether every subgroup of finite index of a finitely generated profinite group is open was known as Serre's problem. It was fairly recently proved to have an affirmative answer by Nik Nikolov and Dan Segal. | |
| Jul 5, 2010 at 4:57 | vote | accept | Zev Chonoles | ||
| Jul 5, 2010 at 3:11 | comment | added | Pete L. Clark | Continued: it was rather recently proven that any finitely generated profinite group has the property that every finite index subgroup is open. I believe this has been known for pro-$p$-groups for a long time: that's probably related to the result of Serre that you're referring to. | |
| Jul 5, 2010 at 3:09 | comment | added | Pete L. Clark | I'm not sure exactly which result of Serre you're referring to, but this is an elementary question so let's be more explicit. Let $G$ be a profinite group and let $f$ be an automorphism of the underlying abstract group. Then $f$ is a homeomorphism iff it is continuous iff $f^{-1}$ is continuous iff $f$ maps open subgroups to open subgroups. Since every group automorphism preserves the index of subgroups and any open subgroup has finite index, a suficient condition for every group automorphism of $G$ to be a homeo is that every finite index subgroup of $G$ is open. This is the case here... | |
| Jul 5, 2010 at 0:34 | history | answered | Yiftach Barnea | CC BY-SA 2.5 |