Timeline for Uncountable group with no proper subgroup of maximal cardinal
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Nov 4, 2019 at 6:36 | comment | added | Dominic van der Zypen | Thanks @PaulPlummer - I appreciate your research! | |
| Oct 31, 2019 at 19:35 | comment | added | user35370 | I have looked around a bit and as far as I can tell that paper by Shelah is the only paper that proves something about existence of uncountable Jónsson groups, so it is probably open if $\aleph_\alpha$ has Jónsson groups for anything other than $\aleph_0,\aleph_1,\aleph_2$ (Shelah mentions that a similar construction can be done for 2 but for general $n$ it seems more complicated) without additional set theoretic assumptions. I wonder, with some advances in small cancellation theories, if some of the construction could be streamlined to be more general? | |
| Oct 31, 2019 at 15:59 | comment | added | user35370 | I remembered incorrectly, Shelah shows $\aleph_1$ (not depending on CH) has a Jónsson group and Jónsson groups of cardinality $\kappa^+=2^\kappa$. | |
| Oct 31, 2019 at 15:45 | comment | added | user35370 | I think that cardinals of Jónsson algebras are different for than cardinals of Jónsson groups so that page isn't quite answering the question. If I remember correctly Shelah actually proved there is a Jónsson group of cardinality continuum, which you can force to have a limit cardinal cofinality which I think answers your question in some models of ZFC (I will have to find the pdf though to double check) | |
| Oct 30, 2019 at 20:25 | comment | added | Dominic van der Zypen | Thanks for your great comments @YCor! I suggest you turn them into an answer? | |
| Oct 30, 2019 at 18:57 | history | edited | YCor | CC BY-SA 4.0 |
wrote title to make the setting more clear
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| Oct 30, 2019 at 18:19 | comment | added | YCor | There are very few known restrictions on (uncountable) cardinals of Jónsson groups. One is called being a "Jónsson cardinal", but very few such cardinals are known and only under assuming existence of large cardinals. See Jónsson cardinal at "Cantor's attic". In particular, whether $\aleph_\omega$ is Jónsson is unknown. | |
| Oct 30, 2019 at 18:13 | comment | added | YCor | Shelah constructed Jónsson groups, but all his examples have successor cardinal, and I'm not aware of any other construction. Reference: S. Shelah. On a problem of Kurosh, Jónsson groups, and applications. Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), pp. 373–394, Stud. Logic Foundations Math., 95, North-Holland, Amsterdam-New York, 1980. | |
| Oct 30, 2019 at 17:58 | comment | added | YCor | Sorry, I misunderstood your question, it's not trivial. The question asks whether there exists a Jonsson group of limit uncountable cardinal. (A group is Jonsson if it is uncountable and all its proper subgroups have smaller cardinal.) | |
| Oct 30, 2019 at 17:56 | history | edited | YCor | CC BY-SA 4.0 |
slightly rephrased to avoid confusion
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| Oct 30, 2019 at 17:53 | history | edited | Dominic van der Zypen | CC BY-SA 4.0 |
replaced |S| by card(S) for more clarity
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| Oct 30, 2019 at 17:53 | comment | added | Dominic van der Zypen | @YCor I don't mean groups with no maximal proper subgroups, but groups with no maximum cardinality proper subgroups. Maybe my wording in the title or post was bad? Apologies if that is the case. I have changed $|S|$ to $\text{card}(S)$ in the definition of the set, in order to add clarity. | |
| Oct 30, 2019 at 17:51 | comment | added | Dominic van der Zypen | @YCor So $G$ could for instance have cardinality $\aleph_\omega$, and $G$ has proper subgroups of cardinality $\geq \aleph_n$ for every $n\in \omega$, but no proper subgroup of cardinality $\aleph_\omega$. Can you give me a further hint, YCor? | |
| Oct 30, 2019 at 17:49 | history | undeleted | Dominic van der Zypen | ||
| Oct 30, 2019 at 17:49 | history | deleted | Dominic van der Zypen | via Vote | |
| Oct 30, 2019 at 17:10 | review | Close votes | |||
| Oct 30, 2019 at 18:00 | |||||
| Oct 30, 2019 at 16:39 | history | asked | Dominic van der Zypen | CC BY-SA 4.0 |