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The Prüfer group $\mathbb{Z}(p^\infty)$, for $p$ prime, has the interesting property that it is infinite, and every proper subgroup is finite, but can be arbitrarily large, so there is no proper subgroup of maximum cardinality.

For any group $G$, let $\text{Sub}(G)$ be the collection of subgroups of $G$. Is there an example of an uncountable group $G$ such that the set of cardinals $$\{\text{card}(S):S\in \text{Sub}(G)\setminus\{G\}\}$$ has no largest member?

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    $\begingroup$ 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.) $\endgroup$
    – YCor
    Commented Oct 30, 2019 at 17:58
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    $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Oct 30, 2019 at 18:13
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    $\begingroup$ 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. $\endgroup$
    – YCor
    Commented Oct 30, 2019 at 18:19
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    $\begingroup$ 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) $\endgroup$
    – user35370
    Commented Oct 31, 2019 at 15:45
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    $\begingroup$ I remembered incorrectly, Shelah shows $\aleph_1$ (not depending on CH) has a Jónsson group and Jónsson groups of cardinality $\kappa^+=2^\kappa$. $\endgroup$
    – user35370
    Commented Oct 31, 2019 at 15:59

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