The Cohen-Lenstra paper says the probability that the odd part of a class group being cyclic is close to 0.98. So I was thinking: can we find any conditions on a finite abelian group so that it cannot be a class group of any number field?
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1$\begingroup$ related discussion math.stackexchange.com/questions/10949/… $\endgroup$– Matthias WendtCommented Jul 16, 2019 at 17:23
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$\begingroup$ @YCor I'd think "old part" probably should be "odd part"... $\endgroup$– paul garrettCommented Jul 16, 2019 at 18:43
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3$\begingroup$ Actually, the passage from Cohen--Lenstra that the question refers to concerns class groups of imaginary quadratic fields. The paper makes no statements about the statistics of class groups of all number fields. $\endgroup$– Alex B.Commented Jul 16, 2019 at 20:04
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It follows from the Cohen-Lenstra heuristic that every finite abelian group is expected to be isomorphic to infinitely many class groups of real quadratic fields (even to a positive proportion of real quadratics, which is stronger), but nothing like this is known.
However, if you take the Galois action into account, then things get interesting: there are Galois modules that are not isomorphic to any class group of a Galois number field with the respective Galois group. See Corollary 4.12 and the discussion following it on page 20 in this paper of mine with Lenstra: https://arxiv.org/abs/1803.06903v4. We show there that if $G$ is a cyclic group of order $58$, then there are finite $\mathbb{Z}[G]$-modules that cannot be realised as the class group of a $G$-extension.
Of course, there are cheap ways of doing that, by considering modules whose fixed submodule is something silly, contradicting the fact that the class group of $\mathbb{Q}$ is trivial, but that is not what is happening in our paper. For example our obstruction cannot be seen by looking at any particular $p$-Sylow of the class group.
