# Is there any conditions on a finite abelian group so that it cannot be class group of any number field?

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?

• related discussion math.stackexchange.com/questions/10949/… – Matthias Wendt Jul 16 at 17:23
• @YCor I'd think "old part" probably should be "odd part"... – paul garrett Jul 16 at 18:43
• 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. – Alex B. Jul 16 at 20:04

## 1 Answer

It follows from the Cohen-Lenstra heuristic that every finite abelian group is 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 4.8 and the bottom of page 17 in this paper of mine with Lenstra: https://arxiv.org/abs/1803.06903v2. You have to read a bit between the lines, but it is shown there that if $$G$$ is a cyclic group of order degree $$58$$, then there are finite $$\mathbb{Q}[G]$$-modules that cannot be realised as the class group of a $$G$$-extension (the "almost all" in the last line of page 17 can be strengthened to "all but one, namely the one for the field $$\mathbb{Q}_{\zeta_{59}}$$").

Of course, there are cheap ways of doing that, by demanding that the 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.