Is the class of modular/distributive groups an axiomatizable class?

Question

I define a modular group to be a group whose lattice of subgroups is a modular lattice, and similarly for distributive groups. My question is, are either of modular and/or distributive groups a first-order axiomatizable class, and if so, are they finitely axiomatizable?

Answer 1

Example 1: Let $Z'$ be a proper elementary extension of $\mathbb{Z}$ , and let $z\in Z'\setminus \mathbb{Z}$. The subgroup lattice of $\mathbb{Z}$ is the dual of the divisibility lattice on $\mathbb{N}$, which is distributive. And in the subgroup lattice of $Z'$, the sublattice generated by $\langle 1\rangle$, $\langle z\rangle$, and $\langle 1+z\rangle$ is isomorphic to M3, giving a failure of distributivity. Thus the class of distributive groups is not first-order axiomatizable.

Example 2: Let $G = Q_8\times (\bigoplus_{n\in \mathbb{Z}} Z/(2n+1)\mathbb{Z})$, and let $G'$ be an elementary extension of $G$ containing an element of infinite order (this is possible because $G$ has elements of arbitrarily large finite order).

A Dedekind group is a group in which every subgroup is normal. It is easy to see that every Dedekind group is modular. Further, it is a theorem of Dedekind and Baer that a non-abelian group is a Dedekind group if and only if it is isomorphic to $Q_8\times B\times D$, where $Q_8$ is the quaternion group, $B$ is an elementary abelian 2-group, and $D$ is a torsion abelian group with all elements of odd order. This classification shows that the group $G$ is Dedekind and hence modular.

On the other hand, it's a theorem of Iwasawa that if a modular group contains an element of infinite order, then the set of all elements of finite order forms an abelian subgroup. $G'$ has an element of infinite order, but it also has a subgroup isomorphic to $Q_8$ (since $G$ does). The set of elements of finite order in $G'$ contains this copy of $Q_8$, so it is not an abelian subgroup of $G'$, and hence $G'$ is not modular. Thus the class of modular groups is not first-order axiomatizable.