**Definition.**
For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say
that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous pregeometry if the following holds:


**(a)** $(\mathcal{A}, cl)$ is a pregeometry,


**(b)** $dim(\mathcal{A})$ is the same as the cardinality of $\mathcal{A}$,


**(c)** If $A \subseteq \mathcal{A}$ is finite and $a, b \in \mathcal{A} \setminus cl(A)$ , then there is $f\in Aut(\mathcal{A}/A)$ such that $f(a) = b$ and for all $B \subseteq \mathcal{A}$, $f(cl(B)) = cl(f(B))$.


**Theorem.** Suppose that $(G, cl)$ is a group carrying an $\omega$-homogeneous
pregeometry. Then either $G$ is commutative or unstable.


**Question.** Are there non-commutative groups that carry an $\omega$-homogeneous pregeometry?


Giving references is appreciated.



Groups and pregeometries
This question has been around for some time. 


Connections of homogeneous pregeomtries, quasiminimal structures and regular types have been studied in a recent [article](http://www1.maths.leeds.ac.uk/~pillay/regular.7.pdf) of Pilay and Tanovic. They show that the generic type of a homogeneous pregeometry is strongly regular (and generically stable) and conversely a global strongly regular type induces a homogeneous pregeomtry (Theorem 3). Earlier they analyse regular groups and ask if every regular group is commutative (which a variant of your question), see the question after Theorem 2.


As far as I know, the question is still open.


