Are there uncountable groups which satisfy the minimal condition on subgroups?

Question

Definition

A group $G$ is said to satisfies the minimal condition on subgroups if every descending chain of subgroups stops after a finite number of steps.

I know this was an unsolved problem in past but I do not know if now it is solved. (Tarski groups are $2$-generated and hence countable.)

Question: Is it true that every group satisfying the minimal condition on subgroups is countable?

Answer 1

There are uncountable artinian groups. The original reference is:

A.Yu. Ol’shanskii, Geometry of defining relations in groups. Kluwer, 1991

Theorem 35.2 states that there are artinian groups of cardinality $\aleph_1$.

PS: In arXiv:1206.3639 (Example 2.6) it is claimed that every artinian group is countable (on the other hand, the abstract and the introduction speak of countable artinian groups). A reference is given to Kurosh's classic text The Theory of groups, page 192. But I couldn't find it there ...