Right Orderable Residually Finite p-Groups and a Kourovka Notebook Problem

Rhemtulla proved that if a group is a residually finite p-group for infinitely many primes p, then it is two-sided orderable. In problem 10.30 of The Kourovka Notebook (14th ed.), N. Ya. Medvedev asked if there is a non-right-orderable group which is a residually finite p-group for at least two different primes p. Using a result of Witte, we will show that many subgroups of finite index in GL3( ) give examples of such groups. On the other hand, we will show that no such example can exist among solvable by finite groups.