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

AbstractRhemtulla 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(Z) give examples of such groups. On the other hand, we will show that no such example can exist among solvable by finite groups