Right orderable residually finite \(p\)-groups and a Kourovka Notebook problem (Q1599083)

It is shown that for any finite set of primes \(\mathcal P\) there exists a non-right-orderable group which is a residually finite \(p\)-group for all \(p\in{\mathcal P}\) (Theorem 1). It is proved that any soluble-by-finite group, which is a residually finite \(p\)-group and a residually finite \(q\)-group, is right orderable (Theorem 2).