On Ruppert's preordering (Q1605615)

For the group topologies \(\tau_1,\tau_2\) on a group \(G\) the preordering \(\tau_1\leq_c\tau_2\) means that every Cauchy ultrafilter on \((G,\tau_2)\) is a Cauchy ultrafilter on \((G,\tau_1)\) [see \textit{Wolfgang A. F. Ruppert}, Semigroup Forum 40, No. 2, 227-237 (1990; Zbl 0689.22003)] and the preordering \(\tau_1\leq_b\tau_2\) means that for every neighborhood \(V\) of the identity in \((G,\tau_1)\) there exists a finite subset \(K\) such that \(KV\) is a neighborhood of the identity in \((G,\tau_2)\) [see \textit{I. V. Protasov}, Sib. Math. J. 34, No. 5, 938-952 (1993); translation from Sib. Mat. Zh. 34, No. 5, 163-180 (1993; Zbl 0828.22002)]. It is proved that the preorders \(\tau_1\leq_c\tau_2\) and \(\tau_1\leq_b\tau_2\) coincide.