No interesting sequential groups

Abstract

We prove that it is consistent with ZFC that no sequential topological groups of intermediate sequential orders exist. This shows that the answer to a 1981 question of P. Nyikos is independent of the standard axioms of set theory. The model constructed also provides consistent answers to several questions of D. Shakhmatov, S. Todorčević and Uzcátegui. In particular, we show that it is consistent with ZFC that every countably compact sequential group is Fréchet–Urysohn.