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.~Todor\v{c}evi\'c and Uzc\'ategui. In particular, we show that it is consistent with ZFC that every countably compact sequential group is Fr\'echet-Urysohn.