Countable Fréchet topological groups under CH (Q1292762)

The classes of spaces defined by \textit{A. V. Arkhangel'skij} (1972) [Trans. Mosc. Math. Soc. 1981, No. 2, 163-200 (1981); translation from Tr. Mosk. Mat. O.-va 40, 171-206 (1979; Zbl 0447.54004)] are now called \(\alpha_i\)-spaces, where \(i=1,1.5,2,3,4\), (here, \(\alpha_1 \Rightarrow \alpha_{1.5} \Rightarrow\alpha_2 \Rightarrow\alpha_3 \Rightarrow\alpha_4)\). Every Fréchet topological group is \(\alpha_4\) (P. Nyikos). But, in a model of ZFC, there exist a Fréchet non-\(\alpha_3\) topological group, and a Fréchet \(\alpha_2\) non-\(\alpha_1\) topological group, etc. (D. B. Shakhmatov). On the other hand, under (CH), there exist two countable Fréchet topological groups with sequential non-Fréchet square (A. Shibakov). In this paper it is shown that, under (CH), there exists a countable Fréchet \(\alpha_3\) non-\(\alpha_2\) topological group, and a countable Fréchet topological group with sequential non-Fréchet square. Also, it is shown that every Fréchet \(\alpha_{1.5}\) topological group is \(\alpha_1\). (These results answer some questions of D. B. Shakhmatov).