KEY SUBGROUPS IN THE POLISH GROUP OF ALL AUTOMORPHISMS OF THE RATIONAL CIRCLE

The paper proves that for G = Aut(Q0) and every u in βG(Q0), the stabilizer H = G_u is inj-key but not co-minimal, first for rational stabilizers (Theorem 3.1) and then for all u (Theorem 3.19). The proof hinges on: (i) a precise description of βG(Q0), its four orbits, and the canonical G-factors (Fact 3.7, Lemma 3.8, Lemma 3.11), (ii) a classification showing there are exactly two admissible coset topologies on G/H, namely τ0/H and τT/H (Lemma 3.12 and surrounding discussion), and (iii) Merson’s Lemma to conclude equality of group topologies once they agree on H and on G/H (Fact 3.14). Non co-minimality follows by pulling back the compact-open topology from an appropriate G-factor (Lemma 3.4). These steps are explicitly present and consistent in the uploaded paper . The candidate solution reproduces the same architecture: describing βG(Q0), listing the two admissible coset topologies, performing a short case analysis, then invoking Merson’s Lemma to obtain inj-key, and finally using the compact-open pullback to witness non co-minimality. Aside from a minor phrasing in the high-level outline about what Merson’s Lemma itself guarantees (the detailed steps correctly use the classification to first synchronize the coset topologies), the model’s solution aligns closely with the paper’s proof.