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We prove that $G\\leftthreetimes P$ is minimal for every closed subgroup $P$ of $Aut(G)$. In case $G$ is abelian, the same is true for every subgroup $P \\subseteq Aut(G)$. We show, in contrast, that there exist a compact two-step nilpotent group $G$ and a subgroup $P$ of $Aut(G)$ such that $G\\leftthr"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1511.07021","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2015-11-22T15:12:03Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"cac14407b38a07116065be72b0a5141b5fd8a0eb7df8af43d52bd5ee286a6bfb","abstract_canon_sha256":"c571246a82a07788fedb298d5e2d27c84c4090548c1a5bddbc716f1615774b00"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:09.544515Z","signature_b64":"N17Z5cGthcjWqr/n1gWNJxS7eQy+6Y241blMJzniXcyEdTyyzpHC7B2+ubQCvu+TMF7mJZ8dENx03vNEw5i+Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"52f6f575ca4361cbb71b9511703ff4553d9c63b58ef83d033f2cb83a8b91b99e","last_reissued_at":"2026-05-18T01:01:09.543831Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:09.543831Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimality of the Semidirect Product","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.GN","authors_text":"Luie Polev, Menachem Shlossberg, Michael Megrelishvili","submitted_at":"2015-11-22T15:12:03Z","abstract_excerpt":"A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. 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