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We give some constructions of kaleidoscopical configurations in an arbitrary $G$-space, develop some kaleidoscopical technique for Abelian groups (considered as $G$-spaces with the action $(g,x)\\mapsto g+x$), and describe kaleidoscopical configurations in the cyclic groups of order $N=p^m$ or $N=p_1... p_k$ where $p$ is prime and $p_1,...,p_k$ are distinct pr"},"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":"1001.0903","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-01-06T14:14:40Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"fbbca4638028e4380958f5d43936ccd013fcb9951dc36aebb04ec96cc74c0d5c","abstract_canon_sha256":"6fb752312a68e0161700c050e468fc326713fb7aaf9e8f6d900bc72b9f8ebe23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:38:14.590255Z","signature_b64":"FIx+XLi7CiQjh/APU6XpTxqNOWnY5aTfc45lUrIzesazJyhHh/eCFu8UXO7viqC4z4LxhdYE8DkaaasInmoXAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8ca77b78d6be3a7194c2b010a7ce533d1ae3f4f26c489e5b802f2e672ad9c582","last_reissued_at":"2026-05-18T03:38:14.589641Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:38:14.589641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Kaleidoscopical Configurations in G-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.CO","authors_text":"I.V. 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