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In this paper it is shown that the kernel Ker$(\\pi)$ of the power function $\\pi$ is a dihedral subgroup of $D_n$ and if $n \\ne 3,$ then the kernel Ker$(\\pi)$ is of order at least $4$. Moreover, all $\\mathcal{M}$ are classified for which Ker$(\\pi)$ is of order $4$. In particular, besides $4$ sporadic maps on $4,4,8$ and $12$ vertices respectively, two infinite families of non-$t$-balanced Cayley maps on $D_n$ are obtained."},"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":"1504.00763","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-03T07:34:53Z","cross_cats_sorted":[],"title_canon_sha256":"4a9d16b2880e57858cc8e41f034284557173109e9946cf7d0a1cd00bdff3d9c8","abstract_canon_sha256":"69548f767691269faa2527bd7c0f7b5589baa6ad0700b2dc31e9d1ebd86c204a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:42.003498Z","signature_b64":"YITfVS0oOHbv5OejcALXjPo0Do/9BArvQVMmy9qrAyu5JpXOdsIy1O5kLEQqtY955hE32b9SKQ+16p3Q6P7lDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf7e09aa5dab9cc2a4afdcf01c864ce9188ee12a783d2583ddbdd95cb5f28528","last_reissued_at":"2026-05-18T02:19:42.002871Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:42.002871Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regular Cayley maps on dihedral groups with the smallest kernel","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Istv\\'an Kov\\'acs, Young Soo Kwon","submitted_at":"2015-04-03T07:34:53Z","abstract_excerpt":"Let $\\mathcal{M}=CM(D_n,X,p)$ be a regular Cayley map on the dihedral group $D_n$ of order $2n, n \\ge 2,$ and let $\\pi$ be the power function associated with $\\mathcal{M}$. In this paper it is shown that the kernel Ker$(\\pi)$ of the power function $\\pi$ is a dihedral subgroup of $D_n$ and if $n \\ne 3,$ then the kernel Ker$(\\pi)$ is of order at least $4$. Moreover, all $\\mathcal{M}$ are classified for which Ker$(\\pi)$ is of order $4$. In particular, besides $4$ sporadic maps on $4,4,8$ and $12$ vertices respectively, two infinite families of non-$t$-balanced Cayley maps on $D_n$ are obtained."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.00763","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1504.00763","created_at":"2026-05-18T02:19:42.002959+00:00"},{"alias_kind":"arxiv_version","alias_value":"1504.00763v1","created_at":"2026-05-18T02:19:42.002959+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.00763","created_at":"2026-05-18T02:19:42.002959+00:00"},{"alias_kind":"pith_short_12","alias_value":"X57ATKS5VOOM","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"X57ATKS5VOOMFJFP","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"X57ATKS5","created_at":"2026-05-18T12:29:47.479230+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/X57ATKS5VOOMFJFP3TYBZBSM5E","json":"https://pith.science/pith/X57ATKS5VOOMFJFP3TYBZBSM5E.json","graph_json":"https://pith.science/api/pith-number/X57ATKS5VOOMFJFP3TYBZBSM5E/graph.json","events_json":"https://pith.science/api/pith-number/X57ATKS5VOOMFJFP3TYBZBSM5E/events.json","paper":"https://pith.science/paper/X57ATKS5"},"agent_actions":{"view_html":"https://pith.science/pith/X57ATKS5VOOMFJFP3TYBZBSM5E","download_json":"https://pith.science/pith/X57ATKS5VOOMFJFP3TYBZBSM5E.json","view_paper":"https://pith.science/paper/X57ATKS5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1504.00763&json=true","fetch_graph":"https://pith.science/api/pith-number/X57ATKS5VOOMFJFP3TYBZBSM5E/graph.json","fetch_events":"https://pith.science/api/pith-number/X57ATKS5VOOMFJFP3TYBZBSM5E/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/X57ATKS5VOOMFJFP3TYBZBSM5E/action/timestamp_anchor","attest_storage":"https://pith.science/pith/X57ATKS5VOOMFJFP3TYBZBSM5E/action/storage_attestation","attest_author":"https://pith.science/pith/X57ATKS5VOOMFJFP3TYBZBSM5E/action/author_attestation","sign_citation":"https://pith.science/pith/X57ATKS5VOOMFJFP3TYBZBSM5E/action/citation_signature","submit_replication":"https://pith.science/pith/X57ATKS5VOOMFJFP3TYBZBSM5E/action/replication_record"}},"created_at":"2026-05-18T02:19:42.002959+00:00","updated_at":"2026-05-18T02:19:42.002959+00:00"}