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If, in addition to having exactly four codegrees, $G$ has two irreducible character degrees, $G$ has largest irreducible character degree $p^2$, $|G:G'|=p^2$, or $G$ has coclass at most 3, then $G$ has nilpotence class at most 4. In the case of coclass at most 3, the order of $G$ is bounded by $p"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1901.07425","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-01-22T15:42:21Z","cross_cats_sorted":[],"title_canon_sha256":"5c474952cd133178fae7afbcd727d31507113299869d861f16ca5522371ccca1","abstract_canon_sha256":"9595d9c0feb2ff8da80a25f221d12613b83c4cd4e7d150128afbb52a299ca398"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:55:44.790496Z","signature_b64":"b1t1No0RaNUE9t+HdfF3Y9yoYqgv2zbbNwLFWoALhEbOgidCrnW8O9h8Xp9xTMR6EU2RpEtJ62RAsgftE2IoDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5b7e7afa8cc5591854a1638e99b33738ae875805216881d2aeac4a1fe45f5b9a","last_reissued_at":"2026-05-17T23:55:44.789834Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:55:44.789834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$p$-groups with exactly four codegrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Mark L. Lewis, Sarah Croome","submitted_at":"2019-01-22T15:42:21Z","abstract_excerpt":"Let $G$ be a $p$-group and let $\\chi$ be an irreducible character of $G$. The codegree of $\\chi$ is given by $|G:\\text{ker}(\\chi)|/\\chi(1)$. Du and Lewis have shown that a $p$-group with exactly three codegrees has nilpotence class at most 2. Here we investigate $p$-groups with exactly four codegrees. If, in addition to having exactly four codegrees, $G$ has two irreducible character degrees, $G$ has largest irreducible character degree $p^2$, $|G:G'|=p^2$, or $G$ has coclass at most 3, then $G$ has nilpotence class at most 4. 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