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In this note, we characterize the $p$-groups of conjugate type $(n,1)$ attaining this maximal number. As a corollary, we characterize $p$-groups having prime order commutator subgroup and maximal number of $z$-classes."},"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":"1605.01166","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-05-04T07:31:46Z","cross_cats_sorted":[],"title_canon_sha256":"85dcaab39230f44a5d9db2793745c9b1ee80818bb7dd4660d4e295ed6475f81d","abstract_canon_sha256":"92292bd5aa6adea1afa4d7d03c30d90e3e277eb6917f5d5c7fd4805cd66ee1da"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:37.640490Z","signature_b64":"HZlu7/PuujI4fp6r+WBVfVUZTcyg4D4PVj26AQKu4ss4par2PqxmBBEOkY9psmfaQdgaumbmD1fZyOPgTQRFAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a02be27112f2840e47e0fc38657da87d6b26ea43452d8e5b4b81e6cb3c463831","last_reissued_at":"2026-05-18T01:15:37.639747Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:37.639747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"z-Classes in finite groups of conjugate type (n,1)","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Krishnendu Gongopadhyay, Shivam Arora","submitted_at":"2016-05-04T07:31:46Z","abstract_excerpt":"Two elements in a group $G$ are said to $z$-equivalent or to be in the same $z$-class if their centralizers are conjugate in $G$. 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