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Then there are at least $r_1+r_2+... +r_s$ non-isomorphic nilpotent groups with class two and the order of each of the group is equal to the order of $G$."},"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":"0906.3645","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2009-06-19T11:41:27Z","cross_cats_sorted":[],"title_canon_sha256":"adbb62e79734b55f56159cd0df631923579bd110be63dadab59cb14141501f0d","abstract_canon_sha256":"e71a7e18f0efd9b92f5c7212681a5f56c4006a430ba2b4e4d43c67f593ca264b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:46.998789Z","signature_b64":"EzJiFs9JeiiTL8NrI4G9zxdeOgYHJZVJYLQzl20EAb1VAOXLy+mVuX6Rg3AM5N4bUTNm1g/CD2XTuqx0Wiy3Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db6cfa20761ab84499a8ff2a00f3019e015af2a39797f10de6fd0327204dbef7","last_reissued_at":"2026-05-18T04:05:46.998415Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:46.998415Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Odd Order Nilpotent Groups With Class 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Vivek Kumar Jain","submitted_at":"2009-06-19T11:41:27Z","abstract_excerpt":"Let $G$ be an odd order nilpotent group with class 2 and $e$ denotes the exponent of its commutator subgroup. Let $e=p_1^{r_1}p_2^{r_2}... p_s^{r_s}$, where $p_i$'s are odd primes and $r_i$'s are non-negative integers. 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