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More specifically, we construct groups $G$ with abelian automorphism group such that $\\gamma_2(G) < \\mathrm{Z}(G) < \\Phi(G)$, where $\\gamma_2(G)$, $\\mathrm{Z}(G)$ and $\\Phi(G)$ denote the commutator subgroup, the center and the Frattini subgroup of $G$ respectively. For a finite $p$-group $G$ with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) $\\mathrm{Z}(G) = \\Phi(G)$ is elementary abelian; (ii) $\\gamma_2"},"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":"1304.1974","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-04-07T09:55:42Z","cross_cats_sorted":[],"title_canon_sha256":"02d98e828507803f4dc3035d91235bb93d48106b491b5c45996c79e3debb7e3b","abstract_canon_sha256":"fe27bf371e771e0c3059a0422ad4ed6531c6e3ff4c626bf93c5dd9c60b4da31d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:19.645043Z","signature_b64":"md0KSSiStsL0GUOh4+0iUTFpu2KLOWSzz+fMNYZn+iMRyrXvpoMSpuAupwex3Q8S48MsRMKoRQbIu1HuvFkIDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ccdc108ed3a742ead1069c5951b9a6f6c2e643a8e1c3c3ba0ec031abb6fe847c","last_reissued_at":"2026-05-18T00:11:19.644541Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:19.644541Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On finite $p$-groups with abelian automorphism group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Manoj K. 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