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We show that if $A_0$ is trivial, $G$ is a rank one group associated to a quadratic Jordan division algebra. If $A_0$ is not trivial (which is always the case if $A$ is not abelian), then $A_0$ defines a subgroup $G_0$ of $G$ which acts quadratically on $V$. We will call $G_0$ the \\textit{quadratic kernel} of $G$. By a result of Timmesfeld we have $G_0 \\con"},"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":"1106.2310","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-06-12T13:37:53Z","cross_cats_sorted":[],"title_canon_sha256":"2a0c327a6050e38c87e166dd2c0bee02e774bcc2375bfdda13e497e5425903b4","abstract_canon_sha256":"62f04881276f8da8e3dfb58dba44f7eaeefefa57357e248b67429c7c5059d376"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:52.124617Z","signature_b64":"hoTC5kZ/wZ7xSi2pLW1Fi7N0tiZ3i4Eww9gxccc8i7BMwVrvZo/reYg5GC7xvw7arE7GmEcxFdsyrpXmwUxjCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"61be4c2d0fe7744f0244103fad883b32dee7c3e651e322896c6c299442174648","last_reissued_at":"2026-05-18T01:11:52.124285Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:52.124285Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On cubic action of a rank one group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Matthias Gr\\\"uninger","submitted_at":"2011-06-12T13:37:53Z","abstract_excerpt":"We consider a rank one group $G = \\langle A,B \\rangle $ which acts cubically on a module $V$, this means $[V,A,A,A] =0$ but $[V,G,G,G] \\ne 0$. We have to distinguish whether the group $A_0 :=C_A([V,A]) \\cap C_A(V/C_V(A))$ is trivial or not. We show that if $A_0$ is trivial, $G$ is a rank one group associated to a quadratic Jordan division algebra. If $A_0$ is not trivial (which is always the case if $A$ is not abelian), then $A_0$ defines a subgroup $G_0$ of $G$ which acts quadratically on $V$. We will call $G_0$ the \\textit{quadratic kernel} of $G$. 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