{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2019:2MRGNVKZGRRYLLAQD4CHOHKD7P","short_pith_number":"pith:2MRGNVKZ","schema_version":"1.0","canonical_sha256":"d32266d559346385ac101f04771d43fbf2c872c33efcbb3af2d201d64ebdc8b7","source":{"kind":"arxiv","id":"1903.03549","version":2},"attestation_state":"computed","paper":{"title":"The fundamental group of the $p$-subgroup complex","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GR","authors_text":"Elias Gabriel Minian, Kevin Ivan Piterman","submitted_at":"2019-03-08T16:51:15Z","abstract_excerpt":"We study the fundamental group of the $p$-subgroup complex of a finite group $G$. We show first that $\\pi_1(A_3(A_{10}))$ is not a free group (here $A_{10}$ is the alternating group on $10$ letters). This is the first concrete example in the literature of a $p$-subgroup complex with non-free fundamental group. We prove that, modulo a well-known conjecture of M. Aschbacher, $\\pi_1(A_p(G)) = \\pi_1(A_p(S_G)) * F$, where $F$ is a free group and $\\pi_1(A_p(S_G))$ is free if $S_G$ is not almost simple. Here $S_G = \\Omega_1(G)/O_{p'}(\\Omega_1(G))$. This result essentially reduces the study of the fun"},"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":"1903.03549","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-03-08T16:51:15Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"12d6122d003d24e2d61bd64aa2737734be138a4fb83c525748eb93cd843f8d2f","abstract_canon_sha256":"21457a55e5d159fad5894a2eaa619c095062f154283f8cb61af88d14588a2b32"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:14.498844Z","signature_b64":"zlV0VisW+8xK5cJRINZHNZqgsPOuzjYbtnVLJtiggXKndUzj1yOo7qXnPPoK3CX6OpLwa957EcMDWI0FbSznCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d32266d559346385ac101f04771d43fbf2c872c33efcbb3af2d201d64ebdc8b7","last_reissued_at":"2026-05-17T23:49:14.498106Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:14.498106Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The fundamental group of the $p$-subgroup complex","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.GR","authors_text":"Elias Gabriel Minian, Kevin Ivan Piterman","submitted_at":"2019-03-08T16:51:15Z","abstract_excerpt":"We study the fundamental group of the $p$-subgroup complex of a finite group $G$. We show first that $\\pi_1(A_3(A_{10}))$ is not a free group (here $A_{10}$ is the alternating group on $10$ letters). This is the first concrete example in the literature of a $p$-subgroup complex with non-free fundamental group. We prove that, modulo a well-known conjecture of M. Aschbacher, $\\pi_1(A_p(G)) = \\pi_1(A_p(S_G)) * F$, where $F$ is a free group and $\\pi_1(A_p(S_G))$ is free if $S_G$ is not almost simple. Here $S_G = \\Omega_1(G)/O_{p'}(\\Omega_1(G))$. 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