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Fatness of an element can be decided efficiently in practice by testing whether its characteristic polynomial has an irreducible factor of degree greater than $d/2$. We show that for groups $G$ with $ \\textrm{SL}(d,q) \\leq G \\leq \\textrm{GL}(d,q) $ most pairs of fat elements from $G$ generate irreducible subgroups, namely we prove that the proportion of pairs of fat elements generating a reducible subgroup, in the set o"},"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":"1903.07083","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2019-03-17T13:32:20Z","cross_cats_sorted":[],"title_canon_sha256":"0c388ea9580dc61299547a8d51273bf14ad14517164623845d72f49ae0be1b39","abstract_canon_sha256":"96a70e0bccfa9b24326a04d214a9785a0d581c0e98e7a6d497dc60e192b5dc2f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:51:04.296978Z","signature_b64":"lrLqO8omdLec/Ra9FXxBvfmxg+RNoc0CfvVCZ89jg6koWam70Bz/4wHmcCcWcv0MGsNYLtH3XheVYS9mqxGoAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4df2f6b2961ae46e5e074ceecbeb851e0f4591c86ac8df78fe2ed163d00b19cf","last_reissued_at":"2026-05-17T23:51:04.296328Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:51:04.296328Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Irreducible linear subgroups generated by pairs of matrices with large irreducible submodules","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alice C. 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