AWhat changed
Hölder proved that any two composition series of a finite group yield the same simple quotient groups up to isomorphism, order and multiplicity. He replaced the earlier invariance of numerical factor orders by invariance of the factor groups themselves, using quotient groups and an induction through intersections of distinct maximal normal subgroups. Applied to algebraic-equation chains, these simple factor groups give unavoidable constituents of the auxiliary equation groups, even when auxiliary irrationalities beyond the original roots are admitted under the paper's rationality-field conventions.
Scientific scope
Hölder proved that any two composition series of a finite group yield the same simple quotient groups up to isomorphism, order and multiplicity. He replaced the earlier invariance of numerical factor orders by invariance of the factor groups themselves, using quotient groups and an induction through intersections of distinct maximal normal subgroups. Applied to algebraic-equation chains, these simple factor groups give unavoidable constituents of the auxiliary equation groups, even when auxiliary irrationalities beyond the original roots are admitted under the paper's rationality-field conventions. Finite groups and finite composition series are the group-theoretic scope. The theorem does not make the subgroup chains unique or determine a group from its multiset of factors: extension problems remain. The algebraic application assumes distinct roots and the explicitly changing rationality fields; this record does not assert a modern inseparable or arbitrary-characteristic extension. The chief-series direct-product statement in Section 11 is expressly given without proof and is not certified here. Jordan's earlier numerical composition-factor invariance, Galois foundations, Dyck's quotient-group precedents and Netto's proof simplifications are retained as antecedents.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Finite groups and finite composition series are the group-theoretic scope. The theorem does not make the subgroup chains unique or determine a group from its multiset of factors: extension problems remain. The algebraic application assumes distinct roots and the explicitly changing rationality fields; this record does not assert a modern inseparable or arbitrary-characteristic extension. The chief-series direct-product statement in Section 11 is expressly given without proof and is not certified here. Jordan's earlier numerical composition-factor invariance, Galois foundations, Dyck's quotient-group precedents and Netto's proof simplifications are retained as antecedents.
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
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