Open Research Index

Arthur742Ramos/KuroshSubgroupTheorem

Arthur Freitas Ramos, David Barros Hulak, Ruy J. G. B. de Queiroz

Palomar Registrypalomar/PALOMAR-2026-08-31-000016Aug 31, 2026math.GRscore 0

Formalizes the sharp Schreier index formula for finite-index subgroups of free groups and the Kurosh subgroup theorem for arbitrary subgroups of indexed free products. The Schreier proof realizes a subgroup as the vertex group of a finite action-groupoid covering, while the Kurosh proof constructs the Bass--Serre tree, a locally bijective graph-of-groups cover, and its path-lifting normal form. The Kurosh factor-only statement is a free product of the nontrivial quotient vertex stabilizers and the free group of the quotient graph; each vertex stabilizer is identified with an intersection…

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