2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/209495We provide elementary proofs of the Nielsen-Schreier Theorem and the Kurosh Subgroup Theorem via wreath products. Our proofs are diagrammatic in nature and work simultaneously in the abstract and profinite categories. A new proof that open subgroups of quasifree profinite groups are quasifree is also given.Group Theory20E06, 20E07, 20E22text