A Wreath Product Approach to Classical Subgroup Theorems

TL;DR

This paper introduces elementary, diagrammatic proofs of key classical subgroup theorems using wreath products, applicable in both abstract and profinite contexts, and extends results to quasifree profinite groups.

Contribution

It provides new, elementary proofs of the Nielsen-Schreier and Kurosh theorems via wreath products, and proves that open subgroups of quasifree profinite groups are quasifree.

Findings

Elementary wreath product proofs of classical theorems

Diagrammatic approach applicable to abstract and profinite groups

Open subgroups of quasifree profinite groups are quasifree

Abstract

We 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.