Imrich, Wilfried; matematični kolokvij marec 2002

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The axiom of choice and trees in group theory

Wilfried Imrich

Univerza v Leobnu, Avstrija

14. marec 2002


The existence of spanning trees in connected graphs is one of the innumerable equivalents of the axiom of choice. After a short introduction about the axiom of choice and the role of the weaker compactness theorem in graph theory we demonstrate how the existence of spanning trees and forests can help to find transparent, surprisingly simple proofs for well known theorems in combinatorial group theory, most prominently for the Nielsen-Schreier Theorem and the subgroup theorems of Grushko and Kurosh. The talk concludes with remarks on results of Howson and Gersten from the point of view of groups acting on trees and connected open problems.

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