Let C be the class of finite nilpotent, solvable, symmetric, simple or semi-simple groups and n be a positive integer. We discuss the following question on group laws: What is the length of the shortest non-trivial law holding for all finite groups from the class C of order less than or equal to n?:Introduction
0 Essentials from group theory
1 The two main tools
1.1 The commutator lemma
1.2 The extension lemma
2 Nilpotent and solvable groups
2.1 Definitions and basic properties
2.2 Short non-trivial words in the derived series of F_2
2.3 Short non-trivial words in the lower central series of F_2
2.4 Laws for finite nilpotent groups
2.5 Laws for finite solvable groups
3 Semi-simple groups
3.1 Definitions and basic facts
3.2 Laws for the symmetric group S_n
3.3 Laws for simple groups
3.4 Laws for finite linear groups
3.5 Returning to semi-simple groups
4 The final conclusion
Index
BibliographySei C die Klasse der endlichen nilpotenten, auflösbaren, symmetrischen oder halbeinfachen Gruppen und n eine positive ganze Zahl. We diskutieren die folgende Frage über Gruppengesetze: Was ist die Länge des kürzesten nicht-trivialen Gesetzes, das für alle endlichen Gruppen der Klasse C gilt, welche die Ordnung höchstens n haben?:Introduction
0 Essentials from group theory
1 The two main tools
1.1 The commutator lemma
1.2 The extension lemma
2 Nilpotent and solvable groups
2.1 Definitions and basic properties
2.2 Short non-trivial words in the derived series of F_2
2.3 Short non-trivial words in the lower central series of F_2
2.4 Laws for finite nilpotent groups
2.5 Laws for finite solvable groups
3 Semi-simple groups
3.1 Definitions and basic facts
3.2 Laws for the symmetric group S_n
3.3 Laws for simple groups
3.4 Laws for finite linear groups
3.5 Returning to semi-simple groups
4 The final conclusion
Index
Bibliograph
