Course Details for A.Y. 2013/2014

Course Objectives

This first module is divided in two parts: 1) Group theory, including actions of groups on sets, Sylow theorems, nilpotent groups, simple groups, alternating groups, free groups and presentations, the Todd-Coxeter algorithm, soluble groups, Schmidt-Iwasawa theorem. 2) Elemetary Galois theory, Galois correspondence andsolution of algebraic equations by radicals.

Course Content

• Actions of groups on sets, orbits, inertia groups, orbit counting, exemples and exercises.
• Sylow theorems and applications, simple groups, nonsimplicity criteria, series, composition and principal series, the Jordan-Hoelder theorem, exemples and exercises.
• Free groups, presentations, the Todd-Coxeter algorithm, exemples and exercises.
• Nilpotent groups, central series, elementary properties of finite nilpotent groups, the Frattini argument, the Frattini subgroup and its properties, exemples and exercises.
• Soluble groups, the derived series, elementary properties, the Theorem by Schmidt Iwasawa.
• The field extension theory, Galois correspondence, normality and stability, splitting fields and splitting closures, separability, solubility of algebraic equations by radicals.

Prerequisites and Learning Activities

An introduction to elementary abstract algebra (Sets: functions, equivalence relations and partitions, products; Groups: subgroups, homomorphism theorem and quotients, permutation groups, groups of invertible matrices; Rings, subrings ideals, quotient rings and homorphism theorem, polynomial rings, euclidean rings, PID, UFD, matrix rings; Fields: fied of fractions of a integral domain, simple extensions, finite fields.

Assessment Methods and Criteria

Oral exam that includes the solution of problems in written form.

Course page updates

• A.Y. 2013/2014
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• A.Y. 2017/2018
• A.Y. 2018/2019