Foundations of Higher Algebra

Credits

Prerequisites

None.

Scientific-disciplinary sector (SSD)

MAT/02 Algebra.

Examination method

Oral exam.

Learning
objectives

The course aims to develop a critical knowledge of the contents and methodologies of modern algebra, with particular regard to group theory, both in its classical results and in some more recent developments. Attention is focused on infinite groups and in particular on the effect that imposing finiteness conditions has on their structure.

Syllabus

The course presents the main results concerning the structure of abelian groups, soluble groups and nilpotent groups. It illustrates the influence exerted on the structure of an infinite group by the imposition of various natural finiteness conditions. The relations between the structure of a group and that of its automorphism group are also examined. Finally, some results from homological algebra that are useful in the study of group extensions are presented.

Expected learning
outcomes

At the end of the course, the student must demonstrate the ability to

know and understand the topics of group theory covered, and in particular the issues related to the classification of groups;
apply the knowledge acquired to construct and compare abstract groups, and present the results studied using the language of the theory;
communicate ideas and solutions clearly, rigorously and effectively to both specialist and non-specialist audiences;
identify the most appropriate methods to analyse and solve a problem related to the topics of the course and correctly interpret the results.

Learning outcomes
to be assessed

Command of the knowledge acquired, clarity of presentation, rigour in the use of language, confidence in using the notions acquired.

Foundations of Higher Algebra