Group Theory

(3 Credit Hours) · Undergraduate

This group theory course develops the theory begun in prelims, and this course will build on that. After recalling basic concepts, the focus will be on two circles of problems. 1. The concept of free group and its universal property allow to define and describe groups in terms of generators and relations. 2. The notion of composition series and the Jordan-Holder Theorem explain how to see, for instance, finite groups as being put together from finitely many simple groups. This leads to the problem of finding and classifying finite simple groups. Conversely, it will be explained how to put together two given groups to get new ones. Moreover, the concept of symmetry will be formulated in terms of group actions and applied to prove some group theoretic statements.