- Lecturer: Andrew Dancer
General Prerequisites:
Course Term: Trinity
Course Lecture Information: 8 lectures
Course Overview:
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-Hölder 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.
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-Hölder 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.
Learning Outcomes:
Students will learn to construct and describe groups. They will learn basic properties of groups and get familiar with important classes of groups. They will understand the crucial concept of simple groups. They will get a better understanding of the notion of symmetry by using group actions.
Course Synopsis:
Free groups. Uniqueness of reduced words and universal mapping property. Normal subgroups of free groups and generators and relations for groups. Examples. [2]
Review of the First Isomorphism Theorem and proof of Second and Third Isomorphism Theorems. Simple groups, statement that \(A_n\) is simple (proof for \(n=5\)). Definition and proof of existence of composition series for finite groups. Statement of the Jordan-Hölder Theorem. Examples. The derived subgroup and solvable groups. [3]
Discussion of semi-direct products and extensions of groups. Examples. [1]
Sylow's three theorems. Applications including classification of groups of small order. [2]
Review of the First Isomorphism Theorem and proof of Second and Third Isomorphism Theorems. Simple groups, statement that \(A_n\) is simple (proof for \(n=5\)). Definition and proof of existence of composition series for finite groups. Statement of the Jordan-Hölder Theorem. Examples. The derived subgroup and solvable groups. [3]
Discussion of semi-direct products and extensions of groups. Examples. [1]
Sylow's three theorems. Applications including classification of groups of small order. [2]