Groups and Rings 2

To build on the basic concepts of group and ring theory covered in the previous semester.

This lesson will cover the following topics:

 Group Theory

    - Group action on a set; quotient of a set under a group action. Cayley’s theorem. Class formula, Burnside’s formula. Application to counting

    - Sylow's theorems and their applications.

    - Distinctive subgroup, group quotient. The isomorphism and factorization theorems. Simple group. The special case of Abelian groups.

    - Group extensions and semidirect products. Examples. The special case of vector spaces.

 Ring Theory

    - Review of ideals and quotients of a ring by an ideal. Isomorphism and factorization theorems. Ideals of a quotient. Application of quotients to the construction of field extensions and (small) finite fields.

    - Prime and maximal ideals. Operations on ideals. The Chinese Remainder Theorem.

The algebra courses for the first and second years of the bachelor's program and the first semester of the third year of the bachelor's program.

Recommended prerequisites: first semester of the third year of a bachelor's degree program

Hourly volumes:

            CM: 22.5

            TD: 22.5

            Practical Work: -

            Land: -