Content:

Section 1 Group Theory:

• Motivating examples: numbers, symmetry groups
  
• Definitions, elementary properties
  
• Subgroups, including subgroups of $Z$
  
• Arithmetic modulo n and the group $Z_n$
  
• Lagrange's Theorem
  
• Permutation groups, odd and even permutations (proof optional)
  
• Normal subgroups and quotient groups

Section 2 Ring Theory:

• Definitions: Commutative and non-commutative rings, integral domains, fields
  
• Examples: $Z, Q, R, C, Z_n$, matrices, polynomials, Gaussian integers

Aims:

To introduce First Year Mathematics students to abstract Algebra, covering Group Theory and Ring Theory.

Objectives:

By the end of the module students should be able to understand:

• the abstract definition of a group, and be familiar with the basic types of examples, including numbers, symmetry groups and groups of permutations and matrices.
• what subgroups are, and be familiar with the proof of Lagrange’s Theorem.
• the definition of various types of ring, and be familiar with a number of examples, including numbers, polynomials, and matrices.
• unit groups of rings, and be able to calculate the unit groups of the integers modulo n.

Books:

Any library book with Abstract Algebra in the title would be useful.