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
To introduce First Year Mathematics students to abstract Algebra, covering Group Theory and Ring Theory.
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.
Any library book with Abstract Algebra in the title would be useful.