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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.