- To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory.

- To give an appreciation of the utility and power of group theory as the study of symmetries.

- To introduce public-key cryptosystems as used in the transmission of confidential data, and also error-correcting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.

(LO1) Be able to apply the Euclidean algorithm to find the greatest common divisor of a pair of positive integers, and use this procedure to find the inverse of an integer modulo a given integer.

(LO2) Be able to solve linear congruences and apply appropriate techniques to solve systems of such congruences.

(LO3) Be able to perform a range of calculations and manipulations with permutations.

(LO4) Recall the definition of a group and a subgroup and be able to identify these in explicit examples.

(LO5) Be able to prove that a given mapping between groups is a homomorphism and identify isomorphic groups.

(LO6) To be able to apply group theoretic ideas to applications with error correcting codes.

(LO7) Engage in project work to investigate applications of the theoretical material covered in the module.

- Naive set theory; sets and maps, subsets, union and intersection. Cartesian product.

- Modular arithmetic. Euclid's algorithm and Bezout's lemma. Linear congruences and invertibility modulo an integer. Fermat's and Euler's theorems. Public key cryptography.

- Bijections, permutations. Cycle notation. Order and sign of permutations.

- Axioms of group theory. Simple examples (including symmetry groups of geometric figures including platonic solids). Subgroups and Lagrange's theorem.

- Homomorphisms and isomorphisms. Examples from modular arithmetic and symmetry groups of geometric figures.

- Error correction and detection for binary codes.