·         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.

Use the division algorithm to construct the greatest common divisor of a pair of positive integers;

Solve linear congruences and find the inverse of an integer modulo a given integer;

Code and decode messages using the public-key method

Manipulate permutations with confidence

Decide when a given set is a group under a specified operation and give formal axiomatic proofs;

Understand the concept of a subgroup and use Lagrange''s theorem;

Understand the concept of a group action, an orbit and a stabiliser subgroup

Understand the concept of a group homomorphism and be able to show (in simple cases) that two groups are isomorphic;

[6] Naive set theory; sets and maps, subsets, union and intersection. Cartesian product.

[9] 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.

[3] Bijections, permutations. Cycle notation. Order and sign of permutations.

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

[3] Group actions, orbits, stabilisers. Orbit-stabiliser theorem and applications to counting symmetries.

[3] Homomorphisms and isomorphisms. Examples from modular arithmetic and symmetry groups of ge ometric figures.

[3] Error correction and detection for binary codes.