On successful completion of this module, students will be able to:
Apply the notions: map/function, sujective/injective/bijective/invertible map, equivalence relation, partition. Give the definition of: group, abelian group,subgroup, normal subgroup, quotient group, direct product of groups, homomorphism, isomorphism, kernal of a homomorphism, cyclic group, order of a group element.
Apply group theory to integer arithmetic: define what the greatest common divisor of two nonzero integers m and n is, compute it and express it as a linear combination of n and m using the extended Euclidan algorithm; Write down the Cayley table of a cyclic group Zn or of the multiplicative group (Zn)x for small n; determine the order of an element of such a group.
Define what a group action is and be able to verify that something is a group action. Apply group theory to describesymmetry; know the three types of rotation symmetry axes of the cube (their 'order' and how may there are of each type); describe the elements of symmetry group of the regular n-gon (the dihedral group D2n) for small values of n and know how to multiply them.
Compute with the symmetric group; determine disjoint cycle form, sign and order of a permutation; multiply two permutations.
Know how to show that a subset of a group is a subgroup or a normal subgroup. State and apply Lagrange's theorem. State and prove the first isomorphism theorem.
To be determined.
None for students admitted to the Mathematics, Theoretical Physics or
This module will be examined in a 2 hour examination in Trinity term. Continuous assessment
will contribute 20% to the final grade for the module at the annual
examination. Supplementals if required will consist of 100% exam.