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 teh multiplicative group (Zn)x for small n; determine the order of an element of such a group.
Define what a grup action is and be able to verify that soemthing 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.
Module Content
Sets and maps. Binary relations, equivalence relations, and partitions. Semigroups, monoids, and groups. Integer division; $ \ IZ_d$ as an additive group and a multiplicative monoid. Remainder modulo $n$ and integer devision.
The symmetric group $S_n$. Parity and the alternating group. Generators for Sn.
Cosets and Lagrange's Theorem. Additive subgroups of $IZ$. Greatest common divisor.
Normal subgroups and quotient groups. Homomorphisms and the first isomorphism theorem for groups. Multiplicative group $IZ_n^*$, Fermat's little theorem and the Chinese Remainder Theorem.
Group actions. A Sylow theorem. The classification of finite abelian groups.
Module Prerequisite
MA1111 - Linear Algebra 1
Assessment Detail
This module will be examined in a 2 hour examination in Trinity term. Continuous assessment
will contribute 10% to the final grade for the module at the annual
examination. Supplementals if required will consist of 100% exam.