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.
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.
MA1111 - Linear Algebra 1
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.