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Module MA1214: Introduction to Group Theory

Credit weighting (ECTS)
5 credits
Semester/term taught
Hilary term 2014-15
Contact Hours
11 weeks, 3 lectures plus 1 tutorial per week
Lecturer
Prof Dmitri Zaitsev
Learning Outcomes
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.
  • Subgroups and generators.
  • Matrix groups: $GL_n, SL_n, O_n, SO_n, U_n, SU_n$.
  • 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.