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Module MA1213: Introduction to group theory
- Credit weighting (ECTS)
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5 credits
- Semester/term taught
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Michaelmas term 2018-19
- Contact Hours
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11 weeks, 2 lectures plus 1 tutorial per week
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- 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 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.
- Module Content
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To be determined.
- Module Prerequisite
- None for students admitted to the Mathematics, Theoretical Physics or
Two-subject Moderatorships.
- Assessment Detail
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This module will be examined in a 2 hour examination in Michaelmas term. Continuous assessment
will contribute 10% to the final grade for the module at the annual
examination.