## Trinity College Dublin, The University of Dublin

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# Module MAU22101: Abstract Algebra 1: Group Theory

Credit weighting (ECTS)
5 credits
Semester/term taught
Michaelmas term 2021-22
Contact Hours
11 weeks, 2 lectures plus 1 tutorial per week
Lecturer
Prof Patrick Fritzsch
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. Define group, abelian group, subgroup, normal subgroup, quotient group, direct product of groups, homomorphism, isomorphism, kernel 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 Euclidean 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 describe symmetry. Know the three types of rotation symmetry axes of the cube (their 'order' and how many there are of each type); describe the elements of the 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 groups. 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.
Text Books

Some standard useful references:
An Introduction to the Theory of Groups - Joseph J. Rotman
Abstract Algebra – David S. Dummit & Richard M. Foote
Algebra – Thomas W. Hungerford

Module Prerequisite
None for students admitted to the Mathematics, Theoretical Physics or Two-subject Moderatorships.
Assessment Detail

This module will be examined in a 2-hour examination in Trinity term. Continuous assessment will contribute 15% to the final grade for the module at the Michaelmas term examination session. Re-assessments if required will consist of 100% exam.

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