MAU22101 Abstract algebra I: Group theory

Assessment Details

• This module is examined in a 2-hour examination at the end of Semester 1.
• Continuous assessment contributes 20% towards the overall mark.
• Re-assessment, if needed, consists of 100% exam.

Contact Hours

11 weeks of teaching with 3 lectures and 1 tutorial per week.

Learning Outcomes

On successful completion of this module, students will be able to

• Define the concepts of group, abelian group, subgroup, normal subgroup, quotient group, direct product, homomorphism, isomorphism, kernel, cyclic group, order of an element.
• Define the greatest common divisor of two nonzero integers m and n, compute it and also express it as a linear combination of m,n using the Euclidean algorithm.
• Write down the Cayley table of the cyclic group Z_n or the multiplicative group (Z_n)^* for small n and determine the order of each element.
• Describe the rotational symmetries of a cube and the elements of the symmetry group of the regular n-gon for small values of n.
• Perform basic computations in the symmetric group: determine the disjoint cycle form, sign and order of a given permutation, as well as multiply two given permutations.
• Verify 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

• Groups, permutations, symmetric group, homomorphisms.
• Subgroups and normal subgroups, cosets, Lagrange's theorem.
• Cyclic groups, dihedral groups, quotient groups, group actions.
• Kernel of a homomorphism, isomorphism theorems.

Recommended Reading

• An introduction to the theory of groups by Joseph J. Rotman.
• Abstract algebra by Dummit and Foote.
• Algebra by Thomas Hungerford.