# Module MAU22101: Abstract Algebra 1: Group Theory

**Credit weighting (ECTS)**- 5 credits
**Semester/term taught**- Michaelmas term 2019-20
**Contact Hours**- 11 weeks, 2 lectures plus 1 tutorial per week
**Lecturer**- Prof. José M. Moreno Fernández
**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.