## MATH2022 Groups and Vector Spaces

### 15 creditsClass Size: 301

Module manager: Dr. Kevin Houston
Email: K.Houston@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2020/21

### Pre-requisite qualifications

(MATH1010 or MATH1005 or MATH1060) and (MATH1025 or MATH1055) or equivalent

### This module is mutually exclusive with

 MATH2020 Algebraic Structures 1 MATH2080 Further Linear Algebra

This module is not approved as a discovery module

### Module summary

The concept of a 'group' developed out of the work of Evariste Galois around 1830 on the insolubility of the general quintic polynomial. Today, group theory is a fundamental branch of mathematics, central also in theoretical physics. From one viewpoint, group theory may be regarded as an abstract study of symmetry. For example, the degree of symmetry of a geometrical figure may be captured by the corresponding group, which tells us not just how many symmetries there are, but also precisely how they interact (the 'structure' of the group). Likewise, we may handle equations (e.g. polynomial or differential equations) using the structure of the group of symmetries of the solutions. Groups, like vector spaces, fields, and rings, are objects of abstract algebra, a branch of mathematics which gains powerful insights by abstracting key mathematical features, e.g. about operations on a set.This course develops basic group theory with a goal, as a highlight, to understand rotations in two and three dimensions. In the process, a deeper understanding will be shed on linear algebra: results in linear algebra such as the Rank and Nullity Theorem will be presented as phenomena of abstract algebra, and linear algebra will furnish important examples of groups of matrices.

### Objectives

On completion of this module, students should be able to:
a) accurately reproduce appropriate definitions;
b) state basic results about groups and vector spaces, reproduce short proofs, and find similar proofs;
c) give examples of groups, identify subgroups and orders of elements in some of these examples;
d) list the cosets of various groups of small order, and describe the structure of certain quotient groups;
e) calculate the symmetry groups of simple geometrical objects;
f) test whether certain maps between vector spaces are linear transformations, calculate ranks and nullity,
and present the Rank and Nulllity Theorem as an example of the First Isomorphism Theorem for vector spaces.

Learning outcomes
The module will both introduce group theory as a beautiful and accessible example of the power of abstract algebra, and reinforce understanding of elementary linear algebra, at a more conceptual level than at Level 1, with emphasis on general vector spaces over arbitrary fields, linear independence and spanning, linear transformations, matrix groups. Students will develop their capacity to handle complex definitions, find proofs, and explain them precisely.

### Syllabus

Definition of group, abelian groups. Examples -- cyclic groups, additive groups, matrix groups and groups of isometries, groups of units, dihedral groups. Direct products. Subgroups, cosets, Lagrange's Theorem. Normal subgroups, quotient groups, homomorphisms, First Isomorphism Theorem for groups. Permutations, symmetric and alternating groups. Vector spaces over fields, linear independence, spanning and bases. Linear transformations, matrix groups, Rank and Nullity as First Isomorphism Theorem for vector spaces, quotient spaces. Determinants of matrices (Leibnitz formula, with links to permutations). Diagonalisability of real symmetric matrices. Groups of rotations in two and three dimensional real Euclidean space.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lectures 22 1.00 22.00 Private study hours 118.00 Total Contact hours 32.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Assessed coursework with feedback to students.

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Coursework
 Assessment type Notes % of formal assessment Written Work * 15.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 hr 30 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated