# Mathematics

## ECM1706 - Numbers, Symmetries and Groups (2015)

MODULE TITLE CREDIT VALUE Numbers, Symmetries and Groups 15 ECM1706 Prof Nigel Byott (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
 Number of Students Taking Module (anticipated) 268
DESCRIPTION - summary of the module content

By taking this module, you will gain an understanding of the theory of groups, up to Lagrange's Theorem, and an appreciation of how groups arise naturally from the consideration of symmetries in the context of geometry and in other settings. Furthermore, you will acquire and be able to demonstrate an understanding of the importance of abstract algebraic structures in unifying and generalising disparate situations exhibiting similar mathematical properties.

AIMS - intentions of the module

The purpose of this module is to provide you with an introduction to axiomatic reasoning in mathematics, particularly in relation to the perspective adopted by modern algebra. Properties of the standard number systems will be developed. Using a number of concrete examples, we will look at the abstract definition of a group, and rigorously prove standard results in the theory of groups.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 demonstrate understanding of the abstract theory of functions and permutations;
2 show comprehension of the theory of groups, up to Lagrange's Theorem, and an appreciation of how groups arise naturally from the consideration of symmetries in the context of geometry and in other settings;
3 understand properties of familiar number systems (N, Z, Z/nZ, Q) and simple logical relations between these properties;
4 identify and use common methods of proof and understand their foundations in the logical and axiomatic basis of modern mathematics.
Discipline Specific Skills and Knowledge:
5 acquire and be able to demonstrate an understanding of the importance of abstract algebraic structures in unifying and generalising disparate situations exhibiting similar mathematical properties;
6 explore open-ended problems independently and clearly state their findings with appropriate justification.
Personal and Key Transferable/ Employment Skills and  Knowledge:
7 formulate and express precise and rigorous arguments, based on explicitly stated assumptions;
8 reason using abstract ideas and communicate reasoning effectively in writing;
9 use learning resources appropriately;
10 exhibit self management and time management skills.

SYLLABUS PLAN - summary of the structure and academic content of the module

- numbers: the standard number systems N, Z, Q, R and C;

- complex numbers, elementary algebraic manipulations and the Argand diagram;

- conjugates;

- modulus and argument;

- exponential notation;

- axioms/construction and properties of the standard number systems N, Z, Z/nZ, Q, R; division with remainder; prime factorisation; fractions in lowest form; examples of irrational numbers; terminating, recurring and non-recurring decimal expansions;

- functions: sets; Cartesian product of sets; definition of a function via its (abstract) graph; injective, surjective and bijective functions; composition of functions, including associativity and (one- and two-sided) inverses;

- permutations: abstract definition of a permutation; the symmetric group; cycle notation and unique factorisation; crossing diagrams, transpositions and elementary transpositions (including non-uniqueness of factorisation); odd and even permutations and the alternating subgroup;

- symmetries: rotations and reflections of regular polygons and polyhedra, including representation as permutations;

- examples of groups: the group axioms; abelian/nonabelian groups; finite/infinite groups; subgroups; standard examples to include the dihedral groups Dn, the symmetric groups Sn, the alternating groups An, the groups Z, Q, R, C (under addition) and Q*, R*, C* (under multiplication), integers modulo n.

- general results on groups: uniqueness of identity and inverses; order of an element; cyclic groups; cosets; Lagrange's Theorem.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 43 107
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures Scheduled learning and teaching activities 10 Tutorials/seminars Guided independent study 107 Reading lecture notes; working exercises

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Tutor-marked assignments Selected questions from 5 sheets, each requiring 3-4 pages. 1-10 Tutorial; model answers provided on ELE and discussed in seminar

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 0 100 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam - Closed book 100 2 hours 1-5, 7, 8 Via SRS

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
Written exam - closed book Ref/Def exam 1-5,7,8 August Ref/Def period

RE-ASSESSMENT NOTES

The module mark is calculated solely from the mark on the referred/deferred exam. For referred candidates, this mark is capped at 40%.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk