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

MODULE TITLE | Numbers, Symmetries and Groups | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECM1706 | MODULE CONVENER | Prof Nigel Byott (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
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DURATION: WEEKS | 11 | 0 | 0 |

Number of Students Taking Module (anticipated) | 268 |
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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.

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.

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.

- 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.

Scheduled Learning & Teaching Activities | 43.00 | Guided Independent Study | 107.00 | Placement / Study Abroad |
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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 |

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 |

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

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 |

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

information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Houston, K | How to think like a mathematician: a companion to undergraduate mathematics | 1st | Cambridge University Press | 2009 | 978-0521719780 | [Library] |

Set | Liebeck M. | A Concise Introduction to Pure Mathematics | 3rd | Chapman & Hall/CRC Press | 2010 | 978-1439835982 | [Library] |

Set | Allenby R.B.J.T. | Numbers and Proofs | Arnold | 1997 | 000-0-340-67653-1 | [Library] | |

Set | Jordan, C. and Jordan, D A. | Groups | Arnold | 1994 | 0-340-61045-X | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
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PRE-REQUISITE MODULES | None |
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CO-REQUISITE MODULES | None |

NQF LEVEL (FHEQ) | 4 | AVAILABLE AS DISTANCE LEARNING | No |
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ORIGIN DATE | Friday 09 January 2015 | LAST REVISION DATE | Friday 09 January 2015 |

KEY WORDS SEARCH | Number systems; symmetries; groups. |
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