Module Specification
The information contained in this module specification was correct at the time of publication but may be subject to change, either during the session because of unforeseen circumstances, or following review of the module at the end of the session. Queries about the module should be directed to the member of staff with responsibility for the module.
1. Module Title Numbers, Groups and Codes
2. Module Code MATH142
3. Year Session 2023-24
4. Originating Department Mathematical Sciences
5. Faculty Fac of Science & Engineering
6. Semester Second Semester
7. CATS Level Level 4 FHEQ
8. CATS Value 15
9. Member of staff with responsibility for the module
Dr EJ Russell Mathematical Sciences Ewan.Russell@liverpool.ac.uk
10. Module Moderator
11. Other Contributing Departments  
12. Other Staff Teaching on this Module
Dr JA Haddley Mathematical Sciences J.A.Haddley@liverpool.ac.uk
Dr SA Fairfax Mathematical Sciences Simon.Fairfax@liverpool.ac.uk
13. Board of Studies
14. Mode of Delivery
15. Location Main Liverpool City Campus
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other TOTAL
16. Study Hours     20

     5

30

 55
17.

Private Study

 100
18.

TOTAL HOURS

 155
 
    Lectures Seminars Tutorials Lab Practicals Fieldwork Placement Other
19. Timetable (if known)            
 
20. Pre-requisites before taking this module (other modules and/or general educational/academic requirements):

 
21. Modules for which this module is a pre-requisite:

 
22. Co-requisite modules:

 
23. Linked Modules:

 
24. Programme(s) (including Year of Study) to which this module is available on a mandatory basis:
25. Programme(s) (including Year of Study) to which this module is available on a required basis:
26. Programme(s) (including Year of Study) to which this module is available on an optional basis:
27. Aims
 

- To provide an introduction to rigorous reasoning in axiomatic systems exemplified by the framework of group theory.

- To give an appreciation of the utility and power of group theory as the study of symmetries.

- To introduce public-key cryptosystems as used in the transmission of confidential data, and also error-correcting codes used to ensure that transmission of data is accurate. Both of these ideas are illustrations of the application of algebraic techniques.
 
28. Learning Outcomes
 

(LO1) Be able to apply the Euclidean algorithm to find the greatest common divisor of a pair of positive integers, and use this procedure to find the inverse of an integer modulo a given integer.
 

(LO2) Be able to solve linear congruences and apply appropriate techniques to solve systems of such congruences.
 

(LO3) Be able to perform a range of calculations and manipulations with permutations.
 

(LO4) Recall the definition of a group and a subgroup and be able to identify these in explicit examples.
 

(LO5) Be able to prove that a given mapping between groups is a homomorphism and identify isomorphic groups.
 

(LO6) To be able to apply group theoretic ideas to applications with error correcting codes.
 

(LO7) Engage in project work to investigate applications of the theoretical material covered in the module.
 
29. Teaching and Learning Strategies
 

Material is provided in advance of classes for students to study asynchronously. The contact hours consist of one 2-hour active learning session and one 2-hour supported study/drop-in session.
 
30. Syllabus
   

- Naive set theory; sets and maps, subsets, union and intersection. Cartesian product.

- Modular arithmetic. Euclid's algorithm and Bezout's lemma. Linear congruences and invertibility modulo an integer. Fermat's and Euler's theorems. Public key cryptography.

- Bijections, permutations. Cycle notation. Order and sign of permutations.

- Axioms of group theory. Simple examples (including symmetry groups of geometric figures including platonic solids). Subgroups and Lagrange's theorem.

- Homomorphisms and isomorphisms. Examples from modular arithmetic and symmetry groups of geometric figures.

- Error correction and detection for binary codes.
 
31. Recommended Texts
  Reading lists are managed at readinglists.liverpool.ac.uk. Click here to access the reading lists for this module.
 

Assessment
32. EXAM Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
  Final Assessment 120 50
33. CONTINUOUS Duration Timing
(Semester)		 % of
final
mark		 Resit/resubmission
opportunity		 Penalty for late
submission		 Notes
  Group Project Standard UoL penalty applies for late submission. 150 25
  Class Test remote open book 0 25