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Module Code - Title:

MB4008 - GROUPS AND ALGEBRAIC STRUCTURES

Year Last Offered:

2025/6

Hours Per Week:

Lecture

2

Lab

0

Tutorial

1

Other

0

Private

7

Credits

6

Grading Type:

N

Prerequisite Modules:

MB4001
MB4002

Rationale and Purpose of the Module:

To develop a broad understanding of algebraic structures especially group structure. To study realizations of group structure in geometry. To study selected applications in Science and Engineering.

Syllabus:

Sets and operations: review of sets, operations; Groupoids and semi-groups: equality, commutativity, associativity, inverses, order; Groups: axioms, properties, sub-groups, cyclic groups, p-groups, permutation groups; Lagrange's theorem: applications to number theory, kernel, isomorphisms, normal subgroups, quotient groups; Sylow's theorems; Group of isometries; group of transformations, enlargements; Group of similarities; Rings: definition; integral domain, fields.

Learning Outcomes:

Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis)

At the end of this module students should be able to: -describe and explain common algebraic structures such as ring, field, and especially the structure of a group -write mathematics clearly with due regard for mathematical conventions and notation -follow and clearly set out a mathematical argument -describe and explain the basic concepts of group theory: sets and operations; axioms; subgroups; cyclic, symmetric and dihedral groups; isomorphism; Lagrange and Sylow theory -use group theory in selected applications e.g. to understand symmetries in geometry -manipulate abstract algebraic expressions.

Affective (Attitudes and Values)

None

Psychomotor (Physical Skills)

None

How the Module will be Taught and what will be the Learning Experiences of the Students:

Normal lecture and tutorial mode of delivery. Examples used should be appropriate and relevant as far as practicable.

Research Findings Incorporated in to the Syllabus (If Relevant):

Prime Texts:

Lederman, Walter (1973) Introduction to group theory , Longman.

Other Relevant Texts:

Jeger, M. (1964) Transformation geometry , Allen and Unwin
Coxeter, P. (1989) Introduction to geometry. 2nd ed. , Wiley.
Herstein, I. (1977) Topics in algebra, 2nd ed , Wiley.
Kim, K.H. and Roush, F. W. (1983) Applied abstract algebra , Ellis Horwood.

Programme(s) in which this Module is Offered:

Semester(s) Module is Offered:

Module Leader:

olivia.fitzmaurice@ul.ie