# Module Information

#### Course Delivery

Delivery Type | Delivery length / details |
---|---|

Lecture | 19 Hours. (19 x 1 hour lectures) |

Seminars / Tutorials | 3 Hours. (3 x 1 hour examples classes) |

#### Assessment

Assessment Type | Assessment length / details | Proportion |
---|---|---|

Semester Exam | 2 Hours (written examination) | 100% |

Supplementary Exam | 2 Hours (written examination) | 100% |

### Learning Outcomes

On completion of this module, a student should be able to:

1. determine whether given algebraic systems are groups;

2. determine whether elements and subsets of a group possess a variety of properties;

3. state and prove some fundamental theorems, selected from the isomorphism theorems, the orbit-stabilizer theorem and the theorems of Lagrange, Cayley and Sylow;

4. represent groups as matrix groups, as permutation groups and with generator-relation presentations, and use these representations to compute within the groups;

5. solve problems in group theory by selecting and applying appropriate theorems and techniques from the general theory.

### Brief description

The concept of a group occurs naturally in situations involving symmetry or in which some quantity is being preserved; for example, various letters such as A, S and I possess different numbers of symmetries and rigid motions preserve distance. This module will introduce the notion of a group as an algebraic object defined by a simple set of axioms. Various techniques for describing groups (presentations, matrix and permutation representation) will be studied. The principal structure theorems for finite groups will be described and applied in a variety of group theoretic contexts.

### Aims

To provide a deeper understanding of the concepts and techniques of abstract algebra, introduced in module MA20310, by focusing on the group concept, starting with an axiomatic development of group theory, establishing a structure theory, mainly in the context of finite groups, and giving brief illustrations of a selection of applications of group theory.

### Content

1. Fundamentals: Definitions and examples. Presentations of groups. Elementary consequences of the definitions. Subgroups. cosets. Lagrange's theorem.

2. Basic structure theory: Normal subgroups and factor groups. Direct products. Homomorphisms. The isomorphism theorems. Automorphism groups.

3. Permutation groups: Symmetric groups. Cycle decomposition. Regular representation. Cayley's theorem. Orbits. Stabilizers. The orbit-stabilizer theorem.

4. Local structure theory: p-subgroups. The Sylow theorems. Classifying groups of small order.

5. Global structure theory: Classification of finite Abelian groups. The Jordan-Holder theorem.

### Reading List

**General Text**

Scott, W. R. (July 1988) Group Theory Primo search

**Recommended Text**

J F Humphreys (2001) A Course in Group Theory Oxford University Press Primo search

**Supplementary Text**

B Baumslag and B Chandler (1968) Theory and Problems of Group Theory McGraw-Hill Primo search C R Jordan and D A Jordan (1994) Group Theory Edward Arnold Primo search D A R Wallace (1998) Groups Rings and Fields Springer Primo search D W Farmer (1996) Groups and Symmetry 6th Ed American Mathematical Society Primo search J B Fraleigh (2003) A First Course in Abstract Algebra 7th Ed Addison-Wesley Primo search J R Durbin (2000) Modern algebra : an introduction 4th Ed John Wiley and Sons, Inc. Primo search

### Notes

This module is at CQFW Level 6