# Gwybodaeth Modiwlau

#### Course Delivery

#### Assessment

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

Semester Exam | 2 Hours (Written Examination) | 100% |

Supplementary Exam | 2 Hours (Written Examination) | 100% |

### Learning Outcomes

On successful completion of this module students 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. compute with permutations in the context of the symmetric groups and of the alternating groups

5. identify group actions and to make use of the orbit-stabilizer theorem, for example in applications to counting with the Burnside lemma

6. 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 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

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

3. Permutation groups: Symmetric groups. Cycle decomposition, Cayley's theorem. Orbits. Stabilizers. The orbit-stabilizer theorem, Burnside lemma and applications to counting.

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

### Module Skills

Skills Type | Skills details |
---|---|

Adaptability and resilience | Students are expected to develop their own approach to time-management and to use the feedback from marked work to support their learning. |

Co-ordinating with others | Students will be encouraged to work in groups to solve problems. |

Creative Problem Solving | The assignments will give the students opportunities to show creativity in finding solutions and develop their problem solving skills. |

Digital capability | Use of the internet, Blackboard, and mathematical packages will be encouraged to enhance their understanding of the module content and examples of application |

Professional communication | Students will be expected to submit clearly written solutions to set exercises. |

Subject Specific Skills | Broadens exposure of students to topics in mathematics, and an area of application that they have not previously encountered. |

### Notes

This module is at CQFW Level 6