# Group Theory (MAE525)

### General

School School of Science Department of Mathematics Undergraduate MAE525 5 Group Theory Lectures, laboratory exercises (Weekly Teaching Hours: 3, Credits: 6) Special background, skills development. - Greek, English Yes See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes Familiarity with: group, abelian group, subgroup, normal subgroup, quotient group, direct product of groups, homomorphism, isomorphism, kernel of a homomorphism.  Apply group theory to describe symmetry, describe the elements of symmetry group of the regular n-gon (the dihedral group D2n). Compute with the symmetric group. Know how to show that a subset of a group is a subgroup or a normal subgroup. State and apply Lagrange's theorem. State and prove the isomorphism theorems. Sylow theorems. The classification of finite abelian groups. Normal series, central series, nilpotent groups. Applications in Geometry. Study particular characteristics of group theory in topology and geometry. Independent and team work. Working in an interdisciplinary.

### Syllabus

• Basic properties in groups.
• Symmetries.
• Subgroups, Direct products, Cosets.
• Symmetric groups.
• Normal Subgroups, Quotient groups.
• Homomorphisms.
• Semidirect product.
• Classification of finite abelian groups.
• Sylow theorems.
• Normal series, Solvable groups. Central series, Nilpotent groups.

### Teaching and Learning Methods - Evaluation

Delivery

Classroom (face-to-face)

Use of Information and Communications Technology

Communication with students

Teaching Methods
Activity Semester Workload
Lectures (13X3) 39
Working independently 78
Exercises-Homeworks 33
Course total 150
Student Performance Evaluation

Written Examination, Oral Presentation, written assignments in Greek (in case of Erasmus students in English) which includes resolving application problems.

### Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

• An Introduction to the Theory of Groups (Graduate Texts in Mathematics) 4th Edition by Joseph Rotman.