Introduction to Group Theory
Course category:2nd year
Please note that this module can be taken by some combined degree 3rd year students with the module code 0530011
At the end of the module you should be able to...
Definition of groups and examples. Axioms, general linear group, symmetric groups, modular groups (revision from Core Algebra).
Consequences of the group axioms.
Subgroups. Special linear group, alternating group, cyclic groups, symmetry groups of plane figures.
Homomorphisms. The concept of a structure preserving map; homomorphisms, isomorphisms, automorphisms (also discussed in Linear Algebra for vector spaces).
Lagrange. Cosets and Lagrange's theorem.
Generators. Cyclic groups, dihedral groups (existence and maximality with respect to presentation).
Fundamental theorem. Normal subgroups, quotient groups and homomorphism theorems.
Products. External and internal direct products, classification of finite abelian groups (statement only).
Classification of groups of small order.
Rotations in 3 dimensions. 3D rotations and special orthogonal group. Finite subgroups. Platonic/regular solids and their symmetries. The ‘belt trick' and example. (if time allows)
Orthogonal and unitary groups. (If time allows. This is discussed more fully in Linear Algebra.)
One and a half hour closed examination in week 1 SpringTerm.
A second course on abstract algebra, assuming a basic knowledge of group theory. Please check prerequisites carefully before asking to take this module as an elective. In choosing this module as an elective it will be assumed that you are familiar with all the material taught in the first year courses Calculus and Core Algebra, or are willing to learn the material if necessary.
Department of Mathematics, University of York, Heslington, York, UK. YO10 5DD