INTRODUCTION TO GROUP THEORY
college of the atlantic
fall 2024
Group theory is an area of mathematics that is concerned with
symmetry. In this context, an object or system is said to be
symmetric if it is unchanged after an action is performed on
it. Examples of such actions include rotations, reflections, and
permutations. Objects' symmetries can be classified by the types
of actions that leave them unchanged. A set of actions, together
with rules for how those actions behave when combined, are
(loosely speaking) what mathematicians refer to as a
group. Group theory has applications in physics, chemistry, and
pure and applied mathematics.
A group is an example of an algebraic structure, which (again
loosely speaking) is a set together with a collection of rules
for combining pairs of elements of that set. Algebraic
structures are classified by the properties of the rules for
combining pairs of elements. In addition to groups, other common
algebraic structures include rings and fields. The broad study
of the properties of different algebraic structures is known as
"abstract algebra".
This course is an introduction to group theory and will also
serve as a springboard to the further study of abstract
algebra. Topics to be covered will include: learning to identify
groups using group axioms; Cayley diagrams; examples of
different classes of groups, including abelian, cyclic,
dihedral, and permutation groups; subgroups and Lagrange's
theorem; products and quotients; group homomorphisms; the
fundamental theorem for abelian groups; and Sylow theory. Other
topics and examples will depend on student interest. This course
will be taught in a seminar style; students will frequently be
asked to prepare examples and proofs for discussion in class and
to work collaboratively on problems, both in and out of class
meetings. Evaluation will be based on problem sets and active
and collaborative class participation. Level:
Intermediate/Advanced. Prerequisites: Calculus II (or the
equivalent), Linear Algebra and Proofs and Mathematical
Structures are both strongly recommended, and permission of
instructor.
Class limit: 5. Lab Fee: None. Meets the following
degree requirements: QR.
