UNIVERSITY OF HAWAI'I AT HILO - MATHEMATICS DEPARTMENT
Math 455 will focus on the algberaic properties of groups.
Compared to rings, groups have a simpler algebraic structure - only one
operations, which is not required to be commutative. Thus, larger
classes of objects satisfy the group axioms and, as a consequence,
there exists much more diversity in the category of groups.
Groups are universally used in Mathematics: Topologists use the
Fundamental Group; Knot Theorists use the Knot Group and the Braid
Group; Algebraic Geometers turn certain varieties into groups; and the
list goes on.
The goal of this course will be to understand on a deep level the
structure of groups using concepts like (normal) subgroups, group
quotients, and conjugacy classes. Towards the end of the
semester, we will unify the concepts from the Fall term by finding a
deep and surprising relationship between field extensions and groups
via Galois Theory.
The successful student will be able to:
- identify when a set and an operation forms a group
- understand well the group and subgroup structure
of various classical groups: Z_n, D_n, S_n, A_n
- understand how to form new groups from old groups
(e.g., direct products, the center of a group)
- take the quotient of a group G by a normal
subgroup N and understand the relationship between G and G/N
- decide when a function between groups is a
- use the properties of group homomorphisms to prove
properties of the domain and co-domain groups
- decide when a set is a vector space over a field
- understand properties of field extensions over
- understand how one can use polynomials in F[x] to
find field extensions over F
- compute the Galois Group of a field extension
- use Galois Theory to understand when a polynomial
has roots that are "solvable by radicals" Homework:
course will have frequent homework assignments. You are not only
allowed - but encouraged -
to work in small groups for these
assignments. However, each student must turn in homework written
his/her own words and must understand all the content of the HW.
Failure to do so may be
interpreted as cheating.
All homework is required to be typeset in LaTeX.
Homework must be printed out prior to class and turned in at the
beginning of lecture on the due date. LaTeX:
use of AMS LaTeX is not only encouraged but will be
required. LaTeX is the
universally accepted scientific typesetting program that allows
mathematicians to write well-formatted mathematical papers. This
open-source compiler and associated interfaces can be found as
freeware. Please see the below links for how to download the
grade will be based largely on homework, midterms, and final.
Furthermore, class participation (e.g., asking questions
class, going to the Math Lab, going to
office hours, asking Bob email
questions, consistent attendeance in class, etc) will also be used in
this computation. The class participation grade is computed at
the discretion of the instructor.
grade will be computed using the following weights:
50% - Homework
40% - Midterms & Final 10% -