Math 455 - Syllabus

 THE UNIVERSITY OF HAWAI'I AT HILO - MATHEMATICS DEPARTMENT

Course Description:

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.

Learning Outcomes:

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 homomorphism
- 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 different fields
- 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:

This 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
in 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:

The 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 appropriate software.

The final grade will be based largely on homework, midterms, and final.
Furthermore, class participation (e.g., asking questions in 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.

Your grade will be computed using the following weights:
50% - Homework
40% - Midterms & Final
10% - Participation

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