#### Thesis Title

#### Date of Graduation

Summer 2008

#### Degree

Master of Science in Mathematics

#### Department

Mathematics

#### Committee Chair

Kishor Shah

#### Keywords

finite groups, invariant polynomials, Reynolds operator, Emmy Noether's Degree Bound theorem, ring of invariants, Molien's theorem, Hilbert series, Macaulay 2

#### Subject Categories

Mathematics

#### Abstract

Invariant theory has been studied by some of the great mathematicians such as Paul Gordan and Emmy Noether. In this thesis, we will prove two main theorems from Invariant theory. The first theorem is Emmy Noether's Degree Bound theorem, which was published in 1916. We provide a proof of this theorem as well as present examples of rings of invariants. The second theorem that we prove is Molien's theorem, published in 1897. After providing a proof of this theorem, we compute the Molien series for certain invariant rings and draw a connection between Molien's theorem and Emmy Noether's Degree Bound Theorem. While all our computations were done by hand, we explain how to use Macaulay 2 to rework some of the examples.

#### Copyright

© Dianne Michelle Twigger

#### Recommended Citation

Twigger, Dianne Michelle, "Rings of Invariants of Finite Groups" (2008). *MSU Graduate Theses*. 1627.

http://bearworks.missouristate.edu/theses/1627