Groups with Noetherian Group Rings

Abstract

In this thesis we will attempt to classify groups and rings based on whether the
associated group rings are Noetherian or not. As this is still an open problem,
we present the current state of research. While there are several classes of
groups proven not to have Noetherian group rings, such as non-amenable and
non-Noetherian groups, so far, virtually polycyclic groups are the only class for
which the group ring is known to be Noetherian. We will show some of these
major results as well as explore the indications as to whether the unsolved cases
might lean one way or another.

Popular Abstract

In a sense, abstract algebra is a study of symmetry; rotations, translations,
reflections and more. More importantly, it considers how those symmetries
interact with each other.
Consider a group of knights sitting at a round table and imagine all the ways
a king could move them around without changing who neighbours whom. The
main two things he could do would be to have each knight move one seat to the
right, or he could have each knight move to the seat across the table. Now, if
the king was feeling bored, he might have them first move to the seat opposite
and then one seat to the right. He could also have them move one seat to the
left and then to the one opposite. A particularly astute king might realize that
these two sets of movements end up with the same exact table configurations!
Letting the knights rest for a bit, one could similarly look at the behaviour
of rotations and translations of an object in 3D space. Or one could look at
ways to shuffle a deck of cards. Is a cut followed by a riffle shuffle the same as
a riffle followed by a cut? Why is cutting a deck 17 times the same as cutting
it once?
To answer these questions, we create ‘groups’ and ‘rings’, among other constructs.
They are sets of things, be they ways to move a 3D object, shuffle cards
or mess with increasingly rebellious knights. Crucially we place some constraints
on how objects in these sets behave. For example, consider how a shuffled deck
can always be rearranged to the original order, like how two perfectly even cuts
do not shuffle the deck at all. Similarly, turning the knights a step to the right
can be undone either by turning them a step to the left, or by turning them to
the right until they complete a full circle. This property is called an inverse and
it is one of the properties we require every element of a group to have.
The purpose of abstract algebra is to find the commonalities in these examples
and find general, abstract results which can then be applied where appropriate.
There is a balance to these constraints, the properties we demand the
constructs to have. The more constraints there are the easier the constructs are
to work with, the more tools we have available. On the other hand, the more
constraints we place, the fewer applications there will be, the fewer situations
we will find that fulfill all the constraints and let us actually apply our results.
Beyond the illustrative examples, the field has proper applications. They
range from algorithms for solving the Rubik’s cube to error correcting codes in
cryptography. As it turns out, our knights of the round table are quite good at
chemistry! Some molecular orbitals follow the same symmetries as the knights
and group theory makes it possible to speed up calculations.
The concept of ‘Noetherianity’ puts a certain size constraint on particularly
abstract groups and rings. If at a table of twelve knights we ignore every other
knight, we notice that the remaining six behave exactly as they would if they
were on a smaller table seating six. Similarly, the shufflings of a small deck of
cards consisting only of spades are contained in the shufflings of a full deck.
These are called subgroups and some truly enormous groups can have infinite
chains of subgroups where the first one is contained in the second, which is contained in the third and so on without stopping, all of them contained in
the original group. Sort of like a Matryoshka doll where if you started in the
innermost one, you would never get out. Groups without such an infinite chain
are called Noetherian and for rings the definition is similar. Most common
groups are, in fact, Noetherian. One particular non-Noetherian group could be
represented by all the different ways of shuffling a deck of infinitely many cards!
The purpose of this thesis is to survey the current research on the Noetherianity
of a specific type of ring, a so-called ‘group ring’.

• author: Proloscic, Marko
• supervisor: Anitha Thillaisundaram
• organization: Mathematics (Faculty of Engineering); Mathematics (Faculty of Sciences); Centre for Mathematical Sciences
• course: MATK11 20222
• year: 2023
• type: M2 - Bachelor Degree
• subject: Mathematics and Statistics
• keywords: group ring, noetherian, amenable, polycylcic, group theory, ring theory
• publication/series: Bachelor's Theses in Mathematical Sciences
• report number: LUNFMA-4143-2023
• ISSN: 1654-6229
• other publication id: 2023:K9
• language: English
• id: 9113857
• date added to LUP: 2024-04-15 16:55:38
• date last changed: 2024-04-15 16:55:38