Project research area: Pure mathematics (Group theory)
Description
The aim of this project is to explore some topics about finite
groups beyond what is covered in Algebra II. The project will be
oriented towards examples and exercises which you can partially
select yourself and which will form an important part of your
project. A main theme is that we can sometimes pin down the
structure of a finite group based only on its order. The simplest
examples of this, which you have seen in Algebra II are: a group of
order \(p\), where \(p\) is a prime, is cyclic, that is, isomorphic
to \(\mathbb{Z}/p\); a group of order \(2p\), where \(p\) is an odd
prime, is isomorphic to either the Dihedral group \(D_p\) or
\(\mathbb{Z}/(2p)\). We will take this further, for example,
determining the possible structures of groups of order \(pq\), where
\(p\) and \(q\) are two distinct primes, as well as several other
situations.
Group Project
- Sylow's theorems. These are generalisations of Cauchy's
theorem, saying that if a power of a prime \(p\) divides the
order of a group, but no higher power of \(p\) divides it, then
there exists a subgroup of that order. Moreover, all such
subgroups are conjugate and we can say how many there are mod
\(p\). They are useful in getting information about the
structure of finite groups. For example,using these theorems and
clever counting argument, we will show that there are exactly 5
groups of order 12 and 4 groups of order 30 and we will be
able to determine the structures of all of these groups, that
is, how they are built up from smaller groups.
- Semidirect products and group extensions. In the above
examples, the way we build groups from smaller ones is through
semidirect products, which are generalisations of the normal
direct product of groups.
Mode of Operation and Evidence of Learning for the group project
The project will revolve around learning through reading with focus
on the underlying theory, mathematical rigour, and the development
of deep conceptual understanding. Students will demonstrate their
understanding by presenting the material in their own way, exploring
examples (existing or new) and theoretical applications of the
material, and clearly communicating it in both written and oral
formats.
Individual Project
The individual project will build on the knowledge we have gained
in the group project and will explore additional advanced topics.
A few examples of topics you will be able to investigate are:
- Simple groups and statement of the classification.
Simple groups are groups without proper non-trivial normal
subgroups. In a certain sense, the simple groups are the
ultimate 'building blocks'/atoms of finite groups. More
precisely, the notion of group extension and the
Jordan-Hölder theorem imply that every finite group is 'built
up' from some uniquely determined simple groups (the difficulty
is how the simple pieces fit together). A basic result is that
the alternating groups \(A_n\) are simple for \(n\geq 5\). This
leads further to finite simple groups of Lie type
(matrix groups like PSL_n over finite fields), which is a big
subject in its own right.
- Solvable, nilpotent and \(p\)-groups. The first two are
important classes of groups which can be defined as having
certain series of subgroups. \(p\)-groups are groups all of
whose elements have order a power of a prime \(p\). Solvable
groups play an important role in Galois theory. There are a
lot of solvable groups. A famous theorem of Feit and
Thompson states that every finite groups of odd order is
solvable (the proof is very long so we will not study that).
Fascinating open problems, such as Higman's PORC conjecture,
seek to find some order in the chaos of finite \(p\)-groups by
predicting that the number of them is given by 'nice' functions.
Mode of Operation and Evidence of Learning for the individual
project
The project will revolve around learning through reading with focus
on the underlying theory, mathematical rigour, and the development
of deep conceptual understanding. Students will demonstrate their
understanding by presenting the material in their own way, exploring
examples (existing or new) and theoretical applications of the
material, and clearly communicating it in both written and oral
formats.
Prerequisites
Algebra II.
Resources
- D. S. Dummit and R. M. Foote, Abstract algebra (3rd
edition or later). A good text book at a level between Algebra
II and more advanced topics.
- J. S. Rose, A Course on Group
Theory.
- G. Smith and O. Tabachnikova, Topics in Group Theory.
- J. F. Humphreys, A Course in Group Theory.
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