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.
The topics we will look at will include all or some of the
following:
- 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. A further generalisation is that of
group extensions and the Jordan-Hölder theorem implies that
every finite group is 'built up' from some uniquely determined
simple groups (the difficulty is how the simple pieces fit
together).
- Simple groups and statement of the classification.
Simple groups are groups without proper non-trivial normal
subgroups. If we consider group extensions, mentioned above,
then the simple groups are the ultimate 'building blocks'/atoms
of finite groups. A basic result which we will study 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).
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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