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 report later.
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.
- 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 studyat 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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