Page 69 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 69
mož Moravec: Some Topics in the Theory of Finite Groups 57
3.1 Introduction
These notes form a background material for a short course on group theory that was
given at 2014 PhD Summer School in Discrete Mathematics and SYGN, Rogla, Slovenia.
Since the summer school was aimed primarily at PhD students who are working in the
latter area and may not necessarily be experts in group theory, the notes give a fairly
general introduction to three main topics: Finite Simple Groups, Extension Theory of
Groups, and Nilpotent groups and Finite p -groups. The choice of the first two topics is
clear from the point of view of classifying all finite groups. It turns out that the knowledge
of all finite simple groups, together with knowing how to “glue” two groups together to
produce new ones, in principle provides a way of constructing all finite groups. The first
problem, classification of finite simple groups (CFSG), has been resolved satisfactory,
and one can operate with a full list of these groups. In these notes we will only touch
this vast area by showing simplicity of alternating groups and projective special linear
groups. We will sketch the classification, but ommit almost all further details. We will
move on to extension theory which tells us how to construct new groups from old. The
extension problem of classifying all possible extensions of one group by another appears
to be hard (impossible?) to solve in general. We will only study a very special case of it.
There are two main reasons why to deal with finite p -groups, i.e., groups whose or-
ders are powers of a prime p . The first is clear to an undergraduate student: finite p -
groups appear as Sylow p -subgroups of finite groups. The second is more delicate and
motivated by a vague statement “Almost all finite groups are p -groups.” We will not make
any attempt of making this statement more precise, but rather develop some basic the-
ory of these groups and indicate their complexity within the universe of all finite groups.
In addition to the above, we include preliminaries that will be needed in subsequent
sections. We collect some basic properties of groups with focus on finite groups. We
also exhibit as many examples as possible in order to illustrate and motivate the the-
ory. A general experience is that most of the students only know some standard types
of groups, such as abelian groups, dihedral groups, symmetric and alternating groups,...
Other groups which do not have clean descriptions are usually put aside. In order to
avoid this, I use GAP (Groups, Algorithms, and Programming), a computational system
designed for constructing and manipulating with groups. GAP is applied in exploring
properties of groups, and even providing proofs of statements. Examples with full GAP
code are be given, but I have decided to leave out all explanations of the syntax and pro-
gramming rules. There are two reasons for this. One is that the reader will mostly find
it easy to figure out what a given line of GAP code does, since the syntax is very much
self-explanatory. The second one is that there is an extensive manual of GAP, together
with tons of tutorials and self-study material available at GAP’s web page [5]. We encour-
age the reader to download GAP (it’s open source) and try out all of the examples in these
3.1 Introduction
These notes form a background material for a short course on group theory that was
given at 2014 PhD Summer School in Discrete Mathematics and SYGN, Rogla, Slovenia.
Since the summer school was aimed primarily at PhD students who are working in the
latter area and may not necessarily be experts in group theory, the notes give a fairly
general introduction to three main topics: Finite Simple Groups, Extension Theory of
Groups, and Nilpotent groups and Finite p -groups. The choice of the first two topics is
clear from the point of view of classifying all finite groups. It turns out that the knowledge
of all finite simple groups, together with knowing how to “glue” two groups together to
produce new ones, in principle provides a way of constructing all finite groups. The first
problem, classification of finite simple groups (CFSG), has been resolved satisfactory,
and one can operate with a full list of these groups. In these notes we will only touch
this vast area by showing simplicity of alternating groups and projective special linear
groups. We will sketch the classification, but ommit almost all further details. We will
move on to extension theory which tells us how to construct new groups from old. The
extension problem of classifying all possible extensions of one group by another appears
to be hard (impossible?) to solve in general. We will only study a very special case of it.
There are two main reasons why to deal with finite p -groups, i.e., groups whose or-
ders are powers of a prime p . The first is clear to an undergraduate student: finite p -
groups appear as Sylow p -subgroups of finite groups. The second is more delicate and
motivated by a vague statement “Almost all finite groups are p -groups.” We will not make
any attempt of making this statement more precise, but rather develop some basic the-
ory of these groups and indicate their complexity within the universe of all finite groups.
In addition to the above, we include preliminaries that will be needed in subsequent
sections. We collect some basic properties of groups with focus on finite groups. We
also exhibit as many examples as possible in order to illustrate and motivate the the-
ory. A general experience is that most of the students only know some standard types
of groups, such as abelian groups, dihedral groups, symmetric and alternating groups,...
Other groups which do not have clean descriptions are usually put aside. In order to
avoid this, I use GAP (Groups, Algorithms, and Programming), a computational system
designed for constructing and manipulating with groups. GAP is applied in exploring
properties of groups, and even providing proofs of statements. Examples with full GAP
code are be given, but I have decided to leave out all explanations of the syntax and pro-
gramming rules. There are two reasons for this. One is that the reader will mostly find
it easy to figure out what a given line of GAP code does, since the syntax is very much
self-explanatory. The second one is that there is an extensive manual of GAP, together
with tons of tutorials and self-study material available at GAP’s web page [5]. We encour-
age the reader to download GAP (it’s open source) and try out all of the examples in these
