One of the major accomplishments of the 20th century was the classification
of the finite simple groups. A group is said to be simple if
its only normal subgroups are the whole group and the group generated by
the identity. This classification is extrodinarily deep and has a number
of technical aspects. The proof of the Feit - Thompson theorem was around
1200 pages, in its original form, by itself. There have been a number of
techniques brought to bear on this problem, from algebraic geometry,
quivers, and those behind Deligne-Lusztig theory.
The first simple groups that one encounters are the cyclic groups of prime
order. A standard exercise in an undergraduate group theory course is
showing that $C_p$ (the cyclic group of order $p$ written multiplicatively)
is simple. One's second encounter with simple groups often comes from the
study of the alternating groups $A_n$, for $n \neq 4$. Incidentally, a
little invariant theory comes up in showing that $A_n$ is not solvable ,
for $n \geq 5$. One then concludes that there is not closed form, in elementary
functions, for the roots of a polynomial of degree five or larger.
The vast majority of the finite simple groups are those of Lie type .
The projective special linear group , denoted $\operatorname{PSL}(n,q)$,
for $n \geq 3$, and $q=p^n$ the power of some prime, provides us with an example
of a simple group of Lie type.
Let $G = \operatorname{GL}(n,q)$.
Here are some of the standard ideas that get one started in this area:
Borel Subgroups: Let $B$ be the set consisting of all invertible
upper triangular matrices in $G$. This is the standard Borel
subgroup . More generally, a Borel subgroup is some
conjugate of $B$.
$(B,N)$-pair: One says that a group $G$ has a $(B,N)$-pair
if there are subgroups $B, N \subseteq G$ such that:
$G$ is generated by $B$ and $N$
$H := B \cap N$ is normal in $N$, and the quotient $W := N/H$
is a group generated by a set $S$ of elements of order 2.
$n_sBn_s \neq B$ if $s \in S$ and $n_s$ a representative of
$s \in N$.
$n_sBn \subseteq Bn_snB \cup BnB$, for any $s \in S$ and $n \in N$.
$\bigcap_{n \in N} nBn^{-1} = H$.
The group $W$ is called the Weyl group of $G$.
Bruhat Decomposition: Suppose that the group $G$ admits a $(B,N)$-pair.
We have the double-coset decomposition:
\[
G = \coprod_{w \in W} Bn_wB.
\]
Frobenius Map: We begin by supposing that $k$ is the algebraic
of a finite field of $p$ elements, for a prime $p$. There is a subfield of $k$
such that there are $q = p^m$ elements, denoted $\mathbb{F}_q$. Then
we can consider the map $F_q: k \rightarrow k$ defiend by $x \mapsto x^q$.
We then have the standard Frobenius map on affine space:
\[
k^n \rightarrow k^n \text{ given by } (x_1, \ldots, x_n) \mapsto (x_1^q, \ldots, x_n^q).
\]
One can then consider the fixed point set:
\[
V^{\mathbb{F}_q} = \{v \in V \mid F_q(v) = v \} = V \cap \mathbb{F}_q^n
\]
where $V \subseteq k^n$ is some closed subset. There's quite a bit of
work to move forward from here. (See Geck's book Ch. 4)
The last three definitions are as given in Geck's book referenced below.
A large amount of work, in this area, was done by Chevalley, Steinberg,
and Tits. I was introduced to the standard examples early in my
mathematical career. It was from my desire to understand a small part of
the classification of finite simple groups, that I sat down and learned
the methods behind the construction of the Suzuki groups. There is much
more to explore in the future!
Notes:
Here are some notes regarding the properties of semidirect products
Semidirect Products
The construction of the Suzuki groups of Lie Type.
Suzuki groups
References:
Alperin, J.L., Bell, Rowen, B., Groups and Representations ,
Graduate Texts in Mathematics, Springer, 1995.
Aschbacher, M., Finite Group Theory , Cambridge Studies
in Advanced Mathematics, Cambridge University Press, 2000.
Geck, Meinolf, An Introduction to Algebraic Geometry and
Algebraic Groups , Oxford Graduate Texts in Mathematics, Oxford
University Press, 2013.
Geck, Meinolf A First Guide to the Character Theory of Finite Groups
of Lie type , arXiv: https://arxiv.org/abs/1705.05083, retrieved
December 2017.
Tao, Terry, Notes on simple groups of Lie type ,
website: https://terrytao.wordpress.com/2013/09/05/notes-on-simple-gropus-of-lie-type/,
retrieved November 2017.