Download FAMILIES OF SIMPLE GROUPS Today we showed that the groups

FAMILIES OF SIMPLE GROUPS
DR. ROSENBERG
Today we showed that the groups An , n ≥ 6, are all simple. All
but finitely many of the other finite simple groups also fall into infinite families, and these families generally consist of invertible matrices
over finite fields such as Fp (the integers mod p, p a prime). Later in
the course we will learn that there is a finite field Fq of order q = pr ,
r ∈ N+ , and that it’s unique up to isomorphism, and essentially everything we say about Fp will work for Fq as well, but for now we’ll focus
on Fp . A square matrix with entries in a field is invertible if and only if
its determinant is non-zero. The group of n×n invertible matrices over
Fp is denoted GLn (Fp ) or GL(n, Fp ). The letters GL stand for “general linear.” The determinant gives a surjective group homomorphism
det : GLn (Fp ) → F×
p (the multiplicative group of non-zero elements of
Fp ), and the kernel is called SLn (Fp ) or SL(n, Fp ). The letters SL
stand for “special linear.” The order of GLn (Fp ) can be computed as
follows. The first row of a matrix in GLn (Fp ) is an n-tuple of elements
of Fp , but it can’t be identically 0, so there are pn −1 possibilities. Once
this row is fixed, the second row can be any n-tuple of elements of Fp
that is not a multiple of the first row, so there are pn − p possibilities.
Similarly there are pn − p2 possibilities for the third row, and so on, so
|GLn (Fp )| = (pn − 1)(pn − p) · · · (pn − pn−1 ).
By the first isomorphism theorem, the order of SLn (Fp ) is this number
divided by p − 1. So the order of SL2 (Fp ) is
|SL2 (Fp )| =
(p2 − 1)(p2 − p)
= p(p − 1)(p + 1) = p3 − p.
p−1
The center of SLn (Fp ) consists of scalar matrices with determinant 1,
i.e., scalar matrices whose diagonal entries are an n-th root of unity in
Fp (if such roots exist). So for p = 2, GLn (F2 ) = SLn (F2 ) has trivial
center. What happens is that P SLn (Fp ), the quotient of SLn (Fp ) by
its center (the P stands for “projective”), is simple except for a finite
number of exceptional cases when n and p are too small. (For example,
GLn (F2 ) = SLn (F2 ) = P SLn (F2 ) has order 6 when n = 2, and is easily
Date: Sept. 30, 2013.
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DR. ROSENBERG
seen to be isomorphic to S3 , which is not simple. For p = 3 and n = 2,
P SL2 (F3 ) has order 12 and again is not simple.)
For the case n = 2, the simplicity is very easy to prove and was
already noted by Galois back in the early 19th century. A simple proof
is available in Paul Monsky, “Frobenius’ Result on Simple Groups of
Order (p3 − p)/2,” The American Mathematical Monthly, Vol. 120, No.
8 (October 2013), pp. 725–732. You can read it at
http://www.jstor.org/stable/10.4169/amer.math.monthly.120.08.725 .