Last update: 1 August 2013
The main purpose of this section is to prove the following theorem:
Theorem 5 (Chevalley, Dickson): Let be an adjoint group and assume is simple indecomposable). If assume is not of type or If assume is not of type Then is simple.
Remark: The cases excluded in Theorem 5 must be excluded. If then has as a normal subgroup of index 2 if is of type respectively. If and is of type then is a normal subgroup of of index 2. Here denotes the alternating group.
A proof of Theorem 5 essentially due to Iwasawa and Tits will be given here in a sequence of lemmas.
Lemma 29: Let be a Chevalley group. If is a minimal expression as a product of simple reflections, then the group generated by and
We know by the minimality of the expression (see Appendix II.19 and II.22). Hence if then Thus, Since and since we may complete the proof by induction.
Lemma 30: If again is any Chevalley group, if is a subset of the set of simple roots, if is the group generated by all and if then
|(a)||is a group.|
|(b)||The groups so obtained are all distinct.|
|(c)||Every subgroup of containing is equal to one of them.|
Part (a) follows from (b) Suppose are distinct subsets of the set of simple roots, say Now and if Thus since simple roots are linearly independent. Hence, and since distinct elements of the Weyl group correspond to distinct double cosets. (c) Let be any subgroup containing Set We shall show Clearly, Since and we need only show implies to get Let a minimal expression of as a product of simple reflections. By Lemma 29, Hence, and
A group conjugate to some is called a parabolic subgroup of We state without proof some further properties of parabolic subgroups which follow from Lemma 29.
|(1)||No two are conjugate.|
|(2)||Each parabolic subgroup is its own normalizer.|
|(4)||is a group if and only if or is a simple reflection.|
Example: If then corresponds to a partition of the matrices into blocks with the diagonal blocks being square matrices. Clearly, there are possibilities for such partitions. is then the subset of of matrices whose subdiagonal blocks are zero.
Lemma 31: Let be simple and let be the adjoint Chevalley group. If is a normal subgroup of then
We first show Suppose and If then for some a contradiction. If then Since is adjoint, it has center 1, and with for some Hence and we are back in the first case.
We now prove the lemma. By Lemma 30(c), for some We must show contains all simple roots. Suppose it does not. Since we see Also since is indecomposable, we can find simple roots with and not orthogonal to Let then with Then by Lemma 25(b). Hence either or Now where Since is not a simple root and so that by Appendix II.20. Hence and are both expressions of minimal length. By Lemma 29, a contradiction. Thus, is the set of all simple roots and
Lemma 32: If and are as in Theorem 5, then the derived group of
Before proving Lemma 32, we first show that Theorem 5 follows from Lemmas 31 and 32. Let be a a normal subgroup of By Lemma 31, so Now equals its derived group and is solvable. Hence and
Instead of proving Lemma 32 directly, we prove the following stronger statement:
Lemma 32': If is as in Theorem 5 then holds in any group in which the relations (R) hold, in fact in which the relations:
Since is generated by the we must show that every We will do this in several steps, excluding as we proceed the cases already treated. The first step takes us almost all the way.
(a) Assume We may choose Then Since and are arbitrary, every
By (a) we may henceforth assume that the rank is at least 2 and that or 3. By the corollary to Lemma 15, we may write the right side of (A) as the factor with having been isolated. We will use the fact that with as in Theorem 1, the maximum number of times one can subtract from and still have a root.
(b) Assume that is a root which can be written so that no other positive integral combination of and is a root and Then as follows at once from (A) with This covers the following cases:
To see this we use the fact that all roots of the same length are congruent under the Weyl group, imbed in an appropriate root system based on a pair of simple roots, and use In all cases but the second cases in (2) and (3) this system can be chosen of type with and roots of the same length as while in those cases it can be chosen of type with and short roots.
Because of the exclusions in the theorem, this leaves the following cases:
(c) If (6) or (7) holds, then in both of these cases we can find roots so that all other roots positive integers) are long, and in (6) we can choose long and short, in (7) both short. Then belongs to by cases already treated, hence so does by (A).
(d) If (8) holds, then Choose roots with long, short, and Since because is short, our assertion will follow from in (A), hence from the next lemma.
Lemma 33: If and form a simple system of type with long and short, then
By Lemma 14, we have
Here and since is not a root. If we multiply this equation by exponentiate, observe that the three factors on the right side commute, and then shift the first of them to the left, we get Lemma 33.
The proof of Theorem 5 is now complete.
In the course of this discussion, we have established the following result.
Corollary: If is indecomposable and of rank and if is any root, then there exist roots and and a positive integer such that and in the relations (A) of Lemma 32'.
Corollary (To Theorem 5): If and is a Chevalley group based on then every solvable normal subgroup of is central and hence finite.
Since the center of a Chevalley group is always finite by Lemma 28(d), we need only prove the first statement. Also we may assume the adjoint group, since by Corollary 5 to Theorem 4', there is a homomorphism of onto with and has center 1. Now we may write where is the adjoint group corresponding to an indecomposable subsystem of By Theorem 5, each is simple. Thus any normal subgroup of is a product of some of the If it also is solvable, the product is empty and the subgroup is 1.
This is a typed excerpt of Lectures on Chevalley groups by Robert Steinberg, Yale University, 1967. Notes prepared by John Faulkner and Robert Wilson. This work was partially supported by Contract ARO-D-336-8230-31-43033.