## Lectures on Chevalley groups

Last update: 1 August 2013

## §4. Simplicity of $G$

The main purpose of this section is to prove the following theorem:

Theorem 5 (Chevalley, Dickson): Let $G$ be an adjoint group and assume $ℒ$ is simple $\text{(}\Sigma$ indecomposable). If $|k|=2,$ assume $ℒ$ is not of type ${A}_{1},{B}_{2},$ or ${G}_{2}\text{.}$ If $|k|=3,$ assume $ℒ$ is not of type ${A}_{1}\text{.}$ Then $G$ is simple.

Remark: The cases excluded in Theorem 5 must be excluded. If $|k|=2,$ then $G$ has ${𝒜}_{3},{𝒜}_{6},{SU}_{3}\left(3\right)$ as a normal subgroup of index 2 if $ℒ$ is of type ${A}_{1},$ ${B}_{2},$ ${G}_{2}$ respectively. If $|k|=3$ and $ℒ$ is of type ${A}_{1},$ then ${𝒜}_{4}$ is a normal subgroup of $G$ of index 2. Here $𝒜$ 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 $G$ be a Chevalley group. If $w\in W,$ $w={w}_{\alpha }{w}_{\beta }\dots$ is a minimal expression as a product of simple reflections, then ${w}_{\alpha },{w}_{\beta },\dots \in {G}_{1},$ the group generated by $B$ and $wB{w}^{-1}\text{.}$

 Proof. We know ${w}^{-1}\alpha <0$ by the minimality of the expression (see Appendix II.19 and II.22). Hence if $\beta =-{w}^{-1}\alpha >0,$ then ${G}_{1}\supseteq w{𝔛}_{\beta }{w}^{-1}={𝔛}_{w\beta }={𝔛}_{-\alpha }\text{.}$ Thus, ${w}_{\alpha }\in {G}_{1}\text{.}$ Since ${w}_{\alpha }wB{w}^{-1}{w}_{\alpha }^{-1}\subseteq {G}_{1}$ and since $\text{length} {w}_{\alpha }w<\text{length} w,$ we may complete the proof by induction. $\square$

Lemma 30: If $G$ again is any Chevalley group, if $\pi$ is a subset of the set of simple roots, if ${W}_{\pi }$ is the group generated by all ${w}_{\alpha },\alpha \in \pi ,$ and if ${G}_{\pi }=\underset{w\in {W}_{\pi }}{\cup }BwB,$ then

 (a) ${G}_{\pi }$ is a group. (b) The ${2}^{\ell }$ groups so obtained are all distinct. (c) Every subgroup of $G$ containing $B$ is equal to one of them.

 Proof. Part (a) follows from $BwB·B{w}_{\alpha }B\subseteq Bw{w}_{\alpha }B\cup BwB\text{.}$ (b) Suppose $\pi ,\pi \prime$ are distinct subsets of the set of simple roots, say $\alpha \in \pi \prime ,\alpha \notin \pi \text{.}$ Now ${w}_{\alpha }\alpha =-\alpha$ and $w\alpha =\alpha +\underset{\beta \in \pi }{\Sigma }{C}_{\beta }\beta$ if $w\in {W}_{\pi }\text{.}$ Thus ${w}_{\alpha }\alpha \ne w\alpha ,$ since simple roots are linearly independent. Hence, ${w}_{\alpha }\notin {W}_{\pi },$ ${W}_{\pi \prime }\ne {W}_{\pi },$ and ${G}_{\pi \prime }\ne {G}_{\pi }$ since distinct elements of the Weyl group correspond to distinct double cosets. (c) Let $A$ be any subgroup containing $B\text{.}$ Set $\pi =\left\{\alpha | \alpha \text{simple,} {w}_{\alpha }\in A\right\}\text{.}$ We shall show $A={G}_{\pi }\text{.}$ Clearly, $A\supseteq {G}_{\pi }\text{.}$ Since $G=\underset{w\in W}{\cup }BwB$ and $A\supseteq B,$ we need only show $w\in A$ implies $w\in {G}_{\pi }$ to get $A\subseteq {G}_{\pi }\text{.}$ Let $w\in A,$ $w={w}_{\alpha }{w}_{\beta }\dots ,$ a minimal expression of $w$ as a product of simple reflections. By Lemma 29, ${w}_{\alpha },{w}_{\beta },\dots \in A\text{.}$ Hence, $\alpha ,\beta ,\dots \in \pi ,$ $w\in {W}_{\pi },$ and $w\in {G}_{\pi }\text{.}$ $\square$

A group conjugate to some $G$ is called a parabolic subgroup of $G\text{.}$ We state without proof some further properties of parabolic subgroups which follow from Lemma 29.

 (1) No two ${G}_{\pi }\text{'s}$ are conjugate. (2) Each parabolic subgroup is its own normalizer. (3) ${G}_{\pi }\cap {G}_{\pi \prime }={G}_{\pi \cap \pi \prime }\text{.}$ (4) $B\cup BwB$ $\left(w\in W\right)$ is a group if and only if $w=1$ or $w$ is a simple reflection.

Example: If $G={SL}_{n},$ then $\pi$ corresponds to a partition of the $n×n$ matrices into blocks with the diagonal blocks being square matrices. Clearly, there are ${2}^{n-1}$ possibilities for such partitions. ${G}_{\pi }$ is then the subset of ${SL}_{n}$ of matrices whose subdiagonal blocks are zero.

Lemma 31: Let $ℒ$ be simple and let $G$ be the adjoint Chevalley group. If $N\ne 1$ is a normal subgroup of $G,$ then $NB=G\text{.}$

 Proof. We first show $N\not\subset B\text{.}$ Suppose $N\subseteq B$ and $1\ne x\in N,$ $x=uh,$ $u\in U,$ $h\in H\text{.}$ If $u\ne 1,$ then for some $w\in W,$ $wx{w}^{-1}\notin B,$ a contradiction. If $u=1,$ then $h\ne 1\text{.}$ Since $G$ is adjoint, it has center 1, and $h{x}_{\alpha }\left(t\right){h}^{-1}={x}_{\alpha }\left(t\prime \right)$ with $t\prime \ne t$ for some $t,t\prime \in k,$ $\alpha \in \Sigma \text{.}$ Hence $\left(h,{x}_{\alpha }\left(t\right)\right)={x}_{\alpha }\left(t\prime -t\right)\in N,$ ${x}_{\alpha }\left(t\prime -t\right)\ne 1,$ and we are back in the first case. We now prove the lemma. By Lemma 30(c), $NB={G}_{\pi }$ for some $\pi \text{.}$ We must show $\pi$ contains all simple roots. Suppose it does not. Since $N⊈B,$ we see $\pi \ne \varnothing \text{.}$ Also since $\Sigma$ is indecomposable, we can find simple roots $\alpha ,\beta$ with $\alpha \in \pi ,$ $\beta \notin \pi$ and $\alpha$ not orthogonal to $\beta \text{.}$ Let ${b}_{1}{w}_{\alpha }{b}_{2}\in N,$ ${b}_{i}\in B,$ then $b{w}_{\alpha }\in N$ with $b={b}_{2}{b}_{1}\in B\text{.}$ Then ${w}_{\beta }b{w}_{\alpha }{w}_{\beta }^{-1}\in N\cap \left(B{w}_{\alpha }{w}_{\beta }B\cup B{w}_{\beta }{w}_{\alpha }{w}_{\beta }B\right)$ by Lemma 25(b). Hence either ${w}_{\alpha }{w}_{\beta }\in {W}_{\pi }$ or ${w}_{\beta }{w}_{\alpha }{w}_{\beta }\in {W}_{\pi }\text{.}$ Now ${w}_{\beta }{w}_{\alpha }{w}_{\beta }=w\gamma ,$ where $\gamma ={w}_{\beta }\alpha =\alpha -⟨\alpha ,\beta ⟩\beta \text{.}$ Since $⟨\alpha ,\beta ⟩\ne 0,\gamma$ is not a simple root and $N\left({w}_{\beta }{w}_{\alpha }{w}_{\beta }\right)\ne 1,$ so that $N\left({w}_{\beta }{w}_{\alpha }{w}_{\beta }\right)\ge 3$ by Appendix II.20. Hence ${w}_{\alpha }{w}_{\beta }$ and ${w}_{\alpha }{w}_{\beta }{w}_{\alpha }$ are both expressions of minimal length. By Lemma 29, ${w}_{\beta }\in {W}_{\pi },$ a contradiction. Thus, $\pi$ is the set of all simple roots and $NB={G}_{\pi }=G\text{.}$ $\square$

Lemma 32: If $ℒ$ and $G$ are as in Theorem 5, then $G=G\prime ,$ the derived group of $G\text{.}$

Before proving Lemma 32, we first show that Theorem 5 follows from Lemmas 31 and 32. Let $N\ne 1$ be a a normal subgroup of $G\text{.}$ By Lemma 31, $NB=G$ so $G/N\simeq B/B\cap N\text{.}$ Now $G/N$ equals its derived group and $B/B\cap N$ is solvable. Hence $G/N=1$ and $N=G\text{.}$

Instead of proving Lemma 32 directly, we prove the following stronger statement:

Lemma 32': If $ℒ$ is as in Theorem 5 then $G\prime =G$ holds in any group $G$ in which the relations (R) hold, in fact in which the relations:

 (A) $\left({x}_{\beta }\left(t\right),{x}_{\gamma }\left(u\right)\right)=\Pi {x}_{i\beta +j\gamma }\left({c}_{ij}{t}^{i}{u}^{j}\right)$ (B) ${h}_{\alpha }\left(t\right){x}_{\alpha }\left(u\right){h}_{\alpha }{\left(t\right)}^{-1}={x}_{\alpha }\left({t}^{2}u\right)$
hold.

Proof.

Since $G$ is generated by the ${𝔛}_{\alpha }\text{'s}$ we must show that every ${𝔛}_{\alpha }\subseteq G\prime \text{.}$ 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 $|k|\ge 4\text{.}$ We may choose $t\in {k}^{*},$ ${t}^{2}\ne 1\text{.}$ Then $\left({h}_{\alpha }\left(t\right),{x}_{\alpha }\left(u\right)\right)={x}_{\alpha }\left(\left({t}^{2}-1\right)u\right)\text{.}$ Since $\alpha$ and $u$ are arbitrary, every ${𝔛}_{\alpha }\subseteq G\prime \text{.}$

By (a) we may henceforth assume that the rank $\ell$ is at least 2 and that $|k|=2$ or 3. By the corollary to Lemma 15, we may write the right side of (A) as ${x}_{\beta +\gamma }\left({N}_{\beta ,\gamma }tu\right)·\Pi \prime ,$ the factor with $i=j=1$ having been isolated. We will use the fact $\left(*\right)$ that ${N}_{\beta ,\gamma }=±\left(r+1\right)$ with $r=r\left(\beta ,\gamma \right)$ as in Theorem 1, the maximum number of times one can subtract $\gamma$ from $\beta$ and still have a root.

(b) Assume that $\alpha$ is a root which can be written $\beta +\gamma$ so that no other positive integral combination of $\beta$ and $\gamma$ is a root and ${N}_{\beta ,\gamma }\ne 0\text{.}$ Then ${𝔛}_{\alpha }\subseteq G\prime ,$ as follows at once from (A) with $\Pi \prime =1\text{.}$ This covers the following cases:

 (1) If all roots have the same length: types ${A}_{\ell },$ ${D}_{\ell },$ ${E}_{\ell }\text{.}$ (2) ${B}_{\ell }$ $\left(\ell \ge 3\right),$ $\alpha$ long; ${B}_{2},$ $\alpha$ long, $|k|=3\text{.}$ (3) ${C}_{\ell }$ $\left(\ell \ge 3\right),$ $\alpha$ short; or $\alpha$ long and $|k|=3\text{.}$ (4) ${F}_{4}\text{.}$ (5) ${G}_{2},$ $\alpha$ long.

To see this we use the fact that all roots of the same length are congruent under the Weyl group, imbed $a$ in an appropriate root system based on a pair of simple roots, and use $\left(*\right)\text{.}$ In all cases but the second cases in (2) and (3) this system can be chosen of type ${A}_{2}$ with $\beta$ and $\gamma$ roots of the same length as $\alpha ,$ while in those cases it can be chosen of type ${B}_{2}$ with $\beta$ and $\gamma$ short roots.

$\square$

Because of the exclusions in the theorem, this leaves the following cases:

 (6) ${B}_{\ell }$ $\left(\ell \ge 2\right),$ $\alpha$ short. (7) ${G}_{2},$ $\alpha$ short, $|k|=3\text{.}$ (8) ${C}_{\ell }$ $\left(\ell \ge 3\right),$ $\alpha$ long, $|k|=2\text{.}$

(c) If (6) or (7) holds, then ${𝔛}_{\alpha }\subseteq G\prime \text{.}$ in both of these cases we can find roots $\beta ,\gamma$ so that $\alpha =\beta +\gamma ,$ all other roots $i\beta +j\gamma$ $\text{(}i,j$ positive integers) are long, and ${N}_{\beta \gamma }\ne 0\text{:}$ in (6) we can choose $\beta$ long and $\gamma$ short, in (7) both short. Then $\Pi \prime$ belongs to $G\prime$ by cases already treated, hence so does ${𝔛}_{\alpha },$ by (A).

(d) If (8) holds, then ${𝔛}_{\alpha }\subseteq G\prime \text{.}$ Choose roots $\beta ,\gamma$ with $\beta$ long, $\gamma$ short, and $\alpha =\beta +2\gamma \text{.}$ Since ${𝔛}_{\beta +\gamma }\subseteq G\prime$ because $\beta +\gamma$ is short, our assertion will follow from ${C}_{12}\ne 0$ in (A), hence from the next lemma.

Lemma 33: If $\beta$ and $\gamma$ form a simple system of type ${B}_{2}$ with $\beta$ long and $\gamma$ short, then $\left({x}_{\beta }\left(t\right),{x}_{\gamma }\left(u\right)\right)={x}_{\beta +\gamma }\left(±tu\right){x}_{\beta +2\gamma }\left(±t{u}^{2}\right)\text{.}$

 Proof. By Lemma 14, we have $xγ(u)Xβ xγ(u)-1= exp (ad uXγ) Xβ=Xβ+u Nγ,βXβ+γ +u2Nγ,β Nγ,β+γ/2 Xβ+2γ.$ Here ${N}_{\gamma ,\beta }=±1$ and ${N}_{\gamma ,\beta +\gamma }=±2$ since $\beta -\gamma$ is not a root. If we multiply this equation by $-t,$ 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. $\square$

The proof of Theorem 5 is now complete.

In the course of this discussion, we have established the following result.

Corollary: If $\Sigma$ is indecomposable and of rank $>1$ and if $\alpha$ is any root, then there exist roots $\beta$ and $\gamma$ and a positive integer $n$ such that $\alpha =\beta +n\gamma$ and ${c}_{1n}\ne 0$ in the relations (A) of Lemma 32'.

Corollary (To Theorem 5): If $|k|\ge 4$ and $G$ is a Chevalley group based on $k,$ then every solvable normal subgroup of $G$ is central and hence finite.

 Proof. 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 $G={G}_{0},$ the adjoint group, since by Corollary 5 to Theorem 4', there is a homomorphism $\phi$ of $G$ onto ${G}_{0}$ with $\text{ker} \phi \subseteq \text{center of} G$ and ${G}_{0}$ has center 1. Now we may write $G={G}_{1}·{G}_{2}\dots {G}_{r}$ where ${G}_{i}$ $i=1,2,\dots ,r$ is the adjoint group corresponding to an indecomposable subsystem of $\Sigma \text{.}$ By Theorem 5, each ${G}_{i}$ is simple. Thus any normal subgroup of $G$ is a product of some of the ${G}_{i}\text{'s.}$ If it also is solvable, the product is empty and the subgroup is 1. $\square$

## Notes and References

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.