Lectures on Chevalley groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 13 July 2013

§11. Some twisted Groups

In this section we study the group Gσ of fixed points of a Chevalley group G under an automorphism σ. We consider only the simplest case, in which σ fixes U,H,U-,N, hence acts on W=N/H and permutes the 𝔛α's. Before launching into the general theory, we consider some examples:

(a) G=SLn. If σ is a nontrivial graph automorphism, it has the form σx=ax-1a-1 (where x is the transpose of x and


εi=±1). We see that σ fixes x if and only if xax=a. If a is skew, we get Gσ=Spn. If a is symmetric, we get Gσ=SOn (split form). The group SO2n in characteristic 2 does not arise here, but it can be recovered as a subgroup of SO2n+1, namely the one "supported" by the long roots.

Let tt be an involutory automorphism of k having k0 as fixed field. If σ is now modified so that σx=ax-1a-1, then Gσ=SUn (split form). This last result holds even if k is a division ring provided tt is an anti-automorphism.

If V is the vector space over generated by the roots and W is the Weyl group, then σ acts on V and W and has fixed point subspaces Vσ and Wσ. Wσ is a reflection group on Vσ with the corresponding "roots" being the projection on Vσ of the original roots. To see these facts, we write n=2m+1 or n=2m and use the indices -m, -(m-1), , m-1, m with the index 0 omitted in case n=2m. If ωi is the weight on H defined by ωi:diag(a-m,,am)ai, then the roots are ωi-ωj (ij) and σωi=-ω-i. Vσ is thus spanned by {ωi=ω-i-ωi|i>0}. Now wWσ if and only if w commutes with σ, i.e., if and only if w(ωi-ωj)=ωk-ω implies w(ω-i-ω-j)=ω-k-ω-. We see that Wσ is the octahedral group acting on Vσ by all permutations and sign changes of the basis {ωi}. The projection of ωi-ωj (i,h0) on Vσ is 12(ωi-ω-i-ωj+ω-j)= 12(±ωk±ω) (k,>0) or ±ωk (k>0). If either i=0 or j=0 the projection of ωi-ωj is ±12ωk (k>0). The projected system is of type Cm if n=2m or BCm (a combination of Bm and Cm) if n=2m+1.

(b) G=SO2n (split form, char k2). We take the group defined by the form f=2Σi=1nxix-i. We will take the graph automorphism to be σx= a1ax-1 a-1a1-1

( a=[11], a1= [ 1 1 01 10 1 1 ] ) .

The corresponding form fixed by elements of Gσ is f= 2Σi=2nxix-i +x12+ x-12. Thus, Gσ fixes f-f=(x1-x-1)2 and hence the hyperplane x1-x-1=0. Gσ on this hyperplane is the group SO2n-1.

If we now combine σ with tt as in (a), the form f is replaced by f= Σi=2n ( xix-i+ x-ixi ) + x1x1+ x-1x-1. If we make the change of coordinates x1 replaced by x1+tx-1x-1 replaced by x1+tx-1 (tk,tt), we see that f is replaced by 2Σi=2nxix-i+ 2 ( x12+ax1 x-1+b x-12 ) and f is replaced by Σi=2n ( xix-i+ x-ixi ) + (2x1x1+ a(x1x-1 +x-1x1) +2bx-1x-1), where a=t+t and b=tt. Since these two forms have the same matrix, Gσ is SO2n over k0 re the new version of f. That is, Gσ is SO2n(k0) for a form of index n-1 which has index n over k.

Example: If n=4, k=, and k0=,Gσ is the Lorentz group (re f=x12-x22-x32-x42). If we observe that D2 corresponds to A1×A1, we see that SL2() and the 0-component of the Lorentz group are isomorphic over their centers. Thus, SL2() is the universal covering group of the connected Lorentz group.

Exercise: Work out D3A3 in the same way.

For other examples see E. Cartan, Oeuvres Complètes, No. 38, especially at the end.

Aside from the specific facts worked out in the above examples we should note the following. In the single root length case, the fixed point set of a graph automorphism yields no new group, only an imbedding of one Chevalley group in another (e.g. Spn or SOn in SLn). To get a new group (e.g. SUn) we must use a field automorphism as well.

Now to start our general development we will consider first the effect of twisting abstract reflection groups and root systems. Let V be a finite dimensional real Euclidean vector space and let Σ be a finite set of nonzero elements of V satisfying

(1) αΣ implies cαΣ if c>0, c1.
(2) wαΣ=Σ for all αΣ where wα is the reflection in the hyperplane orthogonal to α.
(See Appendix I). We pick an ordering on V and let P (respectively Π) be the positive (respectively simple) elements of Σ relative to that ordering. Suppose σ is an automorphism of V which permutes the positive multiples of the elements of each of Σ, P, and Π. It is not required that σ fix Σ, although it will if all elements of Σ have the same length. Let ρ be the corresponding permutation of the roots. Note that σ is of finite order and normalizes W. Let Vσ and Wσ denote the fixed points in V and W respectively. If α is the average of the elements in the σ-orbit of α, then (β,α)=(β,α) for all βVσ. Hence the projection of α on Vσ is α.

Theorem 32: Let Σ,P,Π,σ etc. be as above.

(a) The restriction of Wσ to Vσ is faithful.
(b) Wσ|Vσ is a reflection group.
(c) If Σσ denotes the projection of Σ on Vσ, then Σσ is the corresponding "root system"; i.e., {wα|Vσ,αΣσ} generates Wσ|Vσ and wαΣσ=Σσ. However, (1) may fail for Σσ.
(d) If Πσ is the projection of Π on Vσ, then Πσ is the corresponding "simple system"; i.e. if multiples are cast out (in case (1) fails for Πσ), then Πσ is linearly independent and the positive elements of Σσ are positive linear combinations of elements of Πσ.


Denote the projection of V on Vσ by vv. This commutes with σ and with all elements of Wσ.

(1) If αΣ, then α0; indeed α>0 implies α>0. If α is positive, so are all vectors in the σ-orbit of α. Thus, their average α is also positive. If α<0, then α=-(-α)<0.

(2) Proof of (a). If wWσ, w1, then wα<0 for some root α>0. Thus, wα=wα<0 and α>0. So w|Vσ1.

(3) Let π be a ρ-orbit of simple roots, let Wπ be the group generated by all wα (απ), let Pπ be the corresponding set of positive roots, and let wπ be the unique element of Wπ so that wπPπ=-Pπ. Then wπWσ and wπ|Vσ=wα|Vσ for any root αPπ. To see this, first consider σwπσ-1Wπ. Since σwπσ-1Pπ=Pπ, then σwπσ-1=wπ by uniqueness, and wπWσ. Since ρ permutes the elements of π in a single orbit, the projections on Vσ of the elements of Pπ are all positive multiples of each other. It follows that if α is any element of Pπ, then wπα=-α. If vVσ with (v,α)=0, then 0=(v,β)=(v,β) for βπ. Hence wπv=v. Thus wπ|Vσ= wα|Vσ.

(4) If ν is a ρ-orbit of roots and wWσ then all elements of wν have the same sign. This follows from wσα=σwα for αΣ, wWσ.

(5) {wπ|π a ρ-orbit of simple roots} generates Wσ. Let wWσ with w1 and let α be a simple root such that wα<0. Let π be the ρ-orbit containing α. By (4), wPπ<0 (i.e., wβ<0 for all βPπ). Now wwπPπ>0 and wπ permutes the elements of P-Pπ. Hence, N(wwπ)=N(w)-N(wπ) (see Appendix 11.17). Using induction on N(w), we may thus show that w is a product of wπ's.

(6) If w0 is the element of W such that w0P=-P, then w0Wσ. This follows from σw0σ-1P=-P and the uniqueness of w0.

(7) {wPπ|wWσ,π a ρ-orbit of simple roots} is a partition of Σ. If the wPπ's are called parts, then α,β belong to the same part if and only if α=cβ for some c>0. To prove (7), we consider αΣ, α>0. Now w0α<0 and w0=w1w2wr where each wi=wπ for some ρ-orbit of simple roots π (by (5) and (6)). Choose i so that wi+1wrα>0 and wiwi+1wrα<0. If wi=wπ, then wi+1wrαPπ; i.e., α is in some part. Similarly, if α<0, α is in some part. Now assume α,β belong to the same part, say to wPπ. We may assume α,βPπ. Then α and β are positive multiples of each other, as has been noted in (3). Conversely, assume

8) Σ0 consists of all wα such that wW0 and α is a root whose support lies in a simple ρ-orbit. Now β has its support in π and hence so does β since σ maps simple roots not in π to positive multiples of simple roots not in π. We see then that βPπ, and that any part containing α also contains β. The parts are just the sets of β such that β=cα, c>0 and hence form a partition.

(8) {wα|wWσ,α has support in a ρ-orbit of simple roots}=Σσ.

(9) Parts (b) and (c) follow from (3), (5), and (8).

(10) Proof of (d). We select one root α from each ρ-orbit and form {α}. This set, consisting of elements whose supports in Π are disjoint, is independent since Π is. If α>0 then it is a positive linear combination of the elements of Π. Hence α is a positive linear combination of the elements of Πσ.

Remark: To achieve condition (1) for a root system, we can stick to the set of shortest projections in the various directions.


(a) For σ of order 2, W of type A2n-1, we get Wσ of type Cn. For W of type A2n, we get Wσ of type BCn.
(b) For σ of order 2, W of type Dn, we get Wσ of type Bn-1.
(c) For σ of order 3, W of type D4, we get Wσ of type G2. To see this let α,β,γ,δ be the simple roots with δ connected with α,β, and γ. Then α=13(α+β+γ), δ=δ and α,δ=-1, δ,α=-3, giving Wσ of type G2. Schematically: =
(d) For σ of order 2, W of type E6, we get Wσ of type F4. =
(e) For σ or order 2, W of type C2, we get Wσ of type A1.
(f) For σ or order 2, W of type G2, we get Wσ of type A1.
(g) For σ or order 2, W of type F4, we get Wσ of type 𝒟16 (the dihedral group of order 16). To see this let α β γ δ be the Dynkin diagram of F4, and σα=2δ, σβ=2γ. Since α=12(α+2δ), β=12(β+2γ), we have β,α= -1, α,β= -(2+2). This corresponds to an angle of 7π/8 between α and β. Hence Wσ is of type 𝒟16. Alternatively, we note that wαwβ makes six positive roots negative and that there are 24 positive roots in all, so that w0=(wαwβ)4. Hence, wα2= wβ2= (wαwβ)8 =1 and Wσ is of type 𝒟16. Note that this is the only case of those we have considered in which Wσ fails to be crystallographic (See Appendix V).

In (e), (f), (g) we are assuming that multiples have been cast out.

The partition of Σ in (7) above can be used to define an equivalence relation R on Σ by αβ if and only if α is a positive multiple of β where α is the projection of α on Vσ. Letting Σ/R denote the collection of equivalence classes we have the following:

Corollary: If Σ is crystallographic and indecomposable, then an element of Σ/R is the positive system of roots of a system of one of the following types:

(a) A1n n=1,2, or 3.
(b) A2 (this occurs only if Σ is of type A2n).
(c) C2 (this occurs if Σ is of type C2 or F4).
(d) G2.

Now let G be a Chevalley group over a field k of characteristic p. Let σ be an automorphism of G which is the product of a graph automorphism and a field automorphism θ of k and such that if ρ is the corresponding permutation of the roots then

(1) if ρ preserves lengths, then order θ= order ρ.
(2) if ρ doesn't preserve lengths, then pθ2=1 (where p is the map xxp).
(Condition (1) focuses our attention on the only interesting case. Observe that ρ=id., θ=id. is allowed. Condition (2) could be replaced by θ2=p thereby extending the development to follow, suitably modified, to imperfect fields k.) We know that p=2 if G is of type C2 or F4 and p=3 if G is of type G2. Recall also that

σxα(t)= { xρα (εαtθ) if|α| |ρα| xρα (εαtpθ) if|α|< |ρα| where

εα=±1 and εα=1 if ±α is simple. (See the proof of Theorem 29.)

Now σ preserves U,H,B,U-, and N, and hence N/HW. The action thus induced on W is concordant with the permutation ρ of the roots. Since ρ preserves angles, it agrees up to positive multiples with an isometry on the real space generated by the roots. Thus the results of Theorem 32 may be applied. Also we observe that if n is the order of ρ, then n=1,2, or 3, so that the length of each ρ-orbit is 1 or n.

Lemma 60: Πεα=1 over each ρ-orbit of length n.


Since σn acts on each 𝔛α (±α simple) as a field automorphism, it does so on all of G, whence the lemma.

Lemma 61: If αΣ/R, then 𝔛α,σ1.


Choose αa so that no βa can be added to it to yield another root. If the orbit of α has length 1, set x=xα(1) if εα=1, x=xα(t) with tk, t0 and t+tθ=0 if εα1. Then x𝔛α,σ. If the length is n, we set y=xα(1), then x=y·σy·σ2y over the orbit, and use Lemma 60.

Theorem 33: Let G,σ, etc. be as above.

(a) For each wWσ, the group Uw=Uw-1U-w is fixed by σ.
(b) For each wWσ, there exists nwNσ, indeed nwUσ,Uσ-, so that nwH=w.
(c) If nw (wWσ) is as in (b), then Gσ=wWσBσnwUw,σ with uniqueness of expression on the right.


(a) This is clear since U and w-1U-w are fixed by σ.

(b) We may assume that w=wπ for some ρ-orbit of simple π. By Lemma 61, choose x𝔛-a,σx1, where aΣ/R corresponds to π. Using Theorem 4' we may write x=unwv for some wW where uU, vUw, and nwH=w. Now x=σx=σu·σnwσv and by Theorem 4 and the uniqueness in Theorem 4', we have σw=w, σnw=nw, σu=u, and σv=v. Thus, nwUσ,Uσ-. Since w1, wWσ, and wWπ, we have wα<0 for some απ, wπ<0, and w=wπ.

(c) Let xGσ, say xBwB, Since σ(BwB)=BσwB we have wWσ. Choose nw as in (b) and write x=bnwv with bB and vUw. Applying σ we get bBσ and vUw,σ. Uniqueness follows from Theorem 4'.

Corollary: The conclusions of Theorem 33 are still valid if Gσ and Bσ are replaced by Gσ=Uσ,Uσ- and Bσ=GσBσ. Also since Bσ=UσHσ, we can replace Hσ by Hσ=GσHσ.

Lemma 62: Let a generically denote a class in Σ/R. Let S be a union of classes in Σ/R which is closed under addition and such that if aS then -aS. Then 𝔛S,σ=ΠαS𝔛a,σ with the product taken in any fixed order and there is uniqueness of expression on the right. In particular, Uσ=Πa>0𝔛α,σ and Uw,σ= Πa>0wa0 𝔛a,σ for all wWσ.


We arrange the positive roots in a manner consistent with the order of the a's; i.e., those roots in the first a are first, etc. Now 𝔛S=Πα>0𝔛α in the order just described and with uniqueness of expression on the right by Lemma 17. Hence 𝔛S=Πa>0𝔛a in the given order and again with uniqueness of expression on the right. The lemma follows by considering the fixed points of σ on both sides of the last equation.

Corollary: If a,b are classes in Σ/R with a±b, then (𝔛a,𝔛b) Π𝔛c, where the roots on the right are in the closed subsystem generated by a and b, those of a and b excluded. The condition on c can be stated alternately, in terms of Σσ, that c is in the interior of the (plane) convex cone generated by a and b.

Remark: The exact relations in the above corollary can be quite complicated but generally resemble those in the Chevalley group whose Weyl group is Wσ. For example, if G is of type A3 and σ is of order 2, say α β γ , a={β}, b={α,γ}, c={α+β,β+γ}, d={α+β+γ}, and if we set xα(t)=xβ(t) (tkθ), xb(u)= xα(u)xγ(uθ) (uk), and similarly for c and d, we get ( xa(t), xb(u) ) = xc(±tu) xd(±tuuθ). In C2, the corresponding relation is ( xa(t), xb(u) ) = xa+b(±tu) xa+2b(±tu2).

If G is of type X and σ is of order n, we say Gσ is of type Xn. E.g., the group considered in the above remark is of type A32. The group of type C22 is called the Suzuki group and the groups of type G22 and F42 are called Ree groups. We write GX and GσXn.

Lemma 63: Let a be a class in Σ/R, then 𝔛a,σ has the following structure:

(a) If aA1, then 𝔛a,σ={xα(t)|tkθ}
(b) If aA1n, then 𝔛a,σ= { x·σx|x= xα(t),αa, tk }
(c) If aA2, a={α,β,α+β}, then θ2=1 and 𝔛a,σ= { xα(t) xβ(tθ) xα+β(u) |ttθ+u +uθ=0 }
If (t,u) denotes the given element, then (t,u) (t,u) = ( t+t, u+u- tθt ) .
(d) If aC2, a={α,β,α+β,α+2β}, then 2θ2=1 and 𝔛a,σ= { xα(1) xβ(tθ) xa+2β(u) xα+β (t1+θ+uθ) |t,uk } . If (t,u) denotes the given element, (t,u) (t,u) = ( t+t, u+u+ t2θt ) .
(e) If aG2, a= { α,β,α+β,α+ 2β,α+3β,2α +3β } , then 3θ2=1 and 𝔛a,θ= {xα(t) xβ(tθ) xα+3β(u) xα+β(uθ-t1+θ) x2α+3β(v) xα+2β(vθ-t1+2θ) |t,u,v k}. If (t,u,v) denotes the given element then (t,u,v) (t,u,v)= ( t+t, u+u+t t3θmv+ v-tu+ t2t3θ ) .

Note that in (a) and (b),𝔛a,σ is a one parameter group for the fields kθ and k respectively.


(a) and (b) are easy and we omit their proofs. For (c), normalize the parametrization of 𝔛α+β so that Nα,β=1. Then σxα(t)=xβ(tθ), σxβ(t)=xα(tθ), and σxα+β(u)=xα+β(-uθ). Write x𝔛a,θ as x=xα(t)xβ(v)xα+β(u) and compare the coefficients on both sides of x=σx to get (c). The proof of (d) is similar to that of (c). For (e), first normalize the signs as in Theorem 28, and then complete the proof as in (c) and (d).

Exercise: Complete the details of the above proof.

Remark: The role of the group SL2 in the untwisted case is taken by the groups SL2(kθ), SL2(k), SU3(k,θ) (split form), the Suzuki group, and Ree group of type G2.

Exercise: Determine the structure of Hσ in the case G is universal.

Lemma 64: If G is universal, then Gσ is generated by Uσ and Uσ- except perhaps for the case GσG22 with k infinite.


Let Gσ=Uσ,Uσ- and let Hσ=HσGσ. By the corollary to Theorem 33, it suffices to show HσGσ; i.e., (*) Hσ=Hσ. Since G is universal, H is a direct product of {𝔥α|αsimple} (see the corollary to Lemma 28). These groups are permuted by σ exactly as the roots are. Hence it is enough to prove (*) when there is a single orbit; i.e., when Gσ is one of the types SL2, A22, C22, or G22. For SL2, this is clear.

(1) For xUσ-{1}, write x=u1nu2 with uiUσ-,i=1,2 and n=n(x)NGσ. Then Hσ is generated by {n(x)n(x0)-1|x0 a fixed choice of x}. To see this let Hσ be the group so generated. Consider Gσ= Uσ-Hσ Uσ-Hσ n(x0)Uσ-. This set is closed under multiplication by Uσ-. It is also closed under right multiplication by n(x0)-1. This follows from n(x0)-1= n(x0-1)= n(x0-1) n(x0)-1 n(x0) and n(x0) Uσ-n (x0)-1 =UσGσ since x= u1 (n(x)n(x0)-1) n(x0)u2 for xUσ-{1}. We see that Gσ=Gσ, whence Hσ=Hσ.

(2) If α and β are the simple roots of A2, C2, or G2 labeled as in Lemma 63 (c), (d), or (e) respectively, then Hσ is isomorphic to k* via the map φ:thα(t)hβ(tθ).

(3) Let λ be the weight such that λ,α=1, λ,β=0, let R be a representation of k (obtained from one of by shifting the coefficients to k) having λ as highest weight and let v+ be a corresponding weight vector. Let μ be the lowest weight of R and let v- be a corresponding weight vector. For xUσ-{1}, write xv-=f(x)v++ terms for lower weights. Then f(x)0 and Hσ is isomorphic under φ-1 in (2) to the subgroup m of k* generated by all f(x)f(x0)-1. To prove (3), let xUσ-{1} and write x=u1n(x)u2 as in (1). We see xv-=n(x)v++ terms for lower weights, so n(x)v-=f(x)v+ and n(x)n(x0)-1v+= f(x)f(x0)-1v+. If n(x)n(x0)-1= hα(t)hβ(tθ), then by the choice of λ,f(x)f(x0)-1=t (see Lemma 19 (c)). (3) then follows from (1).

(4) The case GσA22. Here f(x)=-uθ and m=k*. To see this, we note that the representation R of (3) in this case is R:ks3(k) and if x= xα(t) xβ(tθ) xα+β(u) then

x [ 1tu+ttθ 01tθ 001 ] .

Thus, f(x)=u+ttθ=-uθ by Lemma 63 (c). Thus, m is the group generated by ratios of elements (-uθ) of k* whose traces are norms (ttθ). Let uk*. If uθu, set u1=(u-uθ)-1, and if uθ=u, choose u1k* so that u1θ=-u1. Then uu1 and u1 are values of f (their traces are 0 or 1), so that um and m=k*.

(5) The case GσC22. Here f(x)=t2+2θ+u2θ+tu and m=k*. To see this, first note that since the characteristic of k is 2, there is an ideal in k "supported" by short roots. The representation R can be taken as k acting on this ideal, and v+=Xα+β while v-=X-α-β. Letting x= xα(t) xβ(tθ) xα+2β(u) xα+β (uθ+t1+θ) we can determine f(x). By taking t=0 in the expression for f(x) and writing v=(vθ)2θ, we see that m=k*.

(6) The case GσG22. Here f(x)= t4+6θ- u1+3θ- v2+ t3+3θu+ t1+3θu3θ +tv3θ- tuv. The group m is generated by all values of f for which (t,u,v)(0,0,0), and it contains k*2 and -1; hence m=k*, if k is finite. Here the representation R can be taken to be the adjoint representation on k, v+=X2α+3β, and v-=X-2α-3β. Letting x be as in Lemma 63 (e), and working modulo the ideal in k "supported" by the short roots, we can compute f(x). Setting t=u=0, we see that -v2m, hence -1m and k*2m. If k is finite m=k* follows from (*) -1k*2. To show (*), suppose t2=-1 with tk. Then t2θ=-1, so tθ=±t and tθ2=t. Since 3θ2-1, we see t=(tθ2)3=t3. But t3=t2t=-t, so t=0, a contradiction. This proves the lemma.

Corollary: If G is universal, then Gσ=Gσ and Hσ=Hσ except possibly for G22 with k infinite in which case Gσ/Gσ= Hσ/Hσ k*/m with m as in (6) above.


(a) It is not known whether m=k* always if GσG22. One can make the changes in variables vv+tu and then uu-t1+3θ to convert the form f in (6) to t4+6θ-u1+3θ-v2+t2u2+tv3θ. Both before and after this simplification the form satisfies the condition of homogenity: f(t,u,v)= t4+6θf ( 1,u/ t1+3θ,v/ t2+3θ ) ift0.
(b) A corollary of (3) above, is that the forms in (5) and (6) are definite, i.e., f=0 implies t=u(=v)=0. A direct proof in case f is as in (5) can be made as follows: Suppose 0=f(t,u)= t2+2θ+ u2θ+tu with one of t,u nonzero. If t=0, then u=0, so we have t0. We see f(t,u)= t2+2θf (1,u/t2θ+1) using 2θ2=1. Hence we may assume t=1. Thus, 1+u2θ+u=0 or (by applying θ) uθ=1+u. Hence uθ2=1+uθ=u and u=u2θ2=u2. Thus, u=0 or 1, a contradiction. A direct proof in case f is as (6) appears to be quite complicated.
(c) The form in (5) leads to a geometric interpretation of C22. Form the graph v=t2+2θ+u2θ+tu in k3 of the form f(x). Imbed k3 in P3(k) projective 3-space over k, by adding the plane at , and adjoin the point at in the direction (0,0,1) to the graph to obtain a subset Q of P3(k). Q is then an ovoid in P3(k); i.e.
(1) No line meets Q in more than two points.
(2) The lines through any point of Q not meeting Q again always lie in a plane.
The group C22 is then realized as the group of projective transformations of P3(k) fixing Q. For further details as well as a corresponding geometric interpretation of G22 see J. Tits, Séminaire Bourbaki, 210 (1960). For an exhaustive treatment of C22, especially in the finite case, see Luneberg, Springer Lecture Notes 10 (1965).

Theorem 34: Let G and σ be as above with G universal. Excluding the cases: (a) A22(4), (b) B22(2), (c) G22(3), (d) F42(2), we have that Gσ is simple over its center.

Sketch of proof.

Using a calculus of double cosets re Bσ, which can be developed exactly as for the Chevalley groups with Wσ in place of W and Σ/R (or Σσ (see Theorem 32)) in place of Σ, and Theorem 33, the proof can be reduced exactly as for the Chevalley groups to the proof of: Gσ=𝒟Gσ. If k has "enough" elements, so does Hσ by the Corollary to Lemma 64 and the action of Hσ on 𝔛a,σ can be used to show 𝔛a,σ𝒟Gσ. This takes care of nearly everything. If k has "few" elements then the commutator relations within the 𝔛a's and among them can be used. This leads to a number of special calculations. The details are omitted.

Remark: The groups in (a) and (b) above are solvable. The group in (c) contains a normal subgroup of index 3 isomorphic to A1(8). The group in (d) contains a "new" simple normal subgroup of index 2. (See J. Tits, "Algebraic and abstract simple groups," Annals of Math. 1964.)

Exercise: Center of Gσ=(Center ofG)σ.

We now are going to determine the orders of the finite Chevalley groups of twisted type. Let k be a finite field of characteristic p. Let a be minimal such that θ=pa (i.e., such that tθ=tpa for all tk). Then |k|=p2a for An2, Dn2, E62; |k|=p3a for D43; and |k|=p2a+1 for C22, F42, G22. We can write σxα(t)= xρα(εαtq(α)) where q(α) is some power of p less than |k|. If q is the geometric average of q(α) over each ρ-orbit then q=pa except when Gσ is of type C22, F42, or G22 in which case q=pa+1/2.

Let V be the real Euclidean space generated by the roots and let σ0 be the automorphism of V permuting the rays through the roots as ρ permutes the roots. Since σ0 normalizes W, we see that σ0 acts on the space I of polynomials invariant under W. Since σ0 also acts on the subspace of I of homogeneous elements of a given positive degree, we may choose the basic invariants Ij, j=1,,, of Theorem 27 such that σ0Ij=εjIj for some εj (here we have extended the base field to ). As before, we let dj be the degree of Ij, and these are uniquely determined. Since σ0 acts on V, we also have the set {ε0j|j=1,,} of eigenvalues of σ0 on V. We recall also that N denotes the number of positive roots in Σ.

Theorem 35: Let σ,q,N,εj, and dj be as above, and assume G is universal. We have

(a) |Gσ|= qNΠj (qdj-εj).
(b) The order of the corresponding simple group is obtained by dividing |Gσ| by |Cσ| where C is the center of G.

Lemma 65: Let σ,H,U, etc. be as above.

(a) |Uq|=qN, |Uw,σ|=qN(w).
(b) |Hσ|= Πj (q-ε0j).
(c) |Gσ|= qNΠ (q-ε0j) ΣwWσ qN(w).
where N(w) is the number of positive roots in Σ made negative by w.


(a) It suffices to show that |𝔛a,σ|=q|a| aΣ/R by Lemma 62. This is so by Lemma 63. (b) Let π be a ρ-orbit of simple roots. Since σhα(t)= hρα(tq(α)), the contribution to |Hσ| made by elements of Hσ "supported" by π is (Παπq(α)) -1=qm-1 if m=|π|. Since the ε0j's corresponding to π are the roots of the polynomial Xm-1, (b) follows. (c) This follows from (a), (b), and Theorem 33.

Corollary: Uσ is a p-Sylow subgroup.

Lemma 66: We have the following formal identity in t:

ΣwWσ tN(w)=Πj (1-εjtdj)/ (1-ε0jt)


We modify the proof of Theorem 26 as follows:

(a) σ there is replaced by σ0 here.
(b) Σ there is replaced by Σ0 here, where Σ0 is the set of unit vectors in V which lie in the same directions of the roots.
(c) Only those subsets π of Π fixed by σ0 are considered.
(d) (-1)π is now defined to be (-1)k where k is the number of σ0 orbits in π.
(e) W(t) is now defined to be ΣwWσ tN(w).

With these modifications the proof proceeds exactly as before through step (5). Steps (6)-(8) become:

(6') For πΠ, wW, let Nπ be the number of cells in K congruent to Dπ under W and fixed by wσ0. Then Σ(-1)πNπ=detw. (Hint: If V=Vwσ0 and K is the complex on V cut by K, then the cells of K are the intersections with V of the cells of K fixed by wσ0.)
(7') Let x be a character on W,σ0 and xπ the restriction of x to Wπ,σ0 induced up to W,σ0. Then Σ(-1)π xπ(wσ0) = x(wσ0)detw (wW).
(8') Let M be a W,σ0 module, let Iˆ(M) be the space of skew invariants under W, and let Iπ(M) be the space of invariants under Wπ. Then Σ(-1)π tr (σ0,Iπ(M)) =tr (σ0,Iˆ(M)) . The remainder of the proof proceeds as before.

Lemma 67: The εj's form a permutation of the ε0j's.


Set t=1 in Lemma 66. Then (*) 1 has the same multiplicity among the εj's as among the ε0j's. This is so since otherwise the right side of the expression would have either a root or a pole at t=1. Assume σ01, then either σ02=1 and all ε's not 1 are -1 or else σ03=1 and all ε's not 1 are cube roots of 1, coming in conjugate complex pairs since σ0 is real. Thus in all cases (*) implies the lemma.

Proof of Theorem 35.

(a) follows from Lemmas 65, 66, 67. Now let C be the center Gσ. Clearly CCσ. Using the corollary to Theorem 33 and an argument similar to that in the proof of Corollary 1(b) to Theorem 4', we see CHσH. Since H acts "diagonally," we have CC, hence C=Cσ, proving (b).

Corollary: The values of |Gσ| and |Cσ|= |Hom(L0/L1,k*)σ| are as follows:

Gσ εj's1 |Gσ| |Cσ| Chevalley group(σ=1) None (*)qNΠ(qdk-1) |Hom(L0/L1,k*)| An2(n2) -1ifdjis odd Replaceqdj-1byqdj-(-1)djin(*) Same change; i.e.(n+1,q+1) E62 Same asAn2 Same change asAn2 (3,q+1) Dn2 -1for onedj=n Replace oneqn-1byqn+1in(*) (4,qn+1) D43 ω,ω2fordj=4,4 q12(q2-1)(q6-1)(q8+q4+1) 1 C22 -1fordj=4 q4(q2-1)(q4+1) 1 G22 -1fordj=6 q6(q2-1)(q6+1) 1 F42 -1fordj=6,12 q24(q2-1)(q6+1)(q8-1)(q12+1) 1

Here ω denotes a primitive cube root of 1.

Proof (except for |Cσ|).

We consider the cases:

An2. We first note (*) -1Wσ0. To prove (*) we use the standard coordinates {ωi|1in+1} for An. Then σ0 is given by ωi-ωn+2-i. Since W acts via all permutations of {ωi}, we see -1Wσ0. Alternatively, since W is transitive on the simple systems (Appendix II.24), there exists w0W such that w0(-Π)=Π. Hence, -w0(-1)=1 or σ0; i.e., -1W or -1Wσ0. Since there are invariants of odd degree (di=2,3,), -1W. By (*) σ0 fixes the invariants of even degree and changes the signs of those of odd degree.

E62,D2n+12. The second argument to establish (*) in the case An2 may be used here, and the same conclusion holds.

Dn2 (n even or odd). Relative to the standard coordinates {vi|1in}, the basic invariants are the first n-1 elementary symmetric polynomials in {vi2} together with Πvi, and W acts via all permutations and even number of sign changes. Here σ0 can be taken to be the map vivi (1in-1), vn-vn. Hence, only the last invariant changes sign under σ0.

D43. The degrees of the invariants are 2, 4, 6, and 4. By Lemma 67, the εj's are 1, 1, ω, ω2. Since σ0 is real, ω and ω2 must occur in the same dimension. Thus, we replace (q4-1)2 in the usual formula by (q4-ω) (q4-ω2) =q8+q4+1.

C22, G22. In both cases the εj's are 1, -1 by Lemma 67. Since W,σ0 is a finite group, it fixes some nonzero quadratic form, so that εj=1 for dj=2.

F42. The degrees of the invariants are 2, 6, 8, 12 and the εj's are 1, 1, -1, -1. As before there is a quadratic invariant fixed by σ0. Consider I= Σαlong rootα8+ Σβshort root(2β)8. We claim that I is an invariant of degree 8 fixed by σ0 and there is a quadratic invariant fixed by σ0 which does not divide I. The first part is clear since W and σ0 preserve lengths and permute the rays through the roots. To see the second part, choose coordinates {vi|i=1,2,3,4} so that the long roots (respectively, the short roots) are the vectors obtained from 2v1, v1+v2+v3+v4 (respectively, v1+v2) by all permutations and sign changes. The quadratic invariant is v12+ v22+ v32+ v42. To show that this does not divide I, consider the sum of those terms in I which involve only v1 and v2 and note that this is not divisible by v12+v22. Hence, I can be taken as one of the basic invariants, and εj=1 if dj=8.

Remark: |C22| is not divisible by 3. Aside from cyclic groups of prime order, these are the only known finite simple groups with this property.

Now we consider the automorphisms of the twisted groups. As for the untwisted groups diagonal automorphisms and field automorphisms can be defined.

Theorem 36: Let G and σ be as in this section and Gσ the subgroup of G (or Gσ) generated by Uσ and Uσ-. Assume that σ is not the identity. Then every automorphism of Gσ is a product of an inner, a diagonal, and a field automorphism.

Remark: Observe that graph automorphisms are missing. Thus the twisted groups cannot themselves be twisted, at least not in the simple way we have been considering.

Sketch of proof.

As in step (1) of the proof of Theorem 30, the automorphism, call it φ, may be normalized by an inner automorphism so that it fixes Uσ and Uσ- (in the finite case by Sylow's theorem, in the infinite case by arguments from the theory of algebraic groups). Then it also fixes Hσ, and it permutes the 𝔛a's (a simple, aΣ/R; henceforth we write 𝔛a for 𝔛a,σ) and also the 𝔛-a's according to the same permutation, in an angle preserving manner (see step (2)) in terms of the corresponding simple system Πσ of Vσ. By checking cases one sees that the permutation is necessarily the identity: if k is finite, one need only compare the various |𝔛a|'s with each other, while if k is arbitrary further argument is necessary (one can, for example, check which 𝔛a's are Abelian and which are not, thus ruling out all possibilities except for A32, E62, and D43, and then rule out these cases (the first two together) by considering the commutator relations among the 𝔛a's). As in step (4) of the proof of Theorem 30, we need only complete the proof of our theorem when Gσ is one of the groups Ga=𝔛a,𝔛-a, in other words, when Gσ is of one of the types A1, A22, C22 or G22 (with C22(2) and G22(3) excluded, but not A1(2), A1(3), or A22(4)), which we henceforth assume. The case A1 having been treated in §10, we will treat only the other cases, in a sequence of steps. We write x(t,u) or x(t,u,v) for the general element of Uσ as given in Lemma 63 and d(s) for hα(s)hβ(sθ).

(1) We have the equations

d(s) x(t,u) d(s)-1 = x ( s2-θt, s1+θu ) inA22 = x ( s2-2θt, s2θu ) inC22 d(s) x(t,u,v) d(s)-1 = x ( s2-3θt, s-1+3θu, sv ) inG22.

This follows from the definitions and Lemma 20(c).

(2) Let U1,U2 be the subgroups of Uσ obtained by setting t=0, then also u=0. Then UσU1U2=1 is the lower central series Uσ (Uσ,Uσ) (Uσ,(Uσ,Uσ)) for Uσ if the type is A22 or C22, while UσU1U21 is if the type is G22.

Exercise: Prove this.

(3) If the case A22(4) is excluded, then d(s)x(t,) d(s)-1 = x(g(s)t,), with g:k*k* a homomorphism whose image generates k additively.


Consider A22. By (1) we have g(s)=s2-θ, so that g(s)=s for skθ. Since [k:kθ]=2, we need only show that g takes on a value outside of kθ. Now if g doesn't, then s2-θ=(s2-θ)θ so that s3kθ, for all sk*, whence we easily conclude (the reader is asked to supply the proof) that k has at most 4 elements, a contradiction. For C22 and G22 the proof is similar, but easier.

(4) The automorphism φ (of Gσ) can be normalized by a diagonal and a field automorphism to be the identity on Uσ/U1.


Since φ fixes Uσ, it also fixes U1, hence acts on Uσ/U1. Thus there is an additive isomorphism

f:kksuch that φx(t,)=x (f(t),).

By multiplying φ by a diagonal automorphism we may assume f(1)=1. Since φ fixes Hσ, there is an isomorphism i:k*k* such that φd(s)=d(i(s)). Combining these equations with the one in (3) we get

f(g(s)t)= g(i(s))f(t) for allsk*, tk.

Setting t=1, we get (*) f(g(s))=g(i(s)), so that f(g(s)t)=f(g(s))f(t). If the case A22(4) is excluded, then f is multiplicative on k by (3), hence is an automorphism. The same conclusion, however, holds in that case also since f fixes 0 and 1 and permutes the two elements of k not in kθ.

Our object now is to show that once the normalization in (4) has been attained φ is necessarily the identity.

(5) φ fixes each element of U1/U2 and U2, and also some wGσ which represents the nontrivial element of the Weyl group.


The first part easily follows from (2) and (4), then the second follows as in the proof of Theorem 33(b).

(6) If the type is C22 or G22, then φ is the identity.


Consider the type C22. From the equation (*) of (4) and the fact that f=1, we get g(s)=g(i(s)), i.e., s2-2θ=i(s)2-2θ, and then taking the 1+θth power, s=i(s); in other words φ fixes every d(s). By (4) and (5), φx(t,u)=x(t,u+j(t)) with j an additive homomorphism. Conjugating this equation by d(s)=φd(s), using (1), and comparing the new equation with the old, we get j(s2-2θt)= s2θj(t), and on replacing s by s1+θ, j(st)s1+2θj(t). Choosing s0,1, which is possible because C22(2) has been excluded, and replacing s by s+1 and by 1 and combining the three equations, we get (s+s2θ)j(t)=0. Now s+s2θ0, since otherwise we would have s+s2θ=(s+s2θ)2θ, then s=s2, contrary to the choice of s. Thus j(t)=0. In other words φ fixes every element of Uσ. If the type is G22 instead, the argument is similar, requiring one extra step. Since Gσ is generated by Uσ and the element w of (5), φ is the identity.

The preceding argument, slightly modified, barely fails for A22, in fact fails just for the smallest case A22(4). The proof to follow, however, works in all cases.

(7) If the type is A22, then φ is the identity.


Choose w as in (5) and, assuming u0, write wx(t,u)w-1=xnx with x,xUσ, nHσw. A simple calculation in SL3 shows that x=x(atu-1,*) for some ak* depending on w but not on t or u. (Prove this.) If now we write φx(t,u)= x(t,u+j(t)), apply φ to the above equation, and use (4) and (5), we get tu-1= t(u+j(t))-1, so that j(t)=0 and we may complete the proof as before.

It is also possible to determine the isomorphisms among the various Chevalley groups, both twisted and untwisted. We state the results for the finite groups, omitting the proofs.

Theorem 37: (a) Among the finite simple Chevalley groups, their twisted analogues, and the alternating groups 𝒜n (n5), a complete list of isomorphisms is given as follows.

(1) Those independent of k. C1B1A1 C2B2 D2A1×A1 D22 (A1×A1)2 A1 D3A3 D32 A32 A12(q2) A1(q)
(2) Bn(q) Cn(q) if q is even.
(3) Just six other cases, of the indicated orders. A1(4) A1(5) 𝒜5 60 A1(7) A2(2) 168 A1(9) 𝒜6 360 A3(2) 𝒜8 20160 A32(4) B2(3) 25920

(b) In addition there are the following cases in which the Chevalley group just fails to be simple.

The derived group of B2(2)𝒜6 360 G2(2) A22(9) 6048 G22(3) A1(8) 504 F42(2)

The indices in the original group are 2, 3, 2, 2, respectively.


(a) The existence of the isomorphisms in (1) and (2) is easy, and in (3) is proved, e.g., in Dieudonné (Can. J. Math. 1949). There also the first case of (b), considered in the form B2(2)S6 (symmetric group) is proved.
(b) It is natural to include the simple groups 𝒜n in the above comparison since they are the derived groups of the Weyl groups of type An-1 and the Weyl groups in a sense form the skeletons of the corresponding Chevalley groups. We would like to point out that the Weyl groups W(En) are also almost simple and are related to earlier groups as follows.

Proposition: We have the isomorphisms:

𝒟W(E6) B2(3) A32(4) 𝒟W(E7) C3(2) 𝒟W(E8)/C D4(2), withC the center, of order 2.


The proof is similar to the proof of S6=W(A5)B2(2) given near the beginning of §10.

Aside from the cyclic groups of prime order and the groups considered above, only 11 or 12 other finite simple groups are at present (May, 1968) known. We will discuss them briefly.

(a) The five Mathieu groups Mn (n=11,12,22,23,24). These were discovered by Mathieu about a hundred years ago and put on a firm footing by Witt (Hamburger Abh. 12 (1938)). They arise as highly transitive permutation groups on the indicated numbers of letters. Their orders are:

|M11|=7920= 8·9·10·11 |M12|=95040= 8·9·10·11·12 |M22|=443520= 48·20·21·22 |M23|=10200960= 48·20·21·22·23 |M24|=244823040= 48·20·21·22·23·24

(b) The first Janko group J1 discovered by Janko (J. Algebra 3 (1966)) about five years ago. It is a subgroup of G2(11) and can be represented as a permutation group on 266 letters. Its order is

|J1|=175560=11 (11+1) (113-1)= 19·20·21·22= 55·56·57.

The remaining groups were all uncovered last fall, more or less.

(c) The groups J2 and J21/2 of Janko. The existence of J2 was put on a firm basis first by Hall and Wales using a machine, and then by Tits in terms of a "geometry." It has a subgroup of index 100 isomorphic to 𝒟G2(2)A22(9), and is itself of index 416 in G2(4). The group J21/2 has not yet been put on a firm basis, and it appears that it will take a great deal of work to do so (because it does not seem to have any "large" subgroups), but the evidence for its existence is overwhelming. The orders are:

|J2|=604800 |J21/2|=50232960.

(d) The group H of D. Higman and Sims, and the group H of G. Higman. The first group contains M22 as a subgroup of index 100 and was constructed in terms of the automorphism group of a graph with 100 vertices whose existence depends on properties of Steiner systems. Inspired by this construction, G. Higman then constructed his own group in terms of a very special geometry invented for the occasion. The two groups have the same order, and everyone seems to feel that they are isomorphic, but no one has yet proved this. The order is:

|H|=|H| =44352000.

(e) The (latest) group S of Suzuki. This contains G2(4) as a subgroup of index 1782, and is contructed in terms of a graph whose existence depends on the imbedding J2G2(4). It possesses an involutory automorphism whose set of fixed points is exactly J2. Its order is:


(f) The group M of McLaughlin. This group is constructed in terms of a graph and contains A32(9) as a subgroup of index 275. Its order is:


Theorem 38: Among all the finite simple groups above (i.e., all that are currently known), the only coincidences in the orders which do not come from isomorphisms are:

(a) Bn(q) and Cn(q) for n3 and q odd.
(b) A2(4) and A3(2)𝒜8.
(c) H and H if they aren't isomorphic.

That the groups in (a) have the same order and are not isomorphic has been proved earlier. The orders in (b) are both equal to 20160 by Theorem 25, and the groups are not isomorphic since relative to the normalizer B of a 2-Sylow subgroup the first group has six double cosets and the second has 24. The proof that (a), (b) and (c) represent the only possibilities depends on an exhaustive analysis of the group orders which can not be undertaken here.

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.

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