8

T. A. SPRINGER

that f(Ua) = Ua, or that df(Xa) = Xa, (choosing Xa, properly). We then have

Ad(0(O)AV = a(t)Xa,, whencef(f)(a') = a. Then yj0)(a') =

(a')v,

as follows for

example from the equality lsa = say established in the proof of 1.8. This proves (i).

Let A be a basis of 0. One knows that the Ua with aeA together with T generate

G. So an isogeny 0 is completely determined by its restriction to T and to the

U^(a e A). Since/{$) determines T - 7", the only freedom one has when f(f) is

given, is in the choice of the isomorphisms Ua -^ Ua* {a e A). The assertion of (ii)

then readily follows.

2.6. Let 0 be a central isogeny of (G, T) onto (G\ 7"). Then Ker f lies in T. It is

a finite group isomorphic to Hom(A7Im/(0), Q*). Let p be the characteristic ex-

ponent of 0. Then this kernel is isomorphic to the /^-regular part of X/lmf(fa). It

follows that there is a factorization of $: G - i G/Ker ^ A G', where % is the

canonical homomorphism and where p is an isomorphism if/? = 1 and p is a purely

inseparable isogeny if/? 1 (i.e., such that p is an isomorphism of groups). Let t

be the Lie algebra of T; we have Ker(rf^) a t. Now t can be identified with

Xy ®z 0. It follows that Ker(^) is isomorphic to the kernel of /(0) v ® id:

Xy ®

z

Q - (X'Y ®z Q, which is isomorphic to ( ( A ^ X ^ T + Imftyy) ®FpQ.

Hence Ker(^) = 0 if and only if Coker(/(0)) (which is dual to Coker(/(0)))

has order prime to /?.

If p 1 then Ker(rf^) is a central restricted subalgebra of g, which is stable

under Ad(G). Let G/Ktr(dj) be the quotient of G by Kercfy? (see [3, p. 376].) It

follows that we can factor 0, G -^ G/Ker (dfa) -^* G', where a is the canonical

(central) isogeny of [loc. citj. These remarks imply that we can factorize f as

follows:

(1) G=G0 -SU Gx - S - , G2 - ^ - G5_! -^=i* Gs = G',

where j =

TTS_I

o... o ^0. Put fa =

TT5_I

° ••• °

TZ V

(/ ^ 1). Then G2 = G/Ker f, Gt+1

= GJKtTidfa) (1 | i ^ J - 1) and the AT,- are canonical isogenics.

Also, if

G -^-G'

G -*-?'

is a commutative diagram of isogenics, we can arrange the factorizations of ^ and

$ such that there is a diagram with commuting squares

G =

GQ

• Gi • • • Gs_i G5 = G'

(2

i . i i . i

G =

GQ

G\ • • • • Gs-i Gs = G'

Notice that the vertical arrows are uniquely determined once the first one is

given.

2.7.

LEMMA.

Let j and fa be central isogenies of(G, T) onto (G', T) and(G'l9 T{),

respectively. Assume that Imf(fa) = lmf(fa). Then{G\ T')and(G[9 T[) are isomor-

phic.