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.
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