12

R.VOREADOU

tion 2. Clearly, g(l®t)w satisfies all the hypotheses of Proposition 2, ex-

cept, possibly, for the last one. If it does not satisfy the last hypothe-

sis of Proposition 2, then, working as in the last part of the proof of

Proposition 2 (but with h(x,)~ in the role of the h in that proof), we

find that h(x')~ is of the form

([Q,M]SP)SN -^—^- ([F,G]JaEf)KH' f*B1 GBH' g* S ,

where x* is central, f* and g* are allowable, f* cannot be written in the

form and, via x*, [Q,M] is associated with a prime factor of E1 and

[F,G] is associated with a prime factor of P. It is obvious that no prime

factor of E' is constant. By Lemmas 3 and 4, Tf* and Tf are not of type

and Q and F are non-constant. Then h = g*(f*$l)x*xT and, clearly,

[Q,M] is associated with a prime factor of E1 via x*xT(x') = x* and [F,G]

is associated with a prime factor of P via x'(x*xT) . Since neither h nor

hT can be written in any of the forms "central", 8 or ir , it follows that

(hjh'JeW1 , so (h,hf)£W, a contradiction. So g(lKt)w satisfies the last

hypothesis of Proposition 2. So, by Proposition 2, g(lKt)w = r(pKl) and

then h(x')"1 = q(p®l) with Tp = Tf! and Tq = Tg'; by induction, (p,f)^

implies p = ff and (qjg')^ implies q=g'; so, using J12 and 0^5 as in a

similar previous situation, from the hypothesis (h,h')^3V we get h = hf. •

It is clear that (h,hf)£ JfQ implies |~ h = Th'e 1R0 , and (h,hf)e^ im-

plies Th = Th'eTJf, for the classes % and Tfl of [18,§1] (also given here

in §1 of Part Two). So Th = ThV^ implies (h,h')^, and therefore Theo-

rem 2 of [18] is a corollary of Theorem 5 proved above. However, Theorem 5

is of theoretical value only, until we give a finite test for deciding

whether (h,h')^9f . This will be done by using the results of Part TwoQ