10
R.VOREADOU
is a pair of equals and then, by J{ 3 and 0V2, we have
(h,h») = (y(fHg)x,y(f'Bg')x)6^
contrary to the hypothesis. Therefore (f,ff)^3f and (g,g'){W and then
h = h1 .
If not both h and hf are central and none of h and hf is of the form
t r or B, it may be the case that one of h and hf, say h, is central and the
other is of type , hr = g(fBl)x, with x central and f,g allowable natu-
ral transformations. Then hx" is central and h'x = * g(fBl). Also, by
our observation, T and S have no constant prime factors in this case and,
by working directly as in the case "h central" of the proof of Proposition
2 (or of Proposition 7.8 of [11] with hx now replacing the h of [11]),
we find hx = q(pBl) with Tp = Tf, Tq = Tg. By induction, (p,f)/^)f im-
plies p = f, and (q,g)/*)V implies q = g. So, if at least one of (p,f) and
(q,g) is in Jf , then any one of (p,f) and (q,g) which is not in W is a
pair of equals and then, by 3 f 5 and Jfl, we have
(h,h') = (q(pBl)x,g(fBl)x)e T
contrary to the hypothesis. Therefore (p,f)^3K and (q,g)^ and then h = hT.
There remains the case where each one of h and hf is of type and
not of any one of the other types; h = g(fBl)x" and hT = g,(ftBl)xt
with central x" and x1 and allowable f,ff,g,g,# Then no prime factor of T
and S is constant, h(x') is a composite
([Q,M]BP)BN -g- ([B,C]BA)BD f81 CBD—i-»-S ,
with x = x"(x') , and h'fx1) is a composite
([Q,M]BP)BN ft81 MBN g' S ,
and rOCx1)"1) = r(h'(x')_1), (h(x!) " 1,h'(xf)_1) t^T (the latter by ^2
and the hypothesis (h,h,)^0r). Moreover, by Lemma 1, we may assume that
neither f nor f can be written in the form , and, since T and S have no
constant prime factors, it follows that no prime factor of P or of A is
constant. We consider cases according to the possible associations of
[Q,M] and [B,C] via x, following the pattern of the case of h being of type
in the proof of Proposition C of [18],now having h(x*)~ in the role
of the h of [18], rf1 in the role of £ and Tg' in the role of ^ .
Case (i). [Q,M] is associated with [B,C] via x. We work as in Case (i) in
the proof of Proposition C in [18]. Thus, in Subcase I, we arrive at
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