6 R.VOREADOU
u:P -*- ([F,G]SPE)SPH , v:([Q,M]SPE)S(CEBRE) -*• E ,
z:CBR-^ (CHSRH)S(CESRE) , y:PHS(CHBRH) -^ H
and x" = ((lBv)Sy)s((lBu)Bz),where s is the obvious central with domain
([Q,M]S(([F,G]SPE)iSPH))!S((CHSRH)iS(CE!SRE))
and codomain
([F,G]Bl(([Q,M]SPE)S(CEiaRE)))(a(PH!S(CHlSRH)) .
The mate under |~(g(lBt)w) of any element of v(CpBRp) is the same as the
mate under [fn and is in v (M)+v(C)+v(R)+v(S) by the form of |~(g(l®t)w)
(as determined by Th) and in v(E)+v(F) by the form of g(lBt)w (as display-
ed with f" ), so it must be in v(CpBRp). Then the graph of the composite
([Q,M]BPE)B(CE®RE)— ^ E -^-F FBI
is of the form p.Sv for graphs \L: [Q,M]BPE —- F , v:CpBRp —-I. By Proposi-
tion B of [18],b f"v = mSn for allowable natural transformations
m:[Q,M]SPE -^ F , n:CE®RE - I . So f"v = b(mBn). Then
g(l®t)w = g"(ffIBl)x" = g"(fMBl)((lBv)®y)s((l®u)Bz)
= g"(lBy)(b(mBn)Bl)s((lBu)Bz) = g* (mBl)x*
with g*:G®(PH®(C®R)) —- S allowable and
x*:([Q,M]BP)B(CBR) —•- ([F,G]H([Q,M]BPE))B(PH®(CHR))
central. Then
h = g(lBt)w(lB(f®l)) = g*(lB(fBl))(mBl)x* ,
i.e. h is of the form
([Q,M]HP)BN - ^ ([F,G]BEf)HHf m m ^ GBH' -£-^S ,
where E' = [Q,M]SPE , H! = PH!8(CHR) , x* is central, g" = g*(IB(f®1)) and
m are allowable, m cannot be written in the form since £" (and there-
fore f'v = b(mBn) ) cannot be so written and, via x*, [Q,M] is associated
with a prime factor of E' and [F,G] is associated with a prime factor of P.
This contradicts the last hypothesis about h. So g(lBt)w satisfies the last
hypothesis of the present proposition. D
It is now natural to suspect that the method of [11] can be used to
prove that any two allowable natural transformations with (the same domain,
the same codomain and) the same graph are equal, provided that Proposition
2 is applicable to either one of them and in a way compatible with the in-
ductive arguments of [11]. So it seems natural to exclude the class of all
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