4
R.VOREADOU
g:B —- D and central x and y, is done as in [18], where we observe that
(now) the natural transformations, on which the induction hypothesis is
used, clearly satisfy the last hypothesis of the (present) proposition,
since h does so.
The case of h being of type IT is done as in [11]; that (in the nota-
tion of [11])
fa"1:
([Q,M]SP)S(NJSB) -^ C ,
on which the induction hypothesis is used, satisfies (as needed) the last
hypothesis of the present proposition, is proved as follows: Suppose fa
can be written in the form
([Q,M]HP)H(NHB) X" ([F,G]18E)HH
£"m
y GjaH
- 1 ^ C
where x", f", g" are allowable natural transformations,
xM
is central,
fM
cannot be written in the form and, via x", [Q,M] is associated with a
prime factor of E and [F,G] is associated with a prime factor of P. Let Pp
(resp.PH) be an iterated S-product of the prime factors of P associated
via xM with prime factors of E (resp.H). Similarly define Bp and BH. There
are obvious centrals
s:P -^ ([F,G]EPE)18PH , z:B-^BHSB£ , t:([Q,M]EPE)BBE -^ E
and
x" = ((lSt)Sl)u((lSs)18(lSz))
where u is the obvious central with domain
([Q,M]iS(([F,G]SPE)iSPH))S(Nia(BHSBE))
and codomain
([F,G]S(([Q,M]®PE)JSBE))B((PHEN)]SBH)
The mate under [ " (fa ) of any element of v(Bp) is the same as the mate un-
der rf" and is in v(M)+v(N)+v(B)+v(C) by the form of I " (fa" ) (as determined
by Th) and in v(E)+v(F) by the form of fa" (as displayed with f ), so
it must be in v(BF). Then the graph of the composite
-1
([Q,M]SPE)SBE ^ E -^U~F - FSI
is of the form f*Ky for graphs JLL:[Q,M]SPF—• F , v:Bp— ~ I . By Proposition
B of [18], b ft = mSn for allowable natural transformations n:Bp—v I
and m:[Q,M]SPE-^ F . So ft = b(mSn). (Note that this equality also holds
in case there are no prime factors of B associated with prime factors of E
via xM, when = I.) Then
Previous Page Next Page