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a a a a above, above. also and and and and and and and and and and are asbasic be be by

clear conclude Consequently, contrary cotorsion

defined direct divisible divisible, divisible.

ET

lact follows follows follows follows For from from

G. GRÄBE group group group, group,

bave have hence hence

if in is is is is is is is is It it it it it

latter Lemma Let Let let

Moreover,

H=Z(p)H, X)Hom(tG, X/rPZ(p)X=pPZ(p). Ext(Z(p), X),pPZ(p)/XQ(n).X(Q/Z)QZ([2],

GHHXH, mpmρmPiBU. GB, GB. (tG)p, (tG)p. Ext(tG,

GExt(G, X)Ext(tG, X), tGPPtHom((tG)p, Q/Z) (tG)ptir((C(pi))mPi).

B(ρ)=B1...Bi... ; Bi=(C(pi))(mPi), GI|ρZ(p)mρip=1((C(pi))mPi),

X(Q/Z)pPHom((tG)ρ, x(Q/Z))Hom((tG)p, C(p))Z(p)mpII1((C(pi)mPi)i=

UΓIPZ(p)/X(pPZ(p)/pPZ(p))/(x/pPZ(p))

GExt(tG, X)pPHom((tG)p, Q/Z)Γ[Hom((tG)ρPC((p)).ion-trivial now

Df of of

133), 134), 178 2@2 255), 2.6 final (Io)([3], ([5], ([5],

p. p. p. p. 137) P. J. prove

rank recapitulate, reduced. reduced.

5hal1 shows Since since subgroup summand

that that that that that that that that that that the the the the the then then To totorsion-free torsion-free

VILOEN.

We We We we where which