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By R. Göbel, C. Metelli, A. Orsatti, L. Salce

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And A. B(a + m) • pa~. no reason to consider the combination a 8 so For valuated groups it is essential. 1. : n Moreover for each m a + m ?! and and for each a p + p-valuated group B n. there is a valuated p-group B such that 22 R. Hunter- F. Richman- E. Walker pmB(a) ~ 0 pnB(p) ~ 0, and, if PROOF. i i) If so holds. i) generator x holds, then ii) holds. pmB(a) .. pnpm-nB(a) c pnB(a+m-n) c then Now let be cyclic of order B i such that implies ii), ~ vp x • a + i; in particular so and the valuated p-group B m+l p p B(a) ii) holds.

We call a valuated group =A is an equivalence between F( A c B ) is a functor. F We know. from p-group is isomorphic to What remains to show is that if every the p-groups. inherits its values from the height function on every value-bounded P. study of and the category of value-bounded valuated it is clear that in the is finite. The functor P PROOF. A(A) the same as D homomorphism homomorphism from of valuated B I. to B But groups is that applied to some pair F A c B this (RW; Theorem 1] B, and A' c B' A A' ~ are in extends B' true because to P, a is an algebraically compact group, hence injective in the category of valuated groups (RW; Theorem 29].

L} L(N) = U L(N ) N =UN~ U {(f1 ,N1 ) some 'l'l v. I' L(N )) • 'l'l (f't\,N'I'l) is uniquely determined, -notice d), e) in 2)). 6) Let J~ = {(f'l'l,N'I'l): for some (f1 ,N1 ) (l(n), {(f 1 ,N 1 ):l~n}6J'\1 }, and we adopt the conventions of 4). 7) Usually we identify Ji and Ji (for i~~). ), 'tl" n is just such ~Rang(f). 4. ), ~ (B l (CIL) E cf >. non-decreasing. , element will "depend" on few chosen: The bounded below set \ CIL we case) As usual define increasing B~+i and ao£ is in the completion of B0 • (~)() members of J:• and a«.

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