By R.B. Jr. Warfield

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**Example text**

That the condition prove the converse, complete if Z1 in its p-adic If {x n} that and G/Z 1 topology are complete modulo in Zl, in metric on Z1, G. 2 again) Remark. of G. shows that It can be removed if Theorem. 7. 4. subgroup has bounded order). -I pn such that XnXn+ 1 ~ G , then G, then an easy computation sequence follows it suffices and its torsion converges xx-1 n e G pn Z 1. 4. in z-lx. I do not has bounded order can be removed. is odd and the class of For a nilpotent group N, G is 2. the following proper- ties are equivalent: (i) pp(N) = {1} (ll) N pm = {1} (Ill) N such that, if pp(A) and and N/~p(N) N/N pn has a subnormal A is finite.

H and zn into the group to be a (Zn+l)P = Zn, and mapping p. for the equality of pp(G) = {1} by l o o k i n g pp(G) only at are in this direction. Theorem. If G is a nilpotent group and for some positive )pn p~ n, t p (G = {I}, then G = Op(a). Proof. , which we can see by mapping of roots of unity in the complex plane, xi if One should verify that the relations imply that be by generators if First assume that x ~ G and G x = yP, has no p-torslon. then y ~ Gp , In this case, from which the 48 result follows trivially.

Clearly, G = (G p ) and m m pp(G) pp(G p ), which implies our result, since Gp has no P p~ p-torsion. 7. Corollary (Gruenberg torslon-free nilpotent every prime p. 8. Corollary. G is residually Proof. 9. subgroup and flnlte-~ Corollary. is G. G if Then G for nilpotent group, ~, then t~,(G) = {1}. 10. 6), and Lemma. group whose torsion p~(Z l) = {1}. The equivalence all primes be a nilpotent is residually ~-bounded. 6 and the fact that H. is a finitely generated is residually if and only if if and only if Proof.