By Tammo tom Dieck

This publication is a jewel– it explains very important, priceless and deep issues in Algebraic Topology that you simply won`t locate in different places, rigorously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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Then H coincides with the intersection of all the subgroups of ﬁnite index in G which contain H. Notice that this is equivalent to saying that every subgroup of G is closed in the proﬁnite topology on G. Groups with this property are often called subgroup separable. The residual ﬁniteness of polycyclic groups follows on taking H to be 1. 10 If G is abelian, then H G and G/H is ﬁnitely generated and abelian. The result in this case follows at once from the Structure Theorem for ﬁnitely generated abelian groups.

Whether Fit(G) need be nilpotent for any soluble group G with min −n remains an open question. The structure of soluble groups with min −n has been investigated in a number of articles by B. Hartley and D. McDougall. In Hartley (1977) it is shown that if G is a soluble group with min −n which has derived length d ≥ 2, then |G| ≤ ℵd−2 . In particular a metabelian group satisfying min − n is countable (McDougall 1970a). On the other hand, Hartley’s bound is sharp for d = 3, and thus soluble groups with min −n need not be countable.

I) If Z(G) has no non-trivial π-elements, then Zi (G)/Zi−1 (G) has no such elements. (ii) If Z(G) has ﬁnite exponent dividing e, then so does Zi (G)/Zi−1 (G). 19. Thus, in particular, a nilpotent group with torsion-free centre is torsion-free. Also, if a nilpotent group G has class c and Z(G) has exponent dividing e, then G has exponent dividing ec . In addition there is the following result. 21 A ﬁnitely generated nilpotent group whose centre is a π-group is itself a ﬁnite π-group, where π is any set of primes.

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