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• W1996367700 abstract "I am grateful to Professor D. J. S. Robinson for pointing out that Lemma 2.1 and Theorem 2.3 are false. A counterexample is provided by Pontryagin's example of an aperiodic indecomposable abelian group of rank 2 with every non-zero element of type (0, 0, 0, …), which can be found in L. Fuchs's Abelian groups (Oxford 1960), pp. 151–152. There is also an error in the proof of Lemma 2.2, which is nevertheless correct, and the statements of Theorems 3.1 and 3.3 and Corollaries 2.5, 3.2 and 3.4 should be amended by the addition of the following: unless r = 1, when G is abelian. All the subsequent results of the paper depend on corollaries to Theorem 2.3 and alternative proofs are outlined below. Theorems 3.1 and 3.3 and their corollaries can be proved by adding the hypothesis that B is free abelian to Lemma 2.1 and that G is polycyclic in Lemma 2.2, Theorem 2.3 and Corollary 2.4. The given proofs then establish Corollary 2.5, which could alternatively be given a short direct proof. A modified form of Theorem 3.8 can be proved from the following result, which is used in place of Corollary 2.4. LEMMA 2.6. Let G be a completely infinite minimax group which is of abelian section rank r. Let k be the greatest integer less than or equal to r(l +log2 r). Then G is completely infinite of length at most k. Proof. Because G is aperiodic, by [7; vol. 2, 10.32, p. 169] there exists a nilpotent normal subgroup F of G such that G/F is abelian-by-finite. By Corollary 2.5, F is nilpotent of class at most r and by [1; 3.6, p. 606], F ɛ d where d is the derived length of F and hence d r log2 r. Because G is of abelian section rank r the aperiodic rank of F is at most r log2 r and so the aperiodic rank of G is at most r(1 + log2 r). Consequently, G ɛ k. Theorem 3.8 and Corollary 3.9 can now be corrected by replacing “at most r” by the following: at most k, where k is the greatest integer less than or equal to r(1 + log2 r). I wish to thank Dr. R. B. J. T. Allenby for his help with these corrections. Note added in proof. See also the review by D. J. S. Robinson, Math. Reviews, 51 (1976), *13042." @default.
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• W1996367700 date "1976-12-01" @default.
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• W1996367700 title "Residually Nilpotent Groups" @default.
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