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On a result of G. Baumslag. (English) Zbl 0707.20016

The result of G. Baumslag [Compos. Math. 29, 249-252 (1974; Zbl 0309.20007)] referred to in the title is “a recipe for constructing some surprising examples” of finitely generated residually finite groups with isomorphic finite images. Specifically Baumslag extends a cyclic group A of finite order m by infinite cyclic groups B or C which induce on A the same (finite cyclic) automorphism group of order n, say. The resulting extensions G and H, say, are not isomorphic if the generators b of B and c of C induce generators of the automorphism group that are neither equal nor inverse to each other. This requires n to be different from 1, 2, 3, 4, and 6. The authors of the paper under review show, in different notation, that Baumslag’s constructions yields all such pairs G, H of groups such that \(G\times Z\) is isomorphic to \(H\times Z\), where Z is an infinite cyclic group, and only such pairs. They do not address the question when G and H are themselves not isomorphic, as this is settled in Baumslag’s paper. They show that G and H then also have isomorphic automorphism groups.
The last six items of their list of references are not referred to in the text, but provide useful background reading.
{The name Hirshon is misspelled in this list, and the pagination of the 10th item is erroneous.}
Reviewer: B.H.Neumann

MSC:

20E26 Residual properties and generalizations; residually finite groups
20F05 Generators, relations, and presentations of groups
20E22 Extensions, wreath products, and other compositions of groups

Citations:

Zbl 0309.20007
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References:

[1] Baumslag, G. , Residually finite groups with the same finite images . Comp. Math. 29(3) (1974) 249-252. · Zbl 0309.20007
[2] Brigham, R.C. , On the isomorphism problem for just-infinite groups . Comm. Pure and Applied Math. XXIV (1971) 789-796. · Zbl 0217.07703 · doi:10.1002/cpa.3160240605
[3] Cohn, P.M. , The complement of a finitely generated direct summand of an abelian group , Proc. Amer. Math. Soc. 7 (1956) 520-521. · Zbl 0070.25701 · doi:10.2307/2032764
[4] Grunewald, F.J. , Pickel, P.F. and Segal, D. , Polycyclic groups with isomorphic finite quotients . Annals of Math. 111 (1980) 155-195. · Zbl 0431.20033 · doi:10.2307/1971220
[5] Grunewald, F.J. and Segal, D. , On polycyclic groups with isomorphic finite quotients . Math. Proc. Cambridge Phil. Soc. 84 (1978b) 235-46. · Zbl 0388.20033 · doi:10.1017/S0305004100055079
[6] Hirshorn, R. , On cancellation in groups . American Math. Monthly 76 (1969) 1037-1039. · Zbl 0186.03801 · doi:10.2307/2317133
[7] Pickel, P.F. , Finitely generated nilpotent groups with isomorphic finite quotients . Bull. Amer. Soc. 77 (1971)a 216-19. · Zbl 0209.32804 · doi:10.1090/S0002-9904-1971-12687-3
[8] Pickel, P.F. , Finitely generated nilpotent groups with isomorphic finite quotients . Trans. Amer. Math. Soc. 160 (1971b) 327-41. · Zbl 0235.20027 · doi:10.2307/1995809
[9] Pickel, P.F. , Nilpotent-by-finite groups with isomorphic finite quotients . Trans. Amer. Math. Soc. 183 (1973) 313-25. · Zbl 0273.20020 · doi:10.2307/1996471
[10] Pickel, P.F. , Metabelian groups with the same finite quotients . Bull. Austral. Math. Soc. 11 (1974) 115-20. · Zbl 0279.20025 · doi:10.1017/S0004972700043689
[11] Walker, E.A. , Cancellation in direct sums of groups , Proc. Amer. Math. Soc. 7 (1956) 898-902. · Zbl 0071.25203 · doi:10.2307/2033557
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