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Metabelian groups with the same finite quotients

  • P.F. Pickel (a1)
Abstract

Let F(G) denote the set of isomorphism classes of finite quotients of the group G. Two groups G and H are said to have the same finite quotients if F(G) = F(H). We construct infinitely many nonisomorphic finitely presented metabelian groups with the same finite quotients, using modules over a suitably chosen ring. These groups also give an example of infinitely many nonisomorphic split extensions of a fixed finitely presented metabelian group by a fixed finite abelian group, all having the same finite quotients.

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Copyright
COPYRIGHT: © Australian Mathematical Society 1974
References
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[1]Bass, Hyman, Algebraic K-theory (Benjamin, New York, Amsterdam, 1968).
[2]Bass, Hyman and Murthy, M. Pavaman, “Grothendieck groups and Picard groups of abelian group rings”, Ann. of Math. (2) 86 (1967), 16–73.
[3]Baumslag, Gilbert, “A finitely presented metabelian group with a free abelian derived group of infinite rank”, Proc. Amer. Math. Soc. 35 (1972), 61–62.
[4]Brigham, Robert C., “On the isomorphism problem for just-infinite groups”, Comm. Pure Appl. Math. 24 (1971), 789–796.
[5]Dyer, Joan Landman, “On the isomorphism problem for polycyclic groups”, Math. Z. 112 (1969), 145–153.
[6]Hall, P., “Finiteness conditions for soluble groups”, Proc. London Math. Soc. (3) 4 (1954), 419–436.
[7]Hall, P., “On the finiteness of certain soluble groups”, Proc. London Math. Soc. (3) 9 (1959), 595–622.
[8]Murthy, M. Pavaman and Pedrini, Claudio, K 0 and K 1 of polynomial rings”, Algebraic K-theory II, 109–121 (Proc. Conf. Seattle Research Center, Battelle Memorial Institute, 1972. Lecture Notes in Mathematics, 342. Springer-Verlag, Berlin, Heidelberg, New York, 1973).
[9]Pickel, Paul Frederick, “On the isomorphism problem for finitely generated torsion free nilpotent groups”, (PhD thesis, Rice University, Houston, Texas, 1970).
[10]Pickel, P.P., “Nilpotent-by-finite groups with isomorphic finite quotients”, Trans. Amer. Math. Soc. 183 (1973), 313–325.
[11]Ремесленников, B.H. [Remeslennikov, V.N.], “Сопряженность подгрупп в нильпотентных группах” [Conjugacy of subgroups in nilpotent groups”, Algebra i Logika 6 (1967), No. 2, 61–76.
[12]Remeslennikov, V.N., “Groups that are residually finite with respect to conjugacy”, Siberian Math. J. 12 (1971), 783–792.
[13]Zariski, Oscar and Samuel, Pierre, Commutative algebra, Volume I (Van Nostrand, Princeton, New Jersey; Toronto; New York; London; 1958).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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