Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Finitely generated nilpotent groups with isomorphic finite quotients


Author: P. F. Pickel
Journal: Trans. Amer. Math. Soc. 160 (1971), 327-341
MSC: Primary 20F05
DOI: https://doi.org/10.1090/S0002-9947-1971-0291287-3
MathSciNet review: 0291287
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal{F}(G)$ denote the set of isomorphism classes of finite homomorphic images of a group $ G$. We say that groups $ G$ and $ H$ have isomorphic finite quotients if $ \mathcal{F}(G) = \mathcal{F}(H)$. In this paper we show that if $ G$ is a finitely generated nilpotent group, the finitely generated nilpotent groups $ H$ for which $ \mathcal{F}(G) = \mathcal{F}(H)$) lie in only finitely many isomorphism classes. This is done using some finiteness results from the theory of algebraic groups along with some heretofore unpublished results of A. Borel.


References [Enhancements On Off] (What's this?)

  • [1] L. Auslander and G. Baumslag, Automorphism groups of finitely generated nilpotent groups, Bull. Amer. Math. Soc. 73 (1967), 716-717. MR 0217168 (36:259)
  • [2] G. Baumslag, Notes on nilpotent groups, Conference Board Math. Sci., Regional Conference Ser. Math., no. 2, Amer. Math. Soc., Providence, R. I., 1971. MR 0283082 (44:315)
  • [3] N. Blackburn, Conjugacy in nilpotent groups, Proc. Amer. Math. Soc. 16 (1965), 143-148. MR 30 #3140. MR 0172925 (30:3140)
  • [4] A. Borel, Some finiteness properties of adele groups over number fields, Inst. Hautes Études Sci. Publ. Math. No. 16 (1963), 5-30. MR 34 #2578. MR 0202718 (34:2578)
  • [5] A. Borel and J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111-164. MR 31 #5870. MR 0181643 (31:5870)
  • [6] K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29-62. MR 19, 386. MR 0087652 (19:386a)
  • [7] P. Hall, Nilpotent groups, Canadian Mathematical Congress, Summer Seminar, University of Alberta, 1957.
  • [8] K. A. Hirsch, On infinite soluble groups. II, Proc. London Math. Soc. 44 (1938), 336-344.
  • [9] S. A. Jennings, The group ring of a class of infinite nilpotent groups, Canad. J. Math. 7 (1955), 169-187. MR 16, 899. MR 0068540 (16:899a)
  • [10] A. G. Kuroš, The theory of groups. Vol. 2, GITTL, Moscow, 1953; English transl., Chelsea, New York, 1956. MR 15, 501; MR 18, 188.
  • [11] R. C. Lyndon, Groups with parametric exponents, Trans. Amer. Math. Soc. 96 (1960), 518-533. MR 27 #1487. MR 0151502 (27:1487)
  • [12] A. I. Mal'cev, On a class of homogeneous spaces, Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 9-32; English transl., Amer. Math. Soc. Transl. (1) 9 (1962), 276-307. MR 10, 507. MR 0028842 (10:507d)
  • [13] C. C. Moore, Decomposition of unitary representations defined by discrete subgroups of nilpotent groups, Ann. of Math. (2) 82 (1965), 146-182. MR 31 #5928. MR 0181701 (31:5928)
  • [14] P. F. Pickel, Finitely generated nilpotent groups with isomorphic finite quotients, Bull. Amer. Math. Soc. 77 (1971), 216-219. MR 0269739 (42:4634)
  • [15] V. N. Remeslennikov, Conjugacy of subgroups in nilpotent groups, Algebra i Logika Sem. 6 (1967), no. 2, 61-76. (Russian) MR 36 #1545. MR 0218459 (36:1545)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20F05

Retrieve articles in all journals with MSC: 20F05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0291287-3
Keywords: Isomorphic finite quotients, arithmetic groups, algebraic groups, Lie algebras
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society