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)

 
 

 

Representations of free metabelian $ \mathcal{D}_\pi$-groups


Author: John F. Ledlie
Journal: Trans. Amer. Math. Soc. 153 (1971), 307-346
MSC: Primary 20.40
DOI: https://doi.org/10.1090/S0002-9947-1971-0276341-4
MathSciNet review: 0276341
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For $ \pi $ a set of primes, a $ {\mathcal{D}_\pi }$-group is a group $ G$ with the property that, for every element $ g$ in $ G$ and every prime $ p$ in $ \pi ,g$ has a unique $ p$th root in $ G$. Two faithful representations of free metabelian $ {\mathcal{D}_\pi }$-groups are established: the first representation is inside a suitable power series algebra and shows that free metabelian $ {\mathcal{D}_\pi }$-groups are residually torsion-free nilpotent; the second is in terms of two-by-two matrices and is analogous to W. Magnus' representation of free metabelian groups using two-by-two matrices. In a subsequent paper [12], these representations will be used to derive several properties of free metabelian $ {\mathcal{D}_\pi }$-groups.


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

  • [1] G. Baumslag, Some aspects of groups with unique roots, Acta. Math. 104 (1960), 217-303 MR 23 #A191. MR 0122859 (23:A191)
  • [2] -, Some remarks on nilpotent groups with roots, Proc. Amer. Math. Soc. 12 (1961), 262-267. MR 23 #A934. MR 0123609 (23:A934)
  • [3] -, Some subgroup theorems for free $ \mathfrak{B}$-groups, Trans. Amer. Math. Soc. 108 (1963), 516-525. MR 27 #4862. MR 0154919 (27:4862)
  • [4] -, Residual nilpotence and relations in free groups, J. Algebra 2 (1965), 271-282. MR 31 #3487. MR 0179239 (31:3487)
  • [5] -, A representation of the wreath product of two torsion-free abelian groups in a power series ring, Proc. Amer. Math. Soc. 17 (1966), 1159-1165. MR 34 #7623. MR 0207809 (34:7623)
  • [6] G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935), 433-454.
  • [7] S. N. Černikov, Periodic ZA-extensions of complete groups, Mat. Sb. 27 (69) (1950), 117-128. (Russian) MR 12, 156. MR 0036758 (12:156d)
  • [8] P. Hall, Nilpotent groups, Lecture Notes, Canadian Math. Congress, University of Alberta, 1957.
  • [9] G. Higman, Groups and rings having automorphisms without non-trivial fixed elements, J. London Math. Soc. 32 (1957), 321-334. MR 19, 633. MR 0089204 (19:633c)
  • [10] P. G. Kontorovič, Groups with a basis of partition. III, Mat. Sb. 22 (64) (1948), 79-100. (Russian) MR 9, 493. MR 0024435 (9:493b)
  • [11] A. G. Kuroš, The theory of groups, 2nd ed., GITTL, Moscow, 1953; English transl., Vol. II, Chelsea, New York, 1960. MR 15, 501; MR 22 #727. MR 0109842 (22:727)
  • [12] J. F. Ledlie, Properties of free metabelian $ {\mathcal{D}_n}$-groups (to appear).
  • [13] F. W. Levi, Ordered groups, Proc. Indian Acad. Sci. Sect. A. 16 (1942), 256-263. MR 4, 192. MR 0007779 (4:192b)
  • [14] T. MacHenry, Free metabelian $ {\mathcal{D}_\pi }$-groups: A construction, Ph.D. thesis, Adelphi University, Garden City, N. J., 1962.
  • [15] W. Magnus, Beziehungen zwischen Gruppen und Idealen in einem speziellen Ring, Math. Ann. 111 (1935), 259-280. MR 1512992
  • [16] -, On a theorem of Marshall Hall, Ann. of Math. (2) 40 (1939), 764-768. MR 0000262 (1:44b)
  • [17] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Pure and Appl. Math., vol. 13, Interscience, New York, 1966. MR 34 #7617.
  • [18] A. I. Mal'cev, Nilpotent torsion-free groups, Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 201-212. (Russian) MR 10, 507. MR 0028843 (10:507e)
  • [19] B. H. Neumann, Adjunction of elements to groups, J. London Math. Soc. 18 (1943), 4-11. MR 5, 58. MR 0008808 (5:58s)
  • [20] -, Special topics in algebra: Universal algebra, Lecture notes prepared by P. M. Neumann, Courant Inst. Math. Sci., New York University, 1962.
  • [21] H. Neumann, Varieties of groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37, Springer-Verlag, New York, 1967. MR 35 #6734. MR 0215899 (35:6734)
  • [22] J. J. Rotman, The theory of groups. An introduction, Allyn and Bacon, Boston, Mass., 1965. MR 34 #4338. MR 0204499 (34:4338)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20.40

Retrieve articles in all journals with MSC: 20.40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0276341-4
Keywords: Metabelian group, $ \mathcal{D}$-group, unique roots, power series algebra representation, matrix representation
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society