Some results on parafree groups
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- by Yael Roitberg PDF
- Trans. Amer. Math. Soc. 173 (1972), 315-339 Request permission
Abstract:
We obtain some theorems concerning parafree groups in certain varieties, which are analogs of corresponding theorems about free groups in these varieties. Our principal results are: (1) A normal subgroup $N$ of a parafree metabelian group $P$ of rank $\geqslant 2$ such that $N \cdot {\gamma _2}P$ has infinite index in $P$ is not finitely generated unless it is trivial. (2) If $x$ and $y$ are elements of a parafree group $P$ in any variety containing the variety of all metabelian groups which are independent modulo ${\gamma _2}P$, then the commutator $[x,y]$ is not a proper power.References
- Maurice Auslander and R. C. Lyndon, Commutator subgroups of free groups, Amer. J. Math. 77 (1955), 929–931. MR 75204, DOI 10.2307/2372606
- Gilbert Baumslag, Some aspects of groups with unique roots, Acta Math. 104 (1960), 217–303. MR 122859, DOI 10.1007/BF02546390
- Gilbert Baumslag, Some theorems on the free groups of certain product varieties, J. Combinatorial Theory 2 (1967), 77–99. MR 206077, DOI 10.1016/S0021-9800(67)80116-9
- Gilbert Baumslag, Groups with the same lower central sequence as a relatively free group. I. The groups, Trans. Amer. Math. Soc. 129 (1967), 308–321. MR 217157, DOI 10.1090/S0002-9947-1967-0217157-3
- Gilbert Baumslag, Groups with the same lower central sequence as a relatively free group. II. Properties, Trans. Amer. Math. Soc. 142 (1969), 507–538. MR 245653, DOI 10.1090/S0002-9947-1969-0245653-3
- Gilbert Baumslag, B. H. Neumann, Hanna Neumann, and Peter M. Neumann, On varieties generated by a finitely generated group, Math. Z. 86 (1964), 93–122. MR 169895, DOI 10.1007/BF01111331
- P. Hall, Some word-problems, J. London Math. Soc. 33 (1958), 482–496. MR 102540, DOI 10.1112/jlms/s1-33.4.482
- A. Karrass, W. Magnus, and D. Solitar, Elements of finite order in groups with a single defining relation, Comm. Pure Appl. Math. 13 (1960), 57–66. MR 124384, DOI 10.1002/cpa.3160130107
- Hanna Neumann, Varieties of groups, Springer-Verlag New York, Inc., New York, 1967. MR 0215899, DOI 10.1007/978-3-642-88599-0 O. Schreier, Die Untergruppen der freien Gruppen, Abh. Math. Sem. Univ. Hamburg 5 (1928), 161-183.
- Jean-Pierre Serre, Lie algebras and Lie groups, W. A. Benjamin, Inc., New York-Amsterdam, 1965. Lectures given at Harvard University, 1964. MR 0218496
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 173 (1972), 315-339
- MSC: Primary 20E10
- DOI: https://doi.org/10.1090/S0002-9947-1972-0313399-9
- MathSciNet review: 0313399