Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-02T21:08:01.070Z Has data issue: false hasContentIssue false
  • English
  • Français

Subgroups of free solvable groups

Published online by Cambridge University Press:  09 April 2009

Jacques Lewin  and
Tekla Lewin
Jacques Lewin
Affiliation:
Mathematics Department Syracuse UniversitySyracuse, N.Y. 13210, U.S.A.
Tekla Lewin
Affiliation:
Mathematics Department Syracuse UniversitySyracuse, N.Y. 13210, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A consequence of Schreier's formula is that if G is a subgroup of the free group F of rank n < 1 and rank G ≦ n, then G = F or G is of infinite index in F. However, if S is a free sovlvable group of derived length I < 1 and H is a subgroup of S which is free solvable of the same length, then the rank of H does not exceed the rank of S. These observations led G. Baumslag to conjecture that if H is of finite index in S then H = S. In fact, we have sharper results in two directions. If H and S are free solvable of the same length, not only is H of infinite index in S, but δ1−1(S)/δ1−1(H) is torsion-free. In another direction we need not assume that S is free solvable, only that s is torsion-free and of derived length l (l > 1) and that H is not cyclic. Thus Stallings' theorem [11] that a finitely generated torsionfree group with a free subgroup of finite index is itself free has an even stronger counterpart in the variety of groups solvable of length at most l (l > 1): a torsionfree group in that variety with a non-cyclic free subgroup of finite index coincides with this subgroup. The proof relies on the following theorem: If S is a free solvable group, J is the group of automorphisms of S which induce the identity on S/S', and I is the group of inner automorphisms of S, then J/I is torsion-free. The proofs of these theorems form the bulk of the first four sections.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 2 , May 1971 , pp. 145 - 160
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Bachmuth, S., ‘Automorphisms of free metabelian groups, Trans. Amer. Math. Soc. 118 (1965), 93–104.Google Scholar
[2]Baumslag, G., ‘Some subgroup theorems for free b-groups’, Trans. Amer. Math. Soc., 108 (1963), 516–525.Google Scholar
[3]Baumslag, G., ‘Some theorems on the free groups of certain product varieties’, J. of Combinatorial Theory 2 (1967), 77–99.Google Scholar
[4]Baumslag, G., and Taylor, Tekla, ‘The centre of groups with one defining relator’, Math. Annalen 175 (1968), 315–319.CrossRefGoogle Scholar
[5]Hall, P., ‘Finiteness conditions for soluble groups’, Proc. London Math. Soc. (3), 4 (1954), 419–436.Google Scholar
[6]Kaplansky, I., Commutative rings, Queen Mary College mathematics notes (Queen Mary College, London, 1966).Google Scholar
[7]Magnus, W., ‘On a theorem of Marshall Hall’, Ann. of Math. 40, (1939), 764–768.Google Scholar
[8]Neumann, Hanna, Varieties of groups (Springer-Varlag, New York Inc., 1967).CrossRefGoogle Scholar
[9]Northcott, D. G., An introduction to homological algebra, (Cambridge University Press, 1960)CrossRefGoogle Scholar
[10]Smel'kin, A. L., ‘Two notes on free soluble groups’, (Russian), Algebra i. Logika Seminar 6 (1967), 95–109.Google Scholar
[11]Stallings, J., ‘On torsion free groups with infinitely many ends’, Ann. of Math. (2) 88 (1968), 312–334.CrossRefGoogle Scholar
[12]Zariski, O., and Samuel, P., Commutative algebra, Vol. 1 (Van Nostrand, Princeton, 1968).Google Scholar