Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-27T08:56:21.328Z Has data issue: false hasContentIssue false
  • English
  • Français

A Criterion for automorphisms of certain groups to be inner

Published online by Cambridge University Press:  09 April 2009

Joan L. Dyer
Joan L. Dyer
Affiliation:
Institute for Advanced Study Princeton, New Jersey, U.S.A. Lehman CollegeC.U.N.Y. New York, 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.

Let R be a normal subgroup of the free group F, and set G = F/[R, R]. We assume that F/R is a torsion-free group which is either solvable and not cyclic, or has a non-trivial center and is not cyclic-by-periodic. Then any automorphism of G whose restriction to R/[R, R] is trivial is an inner automorphism, determined by some element of R/[R, R]. This result extends a theorem of Šmel'kin (1967).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 21 , Issue 2 , March 1976 , pp. 179 - 184
Copyright
Copyright © Australian Mathematical Society 1976

References

Auslander, L. and Schenkman, E. (1965), ‘Free groups, Hirsch-Plotkin radicals, and applications to geometry’, Proc. Amer. Math. Soc. 16, 784–788.CrossRefGoogle Scholar
Auslander, M. and Lyndon, R. C. (1955), ‘Commutator subgroups of free groups’, Amer. J. Math. 77, 929–931.CrossRefGoogle Scholar
Bachmuth, S. (1965), ‘Automorphisms of free metabelian groups’, Trans. Amer. Math. Soc. 118, 93–104.CrossRefGoogle Scholar
Fox, R. H. (1953), ‘Free differential calculus I’, Ann. of Math. (2) 57, 547–560.CrossRefGoogle Scholar
Magnus, W. (1939), ‘On a theorem of Marshall Hall’, Ann. of Math (2) 40, 764–768.CrossRefGoogle Scholar
Remeslennikov, V. N. and Sokolov, V. G. (1970), ‘Some properties of a Magnus embedding’, Algebra i Logika 9, 566–578. Translation: Algebra and Logic 9, 342–349.Google Scholar
Šmel'kin, A. L. (1967), ‘Two notes on free soluble groups’, Algebra i Logika 6, 95–109.Google Scholar