Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T09:31:44.668Z Has data issue: false hasContentIssue false
  • English
  • Français

Groups whose automorphisms are almost determined by their restriction to a subgroup

Published online by Cambridge University Press:  18 May 2009

Martin R. Pettet
Martin R. Pettet
Affiliation:
Department of Mathematics, University of Toledo, Toledo, Ohio 43606, 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.

The trivial observation that every automorphism of a group is determined by its restriction to a set of generators suggests the converse question: if X is a subset of a group G such that each automorphism of G is determined (or “almost” determined) by its restriction to X, to what extent is the structure of G governed by that of the subgroup which X generates? Is this subgroup in some sense necessarily “large” in G? If the index of the subgroup is used as a measure of largeness, then in the absence of additional hypotheses, the answer to the second question is generally “no”, the additive group of rationals with X = {1} being an obvious counterexample. (More confounding is the existence of uncountable torsion-free abelian groups for which inversion is the only non-trivial automorphism. See, for example, [2], [3], and [4].) However, under certain finiteness assumptions, it seems that some positive conclusions are obtainable. One such example will be considered here.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 28 , Issue 1 , January 1986 , pp. 87 - 93
Copyright
Copyright © Glasgow Mathematical Journal Trust 1986

References

REFERENCES

1.Baer, R., Finite extensions of abelian groups with minimum conditions, Trans. Amer. Math. Soc. 79 (1955), 521–540.CrossRefGoogle Scholar
2.Corner, A. L. S., Endomorphism algebras of large modules with distinguished submodules, J. Algebra 11 (1969), 155–185.CrossRefGoogle Scholar
3.Fuchs, L., The existence of indecomposable abelian groups of arbitrary power, Acta Math. Acad. Sci. Hungar. 10 (1959), 453–457.CrossRefGoogle Scholar
4.Groot, J. de, Indecomposable abelian groups, Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 (1957), 137–145.CrossRefGoogle Scholar
5.Robinson, D. J. S., Finiteness Conditions and Generalized Soluble Groups I (Springer-Verlag, 1972).CrossRefGoogle Scholar
6.Robinson, D. J. S., Splitting theorems for infinite groups, Symposia Mathematica, Vol. XVII (Academic Press, 1976), 441–470.Google Scholar
7.Robinson, D. J. S., A contribution to the theory of groups with finitely many automorphisms, Proc. London Math. Soc. (3) 35 (1977), 34–54.CrossRefGoogle Scholar
8.Robinson, D. J. S., A Course in the Theory of Groups (Springer-Verlag, 1982).CrossRefGoogle Scholar