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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.
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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

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