Geodesic
Groups whose automorphisms are almost determined by their restriction to a subgroup
Glasgow mathematical journal, Tome 28 (1986) no. 1, pp. 87-93

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Pettet, Martin R. Groups whose automorphisms are almost determined by their restriction to a subgroup. Glasgow mathematical journal, Tome 28 (1986) no. 1, pp. 87-93. doi: 10.1017/S0017089500006388
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