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Proceedings of the American Mathematical Society

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A characterization of inner automorphisms

Author: Paul E. Schupp
Journal: Proc. Amer. Math. Soc. 101 (1987), 226-228
MSC: Primary 20E36
MathSciNet review: 902532
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Abstract: It turns out that one can characterize inner automorphisms without mentioning either conjugation or specific elements. We prove the following

Theorem Let $ G$ be a group and let $ \alpha $ be an automorphism of $ G$. The automorphism $ \alpha $ is an inner automorphism of $ G$ if and only if $ \alpha $ has the property that whenever $ G$ is embedded in a group $ H$, then $ \alpha $ extends to some automorphism of $ H$.

References [Enhancements On Off] (What's this?)

  • [1] Charles F. Miller III and Paul E. Schupp, Embeddings into Hopfian groups, J. Algebra 17 (1971), 171–176. MR 0269728
  • [2] Paul E. Schupp, A survey of small cancellation theory, Word problems: decision problems and the Burnside problem in group theory (Conf. on Decision Problems in Group Theory, Univ. California, Irvine, Calif. 1969; dedicated to Hanna Neumann), North-Holland, Amsterdam, 1973, pp. 569–589. Studies in Logic and the Foundations of Math., Vol. 71. MR 0412289

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Article copyright: © Copyright 1987 American Mathematical Society