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Transactions of the American Mathematical Society

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Automorphisms of group extensions


Author: Charles Wells
Journal: Trans. Amer. Math. Soc. 155 (1971), 189-194
MSC: Primary 20.48
DOI: https://doi.org/10.1090/S0002-9947-1971-0272898-8
Erratum: Trans. Amer. Math. Soc. 172 (1972), 507.
MathSciNet review: 0272898
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Abstract: If $ 1 \to G\mathop \to ^\iota \Pi \mathop \to ^\eta 1$ is a group extension, with $ \iota $ an inclusion, any automorphism $ \varphi $ of

Let $ \overline \alpha :\Pi \to $ Out $ G$ be the homomorphism induced by the given extension. A pair $ (\sigma ,\tau ) \in {\rm {Aut }}\Pi \times {\rm {Aut }}G$ is called compatible if $ \sigma $ fixes $ \ker \overline \alpha $, and the automorphism induced by $ \sigma $ on $ \Pi \overline \alpha $ is the same as that induced by the inner automorphism of Out $ G$ determined by $ \tau $. Let $ C < {\rm {Aut }}\Pi \times {\rm {Aut }}G$ be the group of compatible pairs. Let $ {\rm {Aut (}}E;G{\rm {)}}$ denote the group of automorphisms of $ E$ fixing $ G$. The main result of this paper is the construction of an exact sequence

$\displaystyle 1 \to Z_\alpha ^1(\Pi ,ZG) \to \operatorname{Aut} (E;G) \to C \to H_\alpha ^2(\Pi ,ZG).$

The last map is not surjective in general. It is not even a group homomorphism, but the sequence is nevertheless ``exact'' at $ C$ in the obvious sense.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0272898-8
Keywords: Automorphism, extension, Schreier factor function, cohomology group
Article copyright: © Copyright 1971 American Mathematical Society

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