Abstract
Among the subjects we shall touch upon in this chapter there is that of extensions. We have already mentioned this problem in Chapter 2 when talking about Höolder’s program for the classification of finite groups. As we shall see, the solution proposed by Schreier in the years 1920’s allows a classification of the groups that are extensions of an abelian group A by a group π by means of equivalence classes of functions π×π → A. Unfortunately, one does not obtain a system of invariants for the isomorphism classes of the groups obtained in this way, because nonequivalent functions can yield isomorphic groups. No characterization of the isomorphism classes is known.
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Notes
- 1.
It is well known that the endomorphisms of an abelian group form a ring.
- 2.
Since (|N|, |E/N|) = 1 one of the two groups has odd order and therefore, by the Feit-Thompson theorem, already quoted in Chapter 5, is solvable. Hence, any two complements for N are always conjugate.
- 3.
By means of a result known as the “universal coefficient theorem”.
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© 2012 Springer-Verlag Italia
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Machì, A. (2012). Extensions and Cohomology. In: Groups. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2421-2_7
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DOI: https://doi.org/10.1007/978-88-470-2421-2_7
Publisher Name: Springer, Milano
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