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Zbl 0948.20026
Ellis, Graham
The Schur multiplier of a pair of groups. (English)
[J] Appl. Categ. Struct. 6, No.3, 355-371 (1998). ISSN 0927-2852; ISSN 1572-9095/e

In this article the author develops the theory of a Schur multiplier for pairs of groups and he shows that it leads to a more systematic treatment of a number of results on the usual multiplier. In several instances this treatment yields sharper results.\par The Schur multiplier of the pair $(G,N)$, where $N$ is a normal subgroup of $G$, is defined as a functorial Abelian group ${\cal M}(G,N)$ whose principal feature is a natural exact sequence $$\multline H_3(G,\bbfZ)\to H_3(G/N,\bbfZ)\to{\cal M}(G,N)\to{\cal M}(G)\to\\ \to{\cal M}(G/N)\to N/[N,G]\to(G)^{ab}\to(G/N)^{ab}\to 0\endmultline$$ (${\cal M}(K)={\cal M}(K,K)$). The definition is given via classifying spaces of groups but a purely homological algebraic definition and proof of the exactness of the above sequence is also given by the author [in Glasg. Math. J. 29, 13-19 (1987; Zbl 0631.20040)].
[Olympia Talelli (Athens)]

MSC 2000:: *20J05 Homological methods in group theory
19C09 Central extensions, etc.
20E22 Extensions and other compositions of groups

Keywords: Schur multipliers; pairs of groups; classifying spaces of groups

Citations: Zbl 0631.20040

Cited in: Zbl 1261.17010 Zbl 1180.20020 Zbl 1144.20017

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