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The Schur multiplier of a pair of groups. (English) Zbl 0948.20026

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.

The Schur multiplier of the pair $\left(G,N\right)$, where $N$ is a normal subgroup of $G$, is defined as a functorial Abelian group $ℳ\left(G,N\right)$ whose principal feature is a natural exact sequence

$\begin{array}{c}{H}_{3}\left(G,ℤ\right)\to {H}_{3}\left(G/N,ℤ\right)\to ℳ\left(G,N\right)\to ℳ\left(G\right)\to \hfill \\ \hfill \to ℳ\left(G/N\right)\to N/\left[N,G\right]\to {\left(G\right)}^{ab}\to {\left(G/N\right)}^{ab}\to 0\end{array}$

($ℳ\left(K\right)=ℳ\left(K,K\right)$). 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)].

##### MSC:
 20J05 Homological methods in group theory 19C09 Central extensions and Schur multipliers 20E22 Extensions, wreath products, and other compositions of groups