Bounds for the order of the Schur multiplier of a pair of groups (Q6650382)

The notion of multiplier of a group \(G\) was introduced by \textit{I. Schur} in [J. für Math. 127, 20--50 (1904; JFM 35.0155.01)] (the review by Loewy should be read to understand the motivations for introducing this concept).\N\NLet \(G\) be a group and \(N\) a normal subgroup of \(G\). In [\textit{G. Ellis}, Appl. Categ. Struct. 6, No. 3, 355--371 (1998; Zbl 0948.20026)] the following generalization of the Schur multiplier for the pair \((N,G)\) is provided. If \(1 \rightarrow R \rightarrow F \rightarrow G \rightarrow1\) is a free presentation of \(G\) and \(N=S/R\), then \(\mathcal{M}(N,G) \simeq R \cap [S,F]/[R,F]\). If \(N = G\) then \(\mathcal{M}(G,G)=\mathcal{M}(G)\) is the usual Schur multiplier.\N\NIn the paper under review, the authors give some inequalities for the order of the Schur multiplier of a pair \((N,G)\). In particular (see Theorem 2.6), if \((N,G)\) is a pair of finite \(p\)-groups, then\N\[\N\big | \mathcal{M}(N,G) \big | \cdot \big | [N, G] \big | \leq \big |N \big|^{(d(N)+d(G/N))},\N\]\Nwhere \(d(X)\) is the minimal number of generators of a group \(X\).