On the exponent of the Schur multiplier of a pair of finite \(p\)-groups. (Q2847700)

Given a finite group \(G\), a way of defining the Schur multiplier \(M(G)\) of \(G\) is by \(M(G)=H_2(G,\mathbb Z)\), or briefly by \(M(G)=H_2(G)\), that is, by the second homology group of \(G\) with integral coefficients. The notion of Schur multiplier \(M(G,N)\) of a pair \((G,N)\), where \(N\) is a normal subgroup of \(G\), was introduced [in Appl. Categ. Struct. 6, No. 3, 355-371 (1998; Zbl 0948.20026)] by \textit{G. Ellis}. Its definition is functorial and involves the natural exact sequence NEWLINE\[NEWLINE\begin{multlined} H_3(G)\to H_3(G/N)\to M(G,N)\to M(G)\to M(G/N)\to\\ \to N/[N,G]\to G/[G,G]\to G/([G,G]N)\to 0.\end{multlined}NEWLINE\]NEWLINE In particular, one can see that \(M(G,G)=M(G)\) when \(N=G\). The size of \(M(G,N)\) influences strongly the presence of complements in \(G\) and it is interesting to get information on \(M(G,N)\) in order to find results of structure for \(G\).NEWLINENEWLINE The main results of the present paper are Theorems 2.6, 2.7 and 3.11. Recall that the pair \((G,N)\) is nilpotent of class \(k\in\mathbb N\), if the iterated commutator \([N,{_kG}]=[\ldots [[N,\underbrace{G],G]\ldots G]}_{k-\mathrm{times}}\) is trivial. Theorem 2.7 shows that a nilpotent pair \((G,N)\) of class \(k\) with \(N\) of \(\exp(N)=p^e\) for some \(e\in\mathbb N\) has \(\exp(M(G,N))\) dividing \(p^{e+\lfloor\log_pk\rfloor (k-1)}\). Theorem 2.6 is a variation of this bound in terms of covering pairs. Finally, again Theorem 3.11 provides a restriction on \(\exp(M(G,N))\), but this time \((G,N)\) is a pair of finite \(p\)-groups in which \(N\) is powerfully embedded in \(G\) in the sense of \textit{A. Lubotzky} and \textit{A. Mann} [see J. Algebra 105, 484-505 (1987; Zbl 0626.20010)].