Mohammadzadeh, Fahimeh; Hokmabadi, Azam; Mashayekhy, Behrooz On the exponent of the Schur multiplier of a pair of finite \(p\)-groups. (English) Zbl 1284.20017 J. Algebra Appl. 12, No. 8, Article ID 1350053, 11 p. (2013). 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 G. Ellis. Its definition is functorial and involves the natural exact sequence \[ \begin{split} H_3(G)\to H_3(G/N)\to M(G,N)\to M(G)\to M(G/N)\to\\ \to N/[N,G]\to G/[G,G]\to G/([G,G]N)\to 0.\end{split} \] 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 \(G\). 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 A. Lubotzky and A. Mann [see J. Algebra 105, 484-505 (1987; Zbl 0626.20010)]. Reviewer: Francesco G. Russo (Palermo) Cited in 1 Document MSC: 20D15 Finite nilpotent groups, \(p\)-groups 20C25 Projective representations and multipliers 19C09 Central extensions and Schur multipliers 20J05 Homological methods in group theory 20D60 Arithmetic and combinatorial problems involving abstract finite groups Keywords:pairs of groups; Schur multipliers; homology of groups; finite \(p\)-groups; integral homology groups Citations:Zbl 0948.20026; Zbl 0626.20010 PDFBibTeX XMLCite \textit{F. Mohammadzadeh} et al., J. Algebra Appl. 12, No. 8, Article ID 1350053, 11 p. (2013; Zbl 1284.20017) Full Text: DOI arXiv References: [1] DOI: 10.1023/A:1008652316165 · Zbl 0948.20026 · doi:10.1023/A:1008652316165 [2] DOI: 10.1090/S0002-9947-01-02812-4 · Zbl 1032.20023 · doi:10.1090/S0002-9947-01-02812-4 [3] DOI: 10.1090/S0002-9939-1974-0352254-2 · doi:10.1090/S0002-9939-1974-0352254-2 [4] Kayvanfar S., Bull. Iranian Math. Soc. 26 pp 89– (2000) [5] DOI: 10.1016/0021-8693(87)90211-0 · Zbl 0626.20010 · doi:10.1016/0021-8693(87)90211-0 [6] Mashayekhy B., Bull. Iranian Math. Soc. 37 pp 235– (2011) [7] DOI: 10.1016/j.jalgebra.2006.06.035 · Zbl 1120.20034 · doi:10.1016/j.jalgebra.2006.06.035 [8] DOI: 10.4153/CJM-1960-039-x · Zbl 0093.24601 · doi:10.4153/CJM-1960-039-x This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.