University of Isfahan International Journal of Group Theory 2251-7650 2 3 2013 09 01 On the order of the Schur multiplier of a pair of finite p-groups II 1 8 2007 10.22108/ijgt.2013.2007 EN Fahimeh Mohammadzadeh Payame Noor University of Iran Azam Hokmabadi Payame Noor University of Iran Behrooz Mashayekhy Ferdowsi University of Mashhad Journal Article 2012 04 13 ‎Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$ with‎ ‎$|N|=p^n$ and $|G/N|=p^m$‎. ‎A result of Ellis (1998) shows‎ ‎that the order of the Schur multiplier of such a pair $(G,N)$ of finite $p$-groups is bounded‎ ‎by $ p^{\frac{1}{2}n(2m+n-1)}$ and hence it is equal to $‎ ‎p^{\frac{1}{2}n(2m+n-1)-t}$ for some non-negative integer $t$‎. ‎Recently‎, ‎the authors have characterized the structure of $(G,N)$ when $N$ has a complement in $G$ and‎ ‎$t\leq 3$‎. ‎This paper is devoted to classification of pairs‎ ‎$(G,N)$ when $N$ has a normal complement in $G$ and $t=4,5$‎. https://ijgt.ui.ac.ir/article_2007_21cfdbdb330703297453c7ae8d688385.pdf