On the order of the schur multiplier of a pair of finite $p$-groups II

Document Type: Research Paper

Authors

1 Payame Noor University of Iran

2 Ferdowsi University of Mashhad

Abstract

‎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 $t=4,5$‎.

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