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

‎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 (۱۹۹۸) shows‎ ‎that the order of the Schur multiplier of such a pair (G,N) of finite p-groups is bounded‎ ‎by p^{\frac{۱}{۲}n(۲m+n-۱)} and hence it is equal to ‎ ‎p^{\frac{۱}{۲}n(۲m+n-۱)-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 ۳‎. ‎This paper is devoted to classification of pairs‎ ‎(G,N) when N has a normal complement in G and t=۴,۵‎.

Keywords:

Pair of groups , Schur multiplier , finite p-groups