On the Order of the Schur Multiplier of a Pair of Finite p-Groups (original) (raw)

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 pgroups is bounded by p 1 2 n(2m+n−1) and hence it is equal to p 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 ≤ 3. This paper is devoted to classification of pairs (G, N) when N has a normal complement in G and t = 4, 5.