Abstract Let N be a normal subgroup of a given group G, then (N,G) is called a pair of groups. If N and G/N are of orders pn and pm, respectively, then G. Ellis in 1998 showed that the order of the Schur multiplier of (N,G), denoted by M(N,G), is at most p \\2n(n+2m−1). In this talk, we present a necessary condition for the pair (N,G), when |M(N,G) | p 1 2n(n+2m−1)−t, where t = 0 or 1. Some more results in this context are also given.

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