Abstract. In 1998, G. Ellis defined the Schur multiplier of a pair (G, N ) of groups and mentioned that this notion is a useful tool for studying pairs of groups. In this paper, we characterize the structure of a pair of finite p-groups (G, N ) in terms of the order of the Schur multiplier of (G, N ) under some conditions.

In this article, we present an explicit formula for the cth nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most c ≥ 1) of the nth nilpotent product of some cyclic groups G = Z n * · · · n * Z n * Z r 1 n * · · · n * Z r t , (m-copies of Z), where r i+1 |r i for 1 ≤ i ≤ t − 1 and c ≥ n such that ( p, r 1 ) = 1 for all primes p less than or equal to n. Also, we compute the polynilpotent multiplier of the group G with respect to the polynilpotent variety N c 1 ,c 2 ,...,c t , where c 1 ≥ n.

The paper is devoted to finding a homomorphic image for the cnilpotent multiplier of the verbal product of a family of groups with respect to a variety V when V ⊆ N c or N c ⊆ V. Also a structure of the c-nilpotent multiplier of a special case of the verbal product, the nilpotent product, of cyclic groups is given. In fact, we present an explicit formula for the c-nilpotent multiplier of the nth nilpotent product of the group G = Z rt , where r i+1 divides r i for all i, 1 i t −1, and (p, r 1 ) = 1 for any prime p less than or equal to n + c, for all positive integers n, c.

In this paper, we find an upper bound for the exponent of the Schur multiplier of a pair (G, N) of finite p-groups, when N admits a complement in G. As a consequence, we show that the exponent of the Schur multiplier of a pair (G, N) divides exp (N) if (G, N) is a pair of finite p-groups of class at most p – 1. We also prove that if N is powerfully embedded in G, then the exponent of the Schur multiplier of a pair (G, N) divides exp (N).

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