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On the Order of Nilpotent Multipliers of Finite p-Groups

ABSTRACT Let G be a finite p-group of the order p n . Berkovich (1991) proved that G is an elementary abelian p-group if and only if the order of its Schur multiplier, M(G), is at the maximum case. In this article, we first find the upper bound p χ c+1(n) for the order of the c-nilpotent multiplier of G, M (c) (G), where χc+1(i) is the number of basic commutators of weight c + 1 on i letters. Second, we obtain the structure of G, in an abelian case, where , for all 0 ≤ t ≤ n − 1. Finally, by putting a condition on the kernel of the left natural map of the generalized Stallings-Stammbach five–term exact sequence, we show that an arbitrary finite p-group with the c-nilpotent multiplier of maximum order is an elementary abelian p-group.