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 χ_c+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.

Key Words:

• Elementary abelian p-group
• Finite p-group
• Nilpotent multipliers

A.M.S. Classification 2000:

• 20D15
• 20E10
• 20K01

ACKNOWLEDGMENT

This research was in part supported by a grant from IPM.

Notes

^#Communicated by E. Zelmanov.