In this article we show that if V is the variety of polynilpotent
groups of class row (c1β,c2β,...,csβ),Β Nc1β,c2β,...,csββ, and
Gβ ZpΞ±1βββnZpΞ±2βββn...βnZpΞ±tββ
is the nth nilpotent product of some cyclic p-groups, where c1ββ₯n,
Ξ±1ββ₯Ξ±2ββ₯...β₯Ξ±tβ and (q,p)=1 for all primes
q less than or equal to n, then β£Nc1β,c2β,...,csββM(G)β£=pdmβ if and only if Gβ ZpββnZpββn...βnZpβ (m-copies), where
m=βi=1tβΞ±iβ and dmβ=Οcsβ+1β(...(Οc2β+1β(βj=1nβΟc1β+jβ(m)))...). Also, we extend the result to the multiple nilpotent
product Gβ ZpΞ±1βββn1βZpΞ±2βββn2β...βntβ1βZpΞ±tββ, where c1ββ₯n1ββ₯...β₯ntβ1β. Finally a similar result is given
for the c-nilpotent multiplier of Gβ ZpΞ±1βββnZpΞ±2βββn...βnZpΞ±tββ
with the different conditions nβ₯c and (q,p)=1 for all primes q less
than or equal to n+c.Comment: 10 page