On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic $p$-Groups
In this article we show that if ${\cal V}$ is the variety of polynilpotent groups of class row $(c_1,c_2,...,c_s),\ {\mathcal N}_{c_1,c_2,...,c_s}$, and $G\cong{\bf {Z}}_{p^{α_1}}\stackrel{n}{*}{\bf {Z}}_{p^{α_2}}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{p^{α_t} }$ is the $n$th nilpotent product of some cyclic $p$-groups, where $c_1\geq n$, $α_1 \geq α_2 \geq...\geq α_t $ and $ (q,p)=1$ for all primes $q$ less than or equal to $n$, then $|{\mathcal N}_{c_1,c_2,...,c_s}M(G)|=p^{d_m}$ if and only if $G\cong{\bf {Z}}_{p}\stackrel{n}{*}{\bf {Z}}_{p}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{p }$ ($m$-copies), where $m=\sum _{i=1}^t α_i$ and $d_m=χ_{c_s+1}(...(χ_{c_2+1}(\sum_{j=1}^n χ_{c_1+j}(m)))...)$. Also, we extend the result to the multiple nilpotent product $G\cong{\bf {Z}}_{p^{α_1}}\stackrel{n_1}{*}{\bf {Z}}_{p^{α_2}}\stackrel{n_2}{*}...\stackrel{n_{t-1}}{*}{\bf{Z}}_{p^{α_t} }$, where $c_1\geq n_1\geq...\geq n_{t-1}$. Finally a similar result is given for the $c$-nilpotent multiplier of $G\cong{\bf {Z}}_{p^{α_1}}\stackrel{n}{*}{\bf {Z}}_{p^{α_2}}\stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{p^{α_t}}$ with the different conditions $n \geq c$ and $ (q,p)=1$ for all primes $q$ less than or equal to $n+c.$