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 $G\cong {\bf {Z}}_{p^{\alpha_1}} \stackrel{n} {*} {\bf {Z}}_{p^{\alpha_2}} \stackrel{n}{*} ... \stackrel{n} {*} {\bf{Z}}_{p^{\alpha_t} }$ is the $n$th nilpotent product of some cyclic $p$-groups, where $c_1\geq n$, $\alpha_1 \geq \alpha_2 \geq...\geq \alpha_t$ 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 $m=\sum _{i=1}^t \alpha_i$ and $d_m=\chi_{c_s+1} (...(\chi_{c_2+1} (\sum_{j=1}^n \chi_{c_1+j}(m)))...)$. Also, we extend the result to the multiple nilpotent product $G\cong {\bf {Z}}_{p^{\alpha_1}}\stackrel{n_1} {*} {\bf {Z}}_{p^{\alpha_2}} \stackrel{n_2}{*} ... \stackrel{n_{t-1}} {*} {\bf{Z}}_{p^{\alpha_t} }$, where $c_1\geq n_1 \geq...\geq n_{t-1}$. Finally a similar result is given for the $c$-nilpotent multiplier of $G\cong {\bf {Z}}_{p^{\alpha_1}} \stackrel{n} {*} {\bf {Z}}_{p^{\alpha_2}} \stackrel{n}{*}...\stackrel{n}{*}{\bf{Z}}_{p^{\alpha_t}}$ with the different conditions $n \geq c$ and $ (q,p)=1$ for all primes $q$ less than or equal to $n+c.$