On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic $p$-Groups

Abstract

In this article we show that if $\mathcal{V}$ is the variety of polynilpotent groups of class row $(c_1,c_2,...,c_s),{\mathcal{N}}_{c_1,c_2,...,c_s}$, and $G\cong {\mathbf{Z}}_{p^{{\alpha }_1}}\stackrel{n}{\ast }{\mathbf{Z}}_{p^{{\alpha }_2}}\stackrel{n}{\ast }...\stackrel{n}{\ast }{\mathbf{Z}}_{p^{{\alpha }_t}}$ is the $n$th nilpotent product of some cyclic $p$-groups, where $c_1\ge n$, ${\alpha }_1\ge {\alpha }_2\ge ...\ge {\alpha }_t$ 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 $G\cong {\mathbf{Z}}_p\stackrel{n}{\ast }{\mathbf{Z}}_p\stackrel{n}{\ast }...\stackrel{n}{\ast }{\mathbf{Z}}_p$ ($m$-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 $G\cong {\mathbf{Z}}_{p^{{\alpha }_1}}\stackrel{n_1}{\ast }{\mathbf{Z}}_{p^{{\alpha }_2}}\stackrel{n_2}{\ast }...\stackrel{n_{t-1}}{\ast }{\mathbf{Z}}_{p^{{\alpha }_t}}$, where $c_1\ge n_1\ge ...\ge n_{t-1}$. Finally a similar result is given for the $c$-nilpotent multiplier of $G\cong {\mathbf{Z}}_{p^{{\alpha }_1}}\stackrel{n}{\ast }{\mathbf{Z}}_{p^{{\alpha }_2}}\stackrel{n}{\ast }...\stackrel{n}{\ast }{\mathbf{Z}}_{p^{{\alpha }_t}}$ with the different conditions $n\ge c$ and $(q,p)=1$ for all primes $q$ less than or equal to $n+c.$

10 pages

References in corpus (1)

• Polynilpotent Multipliers of some Nilpotent Products of Cyclic Groups II