On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic \(p\)-Groups

• Published in: arXiv.org 2010-11
• Main Authors: Mashayekhy, Behrooz, Mohammadzadeh, Fahimeh
• Format: Article
• Language: English
• Subjects: Multipliers
• ISSN: 2331-8422

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 \(n+c.\)