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

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

2010-11-26

arxiv.org/abs/1011.5711v1

Journal of Advanced Research in Pure Mathematics, Vol. 1, Issue. 1, (2009), 39-48

Mathematics

Group Theory

10 pages

Scientific paper

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 $n+c.$

Affiliated with

Also associated with

No associations

Rating

On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic $p$-Groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic $p$-Groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic $p$-Groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-462528