Multipliers of disposition $p$-groups

Abstract

Let $p$ be a prime number and $c,d$ natural numbers. Up to isomorphism, there is a unique $p$-group $G_d^c$ of least order with rank $d$ and nilpotency class $c$ named disposition group. This group plays an important role in the construction of Galois extensions over number fields with given $p$-group as Galois group. Also, it has a central series with all factors being elementary. Since $G_1^c$ is abelian we consider $d\ge 2$. In this article, first, we determine the order of all its subgroups of lower central series and $n$-th center subgroups of $G_d^c$, $(n\in \mathbb{N})$. Then we deduce these groups are $n$- capable. Also, the structure of the $m$-nilpotent multiplier of $G_d^c$ is determined in two cases $m\ge c$ and $m\le c$. Finally, polynilpotent multiplier of disposition group of class row $(m_1,m_2,\dots ,m_t)$, when $m_1\le c$ is calculated.

arXiv comment7 pages