Multipliers of disposition $p$-groups

TL;DR

This paper investigates the structure and properties of disposition $p$-groups, including subgroup orders, capability, and multipliers, with implications for Galois extensions and group theory classifications.

Contribution

It determines subgroup orders, capability, and multipliers of disposition $p$-groups, providing new insights into their algebraic structure and applications.

Findings

Subgroups of lower central series are explicitly characterized.

Disposition groups are shown to be $n$-capable.

The structure of $m$-nilpotent and polynilpotent multipliers is explicitly calculated.

Abstract

Let $p$ be a prime number and $c, d$ natural numbers. Up to isomorphism, there is a unique $p$-group $G^c_d$ 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 $G^c_1$ is abelian we consider $d\geq 2$. In this article, first, we determine the order of all its subgroups of lower central series and $n$-th center subgroups of $G^c_d$, $(n\in\mathbb N)$. Then we deduce these groups are $n$- capable. Also, the structure of the $m$-nilpotent multiplier of $G^c_d$ is determined in two cases $m\geq c$ and $m\leq c$. Finally, polynilpotent multiplier of disposition group of class row $(m_1,m_2,\ldots,m_t)$, when $m_1\leq c$ is calculated.