# On Covers of Dihedral 2-Groups by Powerful Subgroups

## TL;DR

This paper determines the minimum number of powerful subgroups needed to cover dihedral 2-groups, advancing understanding of subgroup covers in finite p-groups.

## Contribution

It explicitly calculates the powerful covering number for dihedral 2-groups, a specific class of finite p-groups, which was previously unknown.

## Key findings

- The powerful covering number of dihedral 2-groups is established.
- Provides a method to compute covers by powerful subgroups in specific p-groups.
- Enhances understanding of subgroup structures in finite p-groups.

## Abstract

A finite $p$-group $G$ is called \textit{powerful} if either $p$ is odd and $[G,G]\subseteq G^p$ or $p=2$ and $[G,G]\subseteq G^4$. A {\em{cover}} for a group is a collection of subgroups whose union is equal to the entire group. We will discuss covers of $p$-groups by powerful $p$-subgroups. The size of the smallest cover of a $p$-group by powerful $p$-subgroups is called the \textit{powerful covering number}. In this paper we determine the powerful covering number of the dihedral 2-groups.