Normal Subgroups of Powerful $p$ -groups

TL;DR

This paper proves that certain normal subgroups within powerful p-groups are necessarily powerfully nilpotent, extending known results to both odd primes and the case p=2.

Contribution

It establishes new conditions under which normal subgroups of powerful p-groups are powerfully nilpotent, including the case p=2.

Findings

Normal subgroups within G^p are powerfully nilpotent for odd p.

Analogous result for p=2 with N ≤ G^4.

Extends understanding of subgroup structure in powerful p-groups.

Abstract

In this note we show that if $p$ is an odd prime and $G$ is a powerful $p$-group with $N\leq G^{p}$ and $N$ normal in $G$, then $N$ is powerfully nilpotent. An analogous result is proved for $p=2$ when $N\leq G^{4}$.