Omegas of Agemos in Powerful Groups

Abstract

In this note we show that for any powerful $p$-group $G$, the subgroup $\Omega_{i}(G^{p^{j}})$ is powerfully nilpotent for all $i,j\geq1$ when $p$ is an odd prime, and $i\geq1$, $j\geq2$ when $p=2$. We provide an example to show why this modification is needed in the case $p=2$. Furthermore we obtain a bound on the powerful nilpotency class of $\Omega_{i}(G^{p^{j}})$. We give an example to show that powerfully nilpotent characteristic subgroups of powerful $p$-groups need not be strongly powerful.