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 $i,j\ge 1$ when $p$ is an odd prime, and $i\ge 1$, $j\ge 2$ 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.

Accepted and due to appear in the International Journal of Group Theory