Omegas of agemos in powerful groups

In this note we show that for any powerful $p$-group $G$‎, ‎the subgroup‎ ‎$\Omega_{i}(G^{p^{j}})$ is powerfully nilpotent for all $i,j\geq۱$‎ ‎when $p$ is an odd prime‎, ‎and $i\geq۱$‎, ‎$j\geq۲$ when $p=۲$‎. ‎We‎ ‎provide an example to show why this modification is needed in the‎ ‎case $p=۲$‎. ‎Furthermore we obtain a bound on the powerful nilpotency‎ ‎class of $\Omega_{i}(G^{p^{j}})$‎.

Keywords:

powerful , nilpotent , $p$-group