If $n$ is a non-negative integer, and $G$ is a finite $p$-group, then the subgroup ${\mathrm{\℧}}_n\left(G\right)$ is defined to be the subgroup of $G$ generated by elements of $G$ of the form $g^{p^n}$, as $g$ ranges over all elements of $G$.

The AgemoPGroup( n, G ) command computes the subgroup ${\mathrm{\℧}}_n\left(G\right)$ of G, where G is a permutation $p$-group, for some prime $p$.

The first argument n is optional and is equal to $1$ by default. That is, the command AgemoPGroup( G ) is equivalent to AgemoPGroup( 1, G ).

For a $p$-group $G$, and a non-negative integer $n$, the subgroup ${\mathrm{\Ω}}_n\left(G\right)$ is defined to be the subgroup generated by the elements $g$ such that $g^{p^n}$ = 1, for $g\in G$. That is, the subgroup generated by those members of $G$ whose order divides $p^n$.

The OmegaPGroup( n, G ) command computes ${\mathrm{\Ω}}_n\left(G\right)$ for a permutation group G of prime power order.

When called with two arguments, $n$ and $G$, the indicated subgroup ${\mathrm{\Ω}}_n\left(G\right)$ is returned. When called with just one argument $G$, the subgroup ${\mathrm{\Ω}}_1\left(G\right)$ is returned.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{DihedralGroup}\left(8\right)$

$G≔{\mathbf{D}}_8$

$A≔\mathrm{AgemoPGroup}\left(G\right)$

$A≔{\&Agemo;}_1\left({\mathbf{D}}_8\right)$

$\mathrm{IsCyclic}\left(A\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(A\right)$

$4$

$A≔\mathrm{AgemoPGroup}\left(2\,G\right)$

$A≔{\&Agemo;}_2\left({\mathbf{D}}_8\right)$

$\mathrm{GroupOrder}\left(A\right)$

$2$

$\mathrm{AgemoPGroup}\left(0\,G\right)$

${\mathbf{D}}_8$

$G≔\mathrm{CyclicGroup}\left(16807\right)$

$G≔C_{16807}$

$\mathrm{seq}\left(\mathrm{GroupOrder}\left(\mathrm{AgemoPGroup}\left(n\,G\right)\right)\,n\=0..5\right)$

$16807,2401,343,49,7,1$

$G≔\mathrm{QuaternionGroup}\left(5\right)$

$G≔⟨\left(1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\,13\,14\,15\,16\right)\left(17\,18\,19\,20\,21\,22\,23\,24\,25\,26\,27\,28\,29\,30\,31\,32\right)\,\left(1\,31\,9\,23\right)\left(2\,30\,10\,22\right)\left(3\,29\,11\,21\right)\left(4\,28\,12\,20\right)\left(5\,27\,13\,19\right)\left(6\,26\,14\,18\right)\left(7\,25\,15\,17\right)\left(8\,24\,16\,32\right)⟩$

$W≔\mathrm{OmegaPGroup}\left(G\right)$

$W≔{\Omega }_1\left(⟨\left(1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\,13\,14\,15\,16\right)\left(17\,18\,19\,20\,21\,22\,23\,24\,25\,26\,27\,28\,29\,30\,31\,32\right)\,\left(1\,31\,9\,23\right)\left(2\,30\,10\,22\right)\left(3\,29\,11\,21\right)\left(4\,28\,12\,20\right)\left(5\,27\,13\,19\right)\left(6\,26\,14\,18\right)\left(7\,25\,15\,17\right)\left(8\,24\,16\,32\right)⟩\right)$

$\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(2\,G\right)\right)$

$32$

$G≔\mathrm{CyclicGroup}\left(625\right)$

$G≔C_{625}$

$\mathrm{seq}\left(\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(n\,G\right)\right)\,n\=0..4\right)$

$1,5,25,125,625$

$G≔\mathrm{SmallGroup}\left(32\,38\right)\:$

$\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(1\,G\right)\right)$

$16$

$\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(2\,G\right)\right)$

$16$

$\mathrm{GroupOrder}\left(G\right)\=\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(3\,G\right)\right)$

$32=32$

$G≔\mathrm{WreathProduct}\left(\mathrm{SmallGroup}\left(27\,4\right)\,\mathrm{CyclicGroup}\left(3\right)\right)$

$G≔\mathrm{<\; a\; permutation\; group\; on\; 81\; letters\; with\; 4\; generators\; >}$

$W≔\mathrm{OmegaPGroup}\left(G\right)$

$W≔{\Omega }_1\left(\mathrm{<\; a\; permutation\; group\; on\; 81\; letters\; with\; 4\; generators\; >}\right)$

$\mathrm{GroupOrder}\left(W\right)$

$19683$

$\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(2\&comma;G\right)\right)$

$59049$

$G≔\mathrm{DirectProduct}\left(\mathrm{QuaternionGroup}\left(\right)\&dollar;4\&comma;\mathrm{CyclicGroup}\left(4\right)\&comma;\mathrm{DihedralGroup}\left(16\right)\&dollar;3\right)$

$G≔\mathrm{<\; a\; permutation\; group\; on\; 84\; letters\; with\; 15\; generators\; >}$

$\mathrm{GroupOrder}\left(G\right)$

$536870912$

$\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(1\&comma;G\right)\right)$

$1048576$

$\mathrm{GroupOrder}\left(\mathrm{OmegaPGroup}\left(2\&comma;G\right)\right)$

$536870912$