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.

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

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

$G≔{\mathbf{D}}_8$

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

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

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

$\mathrm{true}$

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

$4$

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

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

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

$2$

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

${\mathbf{D}}_8$

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

$G≔C_{16807}$

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

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

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

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

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

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

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

$32$

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

$G≔C_{625}$

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

$1,5,25,125,625$

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

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

$16$

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

$16$

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

$32=32$

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

$G≔⟨\left(1\,2\,8\,6\,11\,15\,7\,12\,3\right)\left(4\,16\,13\,18\,25\,24\,19\,9\,17\right)\left(5\,20\,27\,22\,10\,14\,23\,26\,21\right)\,\left(1\,4\,5\right)\left(2\,9\,10\right)\left(3\,13\,14\right)\left(6\,18\,22\right)\left(7\,19\,23\right)\left(8\,24\,21\right)\left(11\,16\,26\right)\left(12\,25\,20\right)\left(15\,17\,27\right)\,\left(1\,6\,7\right)\left(2\,11\,12\right)\left(3\,8\,15\right)\left(4\,18\,19\right)\left(5\,22\,23\right)\left(9\,16\,25\right)\left(10\,26\,20\right)\left(13\,24\,17\right)\left(14\,21\,27\right)\,\left(1\,28\,55\right)\left(2\,29\,56\right)\left(3\,30\,57\right)\left(4\,31\,58\right)\left(5\,32\,59\right)\left(6\,33\,60\right)\left(7\,34\,61\right)\left(8\,35\,62\right)\left(9\,36\,63\right)\left(10\,37\,64\right)\left(11\,38\,65\right)\left(12\,39\,66\right)\left(13\,40\,67\right)\left(14\,41\,68\right)\left(15\,42\,69\right)\left(16\,43\,70\right)\left(17\,44\,71\right)\left(18\,45\,72\right)\left(19\,46\,73\right)\left(20\,47\,74\right)\left(21\,48\,75\right)\left(22\,49\,76\right)\left(23\,50\,77\right)\left(24\,51\,78\right)\left(25\,52\,79\right)\left(26\,53\,80\right)\left(27\,54\,81\right)⟩$

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

$W≔{\Omega }_1\left(⟨\left(1\,2\,8\,6\,11\,15\,7\,12\,3\right)\left(4\,16\,13\,18\,25\,24\,19\,9\,17\right)\left(5\,20\,27\,22\,10\,14\,23\,26\,21\right)\,\left(1\,4\,5\right)\left(2\,9\,10\right)\left(3\,13\,14\right)\left(6\,18\,22\right)\left(7\,19\,23\right)\left(8\,24\,21\right)\left(11\,16\,26\right)\left(12\,25\,20\right)\left(15\,17\,27\right)\,\left(1\,6\,7\right)\left(2\,11\,12\right)\left(3\,8\,15\right)\left(4\,18\,19\right)\left(5\,22\,23\right)\left(9\,16\,25\right)\left(10\,26\,20\right)\left(13\,24\,17\right)\left(14\,21\,27\right)\,\left(1\,28\,55\right)\left(2\,29\,56\right)\left(3\,30\,57\right)\left(4\,31\,58\right)\left(5\,32\,59\right)\left(6\,33\,60\right)\left(7\,34\,61\right)\left(8\,35\,62\right)\left(9\,36\,63\right)\left(10\,37\,64\right)\left(11\,38\,65\right)\left(12\,39\,66\right)\left(13\,40\,67\right)\left(14\,41\,68\right)\left(15\,42\,69\right)\left(16\,43\,70\right)\left(17\,44\,71\right)\left(18\,45\,72\right)\left(19\,46\,73\right)\left(20\,47\,74\right)\left(21\,48\,75\right)\left(22\,49\,76\right)\left(23\,50\,77\right)\left(24\,51\,78\right)\left(25\,52\,79\right)\left(26\,53\,80\right)\left(27\,54\,81\right)⟩\right)$

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

$19683$

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

$59049$

$G≔\mathrm{DirectProduct}\left(\mathrm{`\$`}\left(\mathrm{QuaternionGroup}\left(\right)\,4\right)\,\mathrm{CyclicGroup}\left(4\right)\,\mathrm{`\$`}\left(\mathrm{DihedralGroup}\left(16\right)\,3\right)\right)$

$G≔⟨\left(1\,2\,3\,4\right)\left(5\,6\,8\,7\right)\,\left(1\,5\,3\,8\right)\left(2\,7\,4\,6\right)\,\left(9\,10\,11\,12\right)\left(13\,14\,16\,15\right)\,\left(9\,13\,11\,16\right)\left(10\,15\,12\,14\right)\,\left(17\,18\,19\,20\right)\left(21\,22\,24\,23\right)\,\left(17\,21\,19\,24\right)\left(18\,23\,20\,22\right)\,\left(25\,26\,27\,28\right)\left(29\,30\,32\,31\right)\,\left(25\,29\,27\,32\right)\left(26\,31\,28\,30\right)\,\left(33\,34\,35\,36\right)\,\left(37\,38\,39\,40\,41\,42\,43\,44\,45\,46\,47\,48\,49\,50\,51\,52\right)\,\left(37\,51\right)\left(38\,50\right)\left(39\,49\right)\left(40\,48\right)\left(41\,47\right)\left(42\,46\right)\left(43\,45\right)\,\left(53\,54\,55\,56\,57\,58\,59\,60\,61\,62\,63\,64\,65\,66\,67\,68\right)\,\left(53\,67\right)\left(54\,66\right)\left(55\,65\right)\left(56\,64\right)\left(57\,63\right)\left(58\,62\right)\left(59\,61\right)\,\left(69\,70\,71\,72\,73\,74\,75\,76\,77\,78\,79\,80\,81\,82\,83\,84\right)\,\left(69\,83\right)\left(70\,82\right)\left(71\,81\right)\left(72\,80\right)\left(73\,79\right)\left(74\,78\right)\left(75\,77\right)⟩$

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

$536870912$

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

$1048576$

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

$536870912$