Let $G$ be a finite soluble group.  A Sylow basis for $G$ is a collection $B$ of Sylow subgroups of $G$, one for each prime divisor of the order of $G$, such that $\mathrm{PQ}=\mathrm{QP}$, for each pair $P,Q$ of Sylow subgroups in $B$.

The existence of a Sylow basis for $G$ is equivalent to the solubility of $G$.

The SylowBasis( G ) command constructs a Sylow basis for the soluble group G. If the group G is not soluble, then an exception is raised. The group G must be an instance of a permutation group.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$B≔\mathrm{SylowBasis}\left(G\right)$

$B≔\left[⟨\left(1\,2\right)\left(3\,4\right)\,\left(1\,3\right)\left(2\,4\right)⟩\,⟨\left(1\,3\,2\right)⟩\right]$

$\mathrm{map}\left(\mathrm{GroupOrder}\,B\right)$

$\left[4\,3\right]$

$\mathrm{evalb}\left(\mathrm{FrobeniusProduct}\left(B\left[1\right]\,B\left[2\right]\,G\right)=\mathrm{FrobeniusProduct}\left(B\left[2\right]\,B\left[1\right]\,G\right)\right)$

$\mathrm{true}$

$G≔\mathrm{DihedralGroup}\left(30\right)$

$G≔{\mathbf{D}}_{30}$

$B≔\mathrm{SylowBasis}\left(G\right)$

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

$\mathrm{map}\left(\mathrm{GroupOrder}\,B\right)$

$\left[5\,3\,4\right]$

$\mathrm{andseq}\left(\mathrm{FrobeniusProduct}\left(S\left[1\right]\,S\left[2\right]\,G\right)=\mathrm{FrobeniusProduct}\left(S\left[2\right]\,S\left[1\right]\,G\right)\,S=\mathrm{combinat}:-\mathrm{choose}\left(B\,2\right)\right)$

$\mathrm{true}$

$G≔\mathrm{FrobeniusGroup}\left(300\,3\right)$

$G≔⟨\left(2\,3\,6\right)\left(4\,40\,57\right)\left(5\,86\,12\right)\left(7\,54\,77\right)\left(8\,93\,27\right)\left(9\,63\,66\right)\left(10\,96\,17\right)\left(11\,23\,59\right)\left(13\,38\,39\right)\left(14\,49\,70\right)\left(15\,91\,20\right)\left(16\,34\,79\right)\left(18\,52\,62\right)\left(19\,43\,68\right)\left(21\,61\,48\right)\left(22\,87\,56\right)\left(24\,95\,41\right)\left(25\,75\,74\right)\left(26\,30\,72\right)\left(28\,47\,53\right)\left(29\,94\,76\right)\left(31\,99\,64\right)\left(32\,85\,89\right)\left(33\,97\,65\right)\left(35\,100\,50\right)\left(36\,90\,81\right)\left(37\,73\,58\right)\left(42\,92\,69\right)\left(44\,98\,55\right)\left(45\,82\,84\right)\left(46\,83\,78\right)\left(51\,88\,67\right)\left(60\,80\,71\right)\,\left(1\,2\right)\left(3\,6\right)\left(4\,7\right)\left(5\,8\right)\left(9\,14\right)\left(10\,15\right)\left(11\,16\right)\left(12\,17\right)\left(13\,18\right)\left(19\,26\right)\left(20\,27\right)\left(21\,28\right)\left(22\,29\right)\left(23\,30\right)\left(24\,31\right)\left(25\,32\right)\left(33\,42\right)\left(34\,43\right)\left(35\,44\right)\left(36\,45\right)\left(37\,46\right)\left(38\,47\right)\left(39\,48\right)\left(40\,49\right)\left(41\,50\right)\left(51\,60\right)\left(52\,61\right)\left(53\,62\right)\left(54\,63\right)\left(55\,64\right)\left(56\,65\right)\left(57\,66\right)\left(58\,67\right)\left(59\,68\right)\left(69\,76\right)\left(70\,77\right)\left(71\,78\right)\left(72\,79\right)\left(73\,80\right)\left(74\,81\right)\left(75\,82\right)\left(83\,88\right)\left(84\,89\right)\left(85\,90\right)\left(86\,91\right)\left(87\,92\right)\left(93\,96\right)\left(94\,97\right)\left(95\,98\right)\left(99\,100\right)\,\left(1\,3\right)\left(2\,6\right)\left(4\,9\right)\left(5\,10\right)\left(7\,14\right)\left(8\,15\right)\left(11\,19\right)\left(12\,20\right)\left(13\,21\right)\left(16\,26\right)\left(17\,27\right)\left(18\,28\right)\left(22\,33\right)\left(23\,34\right)\left(24\,35\right)\left(25\,36\right)\left(29\,42\right)\left(30\,43\right)\left(31\,44\right)\left(32\,45\right)\left(37\,51\right)\left(38\,52\right)\left(39\,53\right)\left(40\,54\right)\left(41\,55\right)\left(46\,60\right)\left(47\,61\right)\left(48\,62\right)\left(49\,63\right)\left(50\,64\right)\left(56\,69\right)\left(57\,70\right)\left(58\,71\right)\left(59\,72\right)\left(65\,76\right)\left(66\,77\right)\left(67\,78\right)\left(68\,79\right)\left(73\,83\right)\left(74\,84\right)\left(75\,85\right)\left(80\,88\right)\left(81\,89\right)\left(82\,90\right)\left(86\,93\right)\left(87\,94\right)\left(91\,96\right)\left(92\,97\right)\left(95\,99\right)\left(98\,100\right)\,\left(1\,4\,11\,22\,37\right)\left(2\,7\,16\,29\,46\right)\left(3\,9\,19\,33\,51\right)\left(5\,12\,23\,38\,56\right)\left(6\,14\,26\,42\,60\right)\left(8\,17\,30\,47\,65\right)\left(10\,20\,34\,52\,69\right)\left(13\,24\,39\,57\,73\right)\left(15\,27\,43\,61\,76\right)\left(18\,31\,48\,66\,80\right)\left(21\,35\,53\,70\,83\right)\left(25\,40\,58\,74\,86\right)\left(28\,44\,62\,77\,88\right)\left(32\,49\,67\,81\,91\right)\left(36\,54\,71\,84\,93\right)\left(41\,59\,75\,87\,95\right)\left(45\,63\,78\,89\,96\right)\left(50\,68\,82\,92\,98\right)\left(55\,72\,85\,94\,99\right)\left(64\,79\,90\,97\,100\right)\,\left(1\,5\,13\,25\,41\right)\left(2\,8\,18\,32\,50\right)\left(3\,10\,21\,36\,55\right)\left(4\,12\,24\,40\,59\right)\left(6\,15\,28\,45\,64\right)\left(7\,17\,31\,49\,68\right)\left(9\,20\,35\,54\,72\right)\left(11\,23\,39\,58\,75\right)\left(14\,27\,44\,63\,79\right)\left(16\,30\,48\,67\,82\right)\left(19\,34\,53\,71\,85\right)\left(22\,38\,57\,74\,87\right)\left(26\,43\,62\,78\,90\right)\left(29\,47\,66\,81\,92\right)\left(33\,52\,70\,84\,94\right)\left(37\,56\,73\,86\,95\right)\left(42\,61\,77\,89\,97\right)\left(46\,65\,80\,91\,98\right)\left(51\,69\,83\,93\,99\right)\left(60\,76\,88\,96\,100\right)⟩$

$B≔\mathrm{SylowBasis}\left(G\right)\:$

$\mathrm{map}\left(\mathrm{GroupOrder}\,B\right)$

$\left[25\,4\,3\right]$

$\mathrm{andseq}\left(\mathrm{FrobeniusProduct}\left(S\left[1\right]\,S\left[2\right]\,G\right)=\mathrm{FrobeniusProduct}\left(S\left[2\right]\,S\left[1\right]\,G\right)\,S=\mathrm{combinat}:-\mathrm{choose}\left(B\,2\right)\right)$

$\mathrm{true}$

$\mathrm{SylowBasis}\left(\mathrm{PSL}\left(4\,3\right)\right)$

Error, (in GroupTheory:-SylowBasis) group must be soluble

$\mathrm{SylowBasis}\left(\mathrm{Symm}\left(5\right)\right)$

Error, (in GroupTheory:-SylowBasis) group must be soluble

The GroupTheory[SylowBasis] command was introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.