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.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$B≔\mathrm{SylowBasis}\left(G\right)$

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

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

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

$\mathrm{evalb}\left(\mathrm{FrobeniusProduct}\left(B_1\,B_2\,G\right)\=\mathrm{FrobeniusProduct}\left(B_2\,B_1\,G\right)\right)$

$\mathrm{true}$

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

$G≔{\mathbf{D}}_{30}$

$B≔\mathrm{SylowBasis}\left(G\right)$

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

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

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

$\mathrm{andseq}\left(\mathrm{FrobeniusProduct}\left(S_1\,S_2\,G\right)\=\mathrm{FrobeniusProduct}\left(S_2\,S_1\,G\right)\,S\=\mathrm{combinat}:-\mathrm{choose}\left(B\,2\right)\right)$

$\mathrm{true}$

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

$G≔\mathrm{<\; a\; permutation\; group\; on\; 100\; letters\; with\; 5\; generators\; >}$

$B≔\mathrm{SylowBasis}\left(G\right)\&colon;$

$\mathrm{map}\left(\mathrm{GroupOrder}\&comma;B\right)$

$\left[25\&comma;4\&comma;3\right]$

$\mathrm{andseq}\left(\mathrm{FrobeniusProduct}\left(S_1\&comma;S_2\&comma;G\right)\&equals;\mathrm{FrobeniusProduct}\left(S_2\&comma;S_1\&comma;G\right)\&comma;S\&equals;\mathrm{combinat}:-\mathrm{choose}\left(B\&comma;2\right)\right)$

$\mathrm{true}$

$\mathrm{SylowBasis}\left(\mathrm{PSL}\left(4\&comma;3\right)\right)$

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.