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.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$B≔\mathrm{SylowBasis}\left(G\right)$

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

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

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

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

$\mathrm{true}$

$B≔\mathrm{SylowBasis}\left(\mathrm{DihedralGroup}\left(15\right)\right)$

$B≔\left[⟨\left(1\,10\,4\,13\,7\right)\left(2\,11\,5\,14\,8\right)\left(3\,12\,6\,15\,9\right)⟩\,⟨\left(1\,11\,6\right)\left(2\,12\,7\right)\left(3\,13\,8\right)\left(4\,14\,9\right)\left(5\,15\,10\right)⟩\,⟨\left(1\,4\right)\left(2\,3\right)\left(5\,15\right)\left(6\,14\right)\left(7\,13\right)\left(8\,12\right)\left(9\,11\right)⟩\right]$

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

$\left[5\,3\,2\right]$

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

Error, (in SylowBasis) group must be soluble

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

Error, (in 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.