SylowBasis - Maple Help

GroupTheory

 SylowBasis
 construct a Sylow basis for a finite soluble group

 Calling Sequence SylowBasis( G )

Parameters

 G - a soluble permutation group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Alt}\left(4\right)$
 ${G}{≔}{{\mathbf{A}}}_{{4}}$ (1)
 > $B≔\mathrm{SylowBasis}\left(G\right)$
 ${B}{≔}\left[⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)⟩{,}⟨\left({1}{,}{3}{,}{2}\right)⟩\right]$ (2)
 > $\mathrm{map}\left(\mathrm{GroupOrder},B\right)$
 $\left[{4}{,}{3}\right]$ (3)
 > $\mathrm{evalb}\left(\mathrm{ComplexProduct}\left(B\left[1\right],B\left[2\right],G\right)=\mathrm{ComplexProduct}\left(B\left[2\right],B\left[1\right],G\right)\right)$
 ${\mathrm{true}}$ (4)
 > $B≔\mathrm{SylowBasis}\left(\mathrm{DihedralGroup}\left(15\right)\right)$
 ${B}{≔}\left[⟨\left({1}{,}{10}{,}{4}{,}{13}{,}{7}\right)\left({2}{,}{11}{,}{5}{,}{14}{,}{8}\right)\left({3}{,}{12}{,}{6}{,}{15}{,}{9}\right)⟩{,}⟨\left({1}{,}{11}{,}{6}\right)\left({2}{,}{12}{,}{7}\right)\left({3}{,}{13}{,}{8}\right)\left({4}{,}{14}{,}{9}\right)\left({5}{,}{15}{,}{10}\right)⟩{,}⟨\left({1}{,}{4}\right)\left({2}{,}{3}\right)\left({5}{,}{15}\right)\left({6}{,}{14}\right)\left({7}{,}{13}\right)\left({8}{,}{12}\right)\left({9}{,}{11}\right)⟩\right]$ (5)
 > $\mathrm{map}\left(\mathrm{GroupOrder},B\right)$
 $\left[{5}{,}{3}{,}{2}\right]$ (6)
 > $\mathrm{SylowBasis}\left(\mathrm{PSL}\left(4,3\right)\right)$
 > $\mathrm{SylowBasis}\left(\mathrm{Symm}\left(5\right)\right)$

Compatibility

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