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)
 > $B≔\mathrm{SylowBasis}\left(G\right)$
 ${B}{≔}\left[⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)⟩{,}⟨\left({1}{,}{3}{,}{2}\right)⟩\right]$ (2)
 > $\mathrm{map}\left(\mathrm{GroupOrder},B\right)$
 $\left[{4}{,}{3}\right]$ (3)
 > $\mathrm{evalb}\left(\mathrm{FrobeniusProduct}\left({B}_{1},{B}_{2},G\right)=\mathrm{FrobeniusProduct}\left({B}_{2},{B}_{1},G\right)\right)$
 ${\mathrm{true}}$ (4)
 > $G≔\mathrm{DihedralGroup}\left(30\right)$
 ${G}{≔}{{\mathbf{D}}}_{{30}}$ (5)
 > $B≔\mathrm{SylowBasis}\left(G\right)$
 ${B}{≔}\left[⟨\left({1}{,}{7}{,}{13}{,}{19}{,}{25}\right)\left({2}{,}{8}{,}{14}{,}{20}{,}{26}\right)\left({3}{,}{9}{,}{15}{,}{21}{,}{27}\right)\left({4}{,}{10}{,}{16}{,}{22}{,}{28}\right)\left({5}{,}{11}{,}{17}{,}{23}{,}{29}\right)\left({6}{,}{12}{,}{18}{,}{24}{,}{30}\right)⟩{,}⟨\left({1}{,}{11}{,}{21}\right)\left({2}{,}{12}{,}{22}\right)\left({3}{,}{13}{,}{23}\right)\left({4}{,}{14}{,}{24}\right)\left({5}{,}{15}{,}{25}\right)\left({6}{,}{16}{,}{26}\right)\left({7}{,}{17}{,}{27}\right)\left({8}{,}{18}{,}{28}\right)\left({9}{,}{19}{,}{29}\right)\left({10}{,}{20}{,}{30}\right)⟩{,}⟨\left({1}{,}{16}\right)\left({2}{,}{17}\right)\left({3}{,}{18}\right)\left({4}{,}{19}\right)\left({5}{,}{20}\right)\left({6}{,}{21}\right)\left({7}{,}{22}\right)\left({8}{,}{23}\right)\left({9}{,}{24}\right)\left({10}{,}{25}\right)\left({11}{,}{26}\right)\left({12}{,}{27}\right)\left({13}{,}{28}\right)\left({14}{,}{29}\right)\left({15}{,}{30}\right){,}\left({1}{,}{9}\right)\left({2}{,}{8}\right)\left({3}{,}{7}\right)\left({4}{,}{6}\right)\left({10}{,}{30}\right)\left({11}{,}{29}\right)\left({12}{,}{28}\right)\left({13}{,}{27}\right)\left({14}{,}{26}\right)\left({15}{,}{25}\right)\left({16}{,}{24}\right)\left({17}{,}{23}\right)\left({18}{,}{22}\right)\left({19}{,}{21}\right)⟩\right]$ (6)
 > $\mathrm{map}\left(\mathrm{GroupOrder},B\right)$
 $\left[{5}{,}{3}{,}{4}\right]$ (7)
 > $\mathrm{andseq}\left(\mathrm{FrobeniusProduct}\left({S}_{1},{S}_{2},G\right)=\mathrm{FrobeniusProduct}\left({S}_{2},{S}_{1},G\right),S=\mathrm{combinat}:-\mathrm{choose}\left(B,2\right)\right)$
 ${\mathrm{true}}$ (8)
 > $G≔\mathrm{FrobeniusGroup}\left(300,3\right)$
 ${G}{≔}{\mathrm{< a permutation group on 100 letters with 5 generators >}}$ (9)
 > $B≔\mathrm{SylowBasis}\left(G\right):$
 > $\mathrm{map}\left(\mathrm{GroupOrder},B\right)$
 $\left[{25}{,}{4}{,}{3}\right]$ (10)
 > $\mathrm{andseq}\left(\mathrm{FrobeniusProduct}\left({S}_{1},{S}_{2},G\right)=\mathrm{FrobeniusProduct}\left({S}_{2},{S}_{1},G\right),S=\mathrm{combinat}:-\mathrm{choose}\left(B,2\right)\right)$
 ${\mathrm{true}}$ (11)
 > $\mathrm{SylowBasis}\left(\mathrm{PSL}\left(4,3\right)\right)$
 > $\mathrm{SylowBasis}\left(\mathrm{Symm}\left(5\right)\right)$

Compatibility

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