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