GroupTheory/HallSystem - Maple Help

GroupTheory

 HallSystem
 compute a Hall system for a finite soluble group

 Calling Sequence HallSystem( G )

Parameters

 G - a finite soluble group

Description

 • Let $G$ be a finite soluble group.  A Hall system for $G$ is a collection $C$ of Hall $\mathrm{\pi }$-subgroups of $G$, one for each subset $\mathrm{\pi }$ of the prime divisors of the order of $G$. Note that this includes both $G$ itself, as well as the trivial subgroup of $G$.
 • A Hall system for $G$ exists provided that $G$ is a soluble group, and conversely.
 • The HallSystem( G ) command constructs a Hall system for the soluble group G. If the group G is not soluble, then an exception is raised.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $C≔\mathrm{HallSystem}\left(\mathrm{Symm}\left(3\right)\right)$
 ${C}{≔}\left\{⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)⟩{,}⟨\left({1}{,}{2}{,}{3}\right)⟩{,}⟨\left({1}{,}{3}\right)⟩{,}⟨⟩\right\}$ (1)
 > $\mathrm{map}\left(\mathrm{GroupOrder},C\right)$
 $\left\{{1}{,}{2}{,}{3}{,}{6}\right\}$ (2)
 > $C≔\mathrm{HallSystem}\left(\mathrm{Alt}\left(4\right)\right)$
 ${C}{≔}\left\{⟨\left({1}{,}{2}{,}{3}\right){,}\left({2}{,}{3}{,}{4}\right)⟩{,}⟨\left({1}{,}{4}{,}{2}\right)⟩{,}⟨\left({1}{,}{2}\right)\left({3}{,}{4}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)⟩{,}⟨⟩\right\}$ (3)
 > $\mathrm{map}\left(\mathrm{GroupOrder},C\right)$
 $\left\{{1}{,}{3}{,}{4}{,}{12}\right\}$ (4)
 > $C≔\mathrm{HallSystem}\left(\mathrm{WreathProduct}\left(\mathrm{Symm}\left(3\right),\mathrm{CyclicGroup}\left(2\right)\right)\right)$
 ${C}{≔}\left\{⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right){,}\left({1}{,}{4}\right)\left({2}{,}{5}\right)\left({3}{,}{6}\right)⟩{,}⟨\left({4}{,}{6}{,}{5}\right){,}\left({1}{,}{3}{,}{2}\right)\left({4}{,}{6}{,}{5}\right)⟩{,}⟨\left({1}{,}{4}\right)\left({2}{,}{5}\right)\left({3}{,}{6}\right){,}\left({1}{,}{6}{,}{3}{,}{4}\right)\left({2}{,}{5}\right){,}\left({1}{,}{3}\right)\left({4}{,}{6}\right)⟩{,}⟨⟩\right\}$ (5)
 > $\mathrm{map}\left(\mathrm{GroupOrder},C\right)$
 $\left\{{1}{,}{8}{,}{9}{,}{72}\right\}$ (6)
 > $G≔\mathrm{FrobeniusGroup}\left(42,1\right)$
 ${G}{≔}⟨\left({2}{,}{7}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right){,}\left({2}{,}{3}{,}{5}\right)\left({4}{,}{7}{,}{6}\right){,}\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}\right)⟩$ (7)
 > $\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$
 $\left({2}\right){}\left({3}\right){}\left({7}\right)$ (8)
 > $C≔\mathrm{HallSystem}\left(G\right)$
 ${C}{≔}\left\{⟨\left({2}{,}{7}\right)\left({3}{,}{6}\right)\left({4}{,}{5}\right){,}\left({2}{,}{3}{,}{5}\right)\left({4}{,}{7}{,}{6}\right){,}\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}\right)⟩{,}⟨\left({1}{,}{4}{,}{2}\right)\left({3}{,}{5}{,}{6}\right){,}\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}\right)⟩{,}⟨\left({1}{,}{6}\right)\left({2}{,}{5}\right)\left({3}{,}{4}\right){,}\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}\right)⟩{,}⟨\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right){,}\left({1}{,}{6}{,}{5}\right)\left({2}{,}{3}{,}{7}\right)⟩{,}⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}\right)⟩{,}⟨\left({1}{,}{6}{,}{5}\right)\left({2}{,}{3}{,}{7}\right)⟩{,}⟨\left({1}{,}{7}\right)\left({2}{,}{6}\right)\left({3}{,}{5}\right)⟩{,}⟨⟩\right\}$ (9)
 > $\mathrm{map}\left(\mathrm{GroupOrder},C\right)$
 $\left\{{1}{,}{2}{,}{3}{,}{6}{,}{7}{,}{14}{,}{21}{,}{42}\right\}$ (10)
 > $G≔\mathrm{DirectProduct}\left(\mathrm{DihedralGroup}\left(15\right),\mathrm{Symm}\left(3\right)\right)$
 ${G}{≔}{\mathrm{< a permutation group on 18 letters with 4 generators >}}$ (11)
 > $\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$
 ${\left({2}\right)}^{{2}}{}{\left({3}\right)}^{{2}}{}\left({5}\right)$ (12)
 > $C≔\mathrm{HallSystem}\left(G\right)$
 ${C}{≔}\left\{{\mathrm{< a permutation group on 18 letters with 4 generators >}}{,}⟨\left({1}{,}{4}{,}{7}{,}{10}{,}{13}\right)\left({2}{,}{5}{,}{8}{,}{11}{,}{14}\right)\left({3}{,}{6}{,}{9}{,}{12}{,}{15}\right){,}\left({1}{,}{6}{,}{11}\right)\left({2}{,}{7}{,}{12}\right)\left({3}{,}{8}{,}{13}\right)\left({4}{,}{9}{,}{14}\right)\left({5}{,}{10}{,}{15}\right){,}\left({16}{,}{17}{,}{18}\right)⟩{,}⟨\left({1}{,}{4}{,}{7}{,}{10}{,}{13}\right)\left({2}{,}{5}{,}{8}{,}{11}{,}{14}\right)\left({3}{,}{6}{,}{9}{,}{12}{,}{15}\right){,}\left({16}{,}{17}\right){,}\left({2}{,}{15}\right)\left({3}{,}{14}\right)\left({4}{,}{13}\right)\left({5}{,}{12}\right)\left({6}{,}{11}\right)\left({7}{,}{10}\right)\left({8}{,}{9}\right)⟩{,}⟨⟩{,}{\mathrm{< a permutation group on 18 letters with 4 generators >}}{,}⟨\left({1}{,}{4}{,}{7}{,}{10}{,}{13}\right)\left({2}{,}{5}{,}{8}{,}{11}{,}{14}\right)\left({3}{,}{6}{,}{9}{,}{12}{,}{15}\right)⟩{,}⟨\left({1}{,}{6}{,}{11}\right)\left({2}{,}{7}{,}{12}\right)\left({3}{,}{8}{,}{13}\right)\left({4}{,}{9}{,}{14}\right)\left({5}{,}{10}{,}{15}\right){,}\left({16}{,}{17}{,}{18}\right)⟩{,}⟨\left({1}{,}{13}\right)\left({2}{,}{12}\right)\left({3}{,}{11}\right)\left({4}{,}{10}\right)\left({5}{,}{9}\right)\left({6}{,}{8}\right)\left({14}{,}{15}\right){,}\left({16}{,}{17}\right)⟩\right\}$ (13)
 > $\mathrm{map}\left(\mathrm{GroupOrder},C\right)$
 $\left\{{1}{,}{4}{,}{5}{,}{9}{,}{20}{,}{36}{,}{45}{,}{180}\right\}$ (14)

Since $GL\left(2,4\right)$ is an insoluble group, attempting to compute a Hall system for this group causes an exception to be raised.

 > $\mathrm{HallSystem}\left(\mathrm{GL}\left(2,4\right)\right)$
 > $\mathrm{IsSoluble}\left(\mathrm{GL}\left(2,4\right)\right)$
 ${\mathrm{false}}$ (15)