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.

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

$C≔\mathrm{HallSystem}\left(\mathrm{Symm}\left(3\right)\right)$

$C≔\left\{⟨\left(1\,2\right)\,\left(1\,2\,3\right)⟩\,⟨\left(1\,2\,3\right)⟩\,⟨\left(1\,3\right)⟩\,⟨⟩\right\}$

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

$\left\{1\,2\,3\,6\right\}$

$C≔\mathrm{HallSystem}\left(\mathrm{Alt}\left(4\right)\right)$

$C≔\left\{⟨\left(1\,2\,3\right)\,\left(2\,3\,4\right)⟩\,⟨\left(1\,2\,4\right)⟩\,⟨\left(1\,2\right)\left(3\,4\right)\,\left(1\,3\right)\left(2\,4\right)⟩\,⟨⟩\right\}$

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

$\left\{1\,3\,4\,12\right\}$

$C≔\mathrm{HallSystem}\left(\mathrm{WreathProduct}\left(\mathrm{Symm}\left(3\right)\,\mathrm{CyclicGroup}\left(2\right)\right)\right)$

$C≔\left\{⟨\left(1\,2\right)\,\left(1\,2\,3\right)\,\left(1\,4\right)\left(2\,5\right)\left(3\,6\right)⟩\,⟨\left(4\,5\,6\right)\,\left(1\,2\,3\right)⟩\,⟨\left(2\,3\right)\left(4\,5\right)\,\left(2\,3\right)\,\left(4\,5\right)\,\left(1\,6\right)\left(2\,5\right)\left(3\,4\right)⟩\,⟨⟩\right\}$

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

$\left\{1\,8\,9\,72\right\}$

$G≔\mathrm{FrobeniusGroup}\left(42\,1\right)$

$G≔⟨\left(2\,7\right)\left(3\,6\right)\left(4\,5\right)\,\left(2\,3\,5\right)\left(4\,7\,6\right)\,\left(1\,2\,3\,4\,5\,6\,7\right)⟩$

$\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$

$\left(2\right)\left(3\right)\left(7\right)$

$C≔\mathrm{HallSystem}\left(G\right)$

$C≔\left\{⟨\left(2\,7\right)\left(3\,6\right)\left(4\,5\right)\,\left(2\,3\,5\right)\left(4\,7\,6\right)\,\left(1\,2\,3\,4\,5\,6\,7\right)⟩\,⟨\left(1\,4\,7\,3\,6\,2\,5\right)\,\left(1\,4\,2\right)\left(3\,5\,6\right)⟩\,⟨\left(1\,4\,7\,3\,6\,2\,5\right)\,\left(1\,6\right)\left(2\,5\right)\left(3\,4\right)⟩\,⟨\left(1\,3\,4\right)\left(2\,7\,6\right)\,\left(1\,2\right)\left(3\,7\right)\left(4\,6\right)⟩\,⟨\left(1\,4\,7\,3\,6\,2\,5\right)⟩\,⟨\left(1\,3\,4\right)\left(2\,7\,6\right)⟩\,⟨\left(1\,2\right)\left(3\,7\right)\left(4\,6\right)⟩\,⟨⟩\right\}$

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

$\left\{1\,2\,3\,6\,7\,14\,21\,42\right\}$

$G≔\mathrm{DirectProduct}\left(\mathrm{DihedralGroup}\left(15\right)\,\mathrm{Symm}\left(3\right)\right)$

$G≔⟨\left(1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\,13\,14\,15\right)\,\left(1\,14\right)\left(2\,13\right)\left(3\,12\right)\left(4\,11\right)\left(5\,10\right)\left(6\,9\right)\left(7\,8\right)\,\left(16\,17\right)\,\left(16\,17\,18\right)⟩$

$\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$

${\left(2\right)}^2{\left(3\right)}^2\left(5\right)$

$C≔\mathrm{HallSystem}\left(G\right)$

$C≔\left\{⟨\left(1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\,13\,14\,15\right)\,\left(1\,14\right)\left(2\,13\right)\left(3\,12\right)\left(4\,11\right)\left(5\,10\right)\left(6\,9\right)\left(7\,8\right)\,\left(16\,17\right)\,\left(16\,17\,18\right)⟩\,⟨\left(1\,6\,11\right)\left(2\,7\,12\right)\left(3\,8\,13\right)\left(4\,9\,14\right)\left(5\,10\,15\right)\,\left(1\,7\,13\,4\,10\right)\left(2\,8\,14\,5\,11\right)\left(3\,9\,15\,6\,12\right)\,\left(16\,17\,18\right)⟩\,⟨\left(1\,10\,4\,13\,7\right)\left(2\,11\,5\,14\,8\right)\left(3\,12\,6\,15\,9\right)\,\left(17\,18\right)\,\left(1\,7\right)\left(2\,6\right)\left(3\,5\right)\left(8\,15\right)\left(9\,14\right)\left(10\,13\right)\left(11\,12\right)⟩\,⟨\left(1\,11\,6\right)\left(2\,12\,7\right)\left(3\,13\,8\right)\left(4\,14\,9\right)\left(5\,15\,10\right)\,\left(16\,17\right)\,\left(16\,17\,18\right)\,\left(1\,3\right)\left(4\,15\right)\left(5\,14\right)\left(6\,13\right)\left(7\,12\right)\left(8\,11\right)\left(9\,10\right)⟩\,⟨\left(1\,7\,13\,4\,10\right)\left(2\,8\,14\,5\,11\right)\left(3\,9\,15\,6\,12\right)⟩\,⟨\left(1\,6\,11\right)\left(2\,7\,12\right)\left(3\,8\,13\right)\left(4\,9\,14\right)\left(5\,10\,15\right)\,\left(16\,17\,18\right)⟩\,⟨\left(1\,13\right)\left(2\,12\right)\left(3\,11\right)\left(4\,10\right)\left(5\,9\right)\left(6\,8\right)\left(14\,15\right)\,\left(17\,18\right)⟩\,⟨⟩\right\}$

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

$\left\{1\,4\,5\,9\,20\,36\,45\,180\right\}$

$\mathrm{HallSystem}\left(\mathrm{GL}\left(2\,4\right)\right)$

Error, (in GroupTheory:-HallSystem) group must be soluble

$\mathrm{IsSoluble}\left(\mathrm{GL}\left(2\,4\right)\right)$

$\mathrm{false}$