Let $G$ be a finite group, and let $\mathrm{pi}$ be a set of (positive, rational) primes. A Hall $\mathrm{pi}$-subgroup of $G$ is a maximal $\mathrm{pi}$-subgroup of $G$ where, by a $\mathrm{pi}$-subgroup, we mean a subgroup whose order is a $\mathrm{pi}$-number (one whose prime divisors all belong to $\mathrm{pi}$). Equivalently, a subgroup $H$ of a finite group $G$ is a Hall-subgroup if its order and index are relatively prime.

If $\mathrm{pi}$ consists of a single prime number $p$, then a Hall $\mathrm{pi}$-subgroup of $G$ is just a Sylow $p$-subgroup of $G$.

A finite group $G$ is soluble if, and only if, for each set $\mathrm{pi}$ of primes, $G$ has a Hall $\mathrm{pi}$-subgroup. Moreover, any two Hall $\mathrm{pi}$-subgroups of $G$ are conjugate in $G$, and every $\mathrm{pi}$-subgroup of $G$ is contained in a Hall $\mathrm{pi}$subgroup.

A finite insoluble group may, or may not, have Hall subgroups.

The HallSubgroup( pi, G ) command constructs a Hall pi-subgroup of a finite soluble group G. The group G must be an instance of a permutation group. Apart from a handful of exceptions, the permutation group G must be soluble; otherwise, an exception is raised.

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

$G≔\mathrm{DihedralGroup}\left(30\right)$

$G≔{\mathbf{D}}_{30}$

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

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

$H≔\mathrm{HallSubgroup}\left(\left\{2\,5\right\}\,G\right)$

$H≔⟨\left(2\,30\right)\left(3\,29\right)\left(4\,28\right)\left(5\,27\right)\left(6\,26\right)\left(7\,25\right)\left(8\,24\right)\left(9\,23\right)\left(10\,22\right)\left(11\,21\right)\left(12\,20\right)\left(13\,19\right)\left(14\,18\right)\left(15\,17\right)\,\left(1\,16\right)\left(2\,15\right)\left(3\,14\right)\left(4\,13\right)\left(5\,12\right)\left(6\,11\right)\left(7\,10\right)\left(8\,9\right)\left(17\,30\right)\left(18\,29\right)\left(19\,28\right)\left(20\,27\right)\left(21\,26\right)\left(22\,25\right)\left(23\,24\right)\,\left(1\,19\,7\,25\,13\right)\left(2\,20\,8\,26\,14\right)\left(3\,21\,9\,27\,15\right)\left(4\,22\,10\,28\,16\right)\left(5\,23\,11\,29\,17\right)\left(6\,24\,12\,30\,18\right)⟩$

$\mathrm{igcd}\left(\mathrm{GroupOrder}\left(H\right)\,\mathrm{Index}\left(H\,G\right)\right)$

$1$

$\mathrm{HallSubgroup}\left(\left\{2\,3\right\}\,\mathrm{Symm}\left(5\right)\right)$

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

$\mathrm{HallSubgroup}\left(\left\{5\right\}\,\mathrm{Symm}\left(5\right)\right)$

$⟨\left(1\,2\,3\,4\,5\right)⟩$

$\mathrm{HallSubgroup}\left(\varnothing \,\mathrm{Symm}\left(5\right)\right)$

$\mathrm{HallSubgroup}\left(\left\{2\,3\,5\right\}\,\mathrm{Symm}\left(5\right)\right)$

${\mathbf{S}}_5$

The GroupTheory[HallSubgroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.