GroupTheory/HallSubgroup - Maple Help

GroupTheory

 HallSubgroup
 construct a Hall subgroup of a finite soluble group

 Calling Sequence HallSubgroup( pi, G )

Parameters

 pi - a list or set of primes G - a soluble permutation group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{DihedralGroup}\left(30\right)$
 ${G}{≔}{{\mathbf{D}}}_{{30}}$ (1)
 > $\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$
 ${\left({2}\right)}^{{2}}{}\left({3}\right){}\left({5}\right)$ (2)
 > $H≔\mathrm{HallSubgroup}\left(\left\{2,5\right\},G\right)$
 ${H}{≔}⟨\left({1}{,}{20}\right)\left({2}{,}{19}\right)\left({3}{,}{18}\right)\left({4}{,}{17}\right)\left({5}{,}{16}\right)\left({6}{,}{15}\right)\left({7}{,}{14}\right)\left({8}{,}{13}\right)\left({9}{,}{12}\right)\left({10}{,}{11}\right)\left({21}{,}{30}\right)\left({22}{,}{29}\right)\left({23}{,}{28}\right)\left({24}{,}{27}\right)\left({25}{,}{26}\right){,}\left({1}{,}{16}\right)\left({2}{,}{17}\right)\left({3}{,}{18}\right)\left({4}{,}{19}\right)\left({5}{,}{20}\right)\left({6}{,}{21}\right)\left({7}{,}{22}\right)\left({8}{,}{23}\right)\left({9}{,}{24}\right)\left({10}{,}{25}\right)\left({11}{,}{26}\right)\left({12}{,}{27}\right)\left({13}{,}{28}\right)\left({14}{,}{29}\right)\left({15}{,}{30}\right){,}\left({1}{,}{19}{,}{7}{,}{25}{,}{13}\right)\left({2}{,}{20}{,}{8}{,}{26}{,}{14}\right)\left({3}{,}{21}{,}{9}{,}{27}{,}{15}\right)\left({4}{,}{22}{,}{10}{,}{28}{,}{16}\right)\left({5}{,}{23}{,}{11}{,}{29}{,}{17}\right)\left({6}{,}{24}{,}{12}{,}{30}{,}{18}\right)⟩$ (3)
 > $\mathrm{igcd}\left(\mathrm{GroupOrder}\left(H\right),\mathrm{Index}\left(H,G\right)\right)$
 ${1}$ (4)

Hall subgroups can only be computed for soluble groups, in general, so the following example cause an exception to be raised.

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

However, for certain special cases, a Hall subgroup is returned without exception.

 > $\mathrm{HallSubgroup}\left(\left\{5\right\},\mathrm{Symm}\left(5\right)\right)$
 $⟨\left({1}{,}{2}{,}{3}{,}{4}{,}{5}\right)⟩$ (5)
 > $\mathrm{HallSubgroup}\left(\varnothing ,\mathrm{Symm}\left(5\right)\right)$
 $⟨⟩$ (6)
 > $\mathrm{HallSubgroup}\left(\left\{2,3,5\right\},\mathrm{Symm}\left(5\right)\right)$
 ${{\mathbf{S}}}_{{5}}$ (7)

Compatibility

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