SylowSubgroup - Maple Help
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GroupTheory

 SylowSubgroup
 construct a Sylow subgroup of a group

 Calling Sequence SylowSubgroup( p, G )

Parameters

 p - a positive rational prime G - a permutation group or Cayley table group

Description

 • Let $G$ be a finite group, and let $p$ be a positive (rational) prime.  A Sylow $p$-subgroup of $G$ is a maximal $p$-subgroup of $G$ where, by a $p$-subgroup, we mean a subgroup whose order is a power of $p$. The Sylow theorems assert that, for a prime divisor $p$ of the order of a finite group $G$, there is a Sylow $p$-subgroup of $G$ and that all Sylow $p$-subgroups of $G$ are conjugate in $G$.  Moreover, the number of Sylow $p$-subgroups of $G$ is congruent to $1$ modulo $p$.
 • The SylowSubgroup( p, G ) command constructs a Sylow p-subgroup of a group G. The group G must be an instance of a permutation group or a Cayley table group.
 • Note that, if p is not a divisor of the order of G, then the trivial subgroup of G is returned.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SL}\left(2,5\right):$
 > $\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$
 ${\left({2}\right)}^{{3}}{}\left({3}\right){}\left({5}\right)$ (1)
 > $\mathrm{P2}≔\mathrm{SylowSubgroup}\left(2,G\right)$
 ${\mathrm{P2}}{≔}⟨\left({1}{,}{2}{,}{3}{,}{4}\right)\left({5}{,}{20}{,}{15}{,}{10}\right)\left({6}{,}{22}{,}{18}{,}{14}\right)\left({7}{,}{23}{,}{19}{,}{11}\right)\left({8}{,}{24}{,}{16}{,}{12}\right)\left({9}{,}{21}{,}{17}{,}{13}\right){,}\left({1}{,}{5}{,}{3}{,}{15}\right)\left({2}{,}{10}{,}{4}{,}{20}\right)\left({6}{,}{8}{,}{18}{,}{16}\right)\left({7}{,}{13}{,}{19}{,}{21}\right)\left({9}{,}{23}{,}{17}{,}{11}\right)\left({12}{,}{14}{,}{24}{,}{22}\right)⟩$ (2)
 > $\mathrm{GroupOrder}\left(\mathrm{P2}\right)$
 ${8}$ (3)
 > $\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(3,G\right)\right)$
 ${3}$ (4)
 > $\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(5,G\right)\right)$
 ${5}$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(7,G\right)\right)$
 ${1}$ (6)
 > $G≔\mathrm{CayleyTableGroup}\left(\mathrm{Symm}\left(4\right)\right)$
 ${G}{≔}{\mathrm{< a Cayley table group with 24 elements >}}$ (7)
 > $P≔\mathrm{SylowSubgroup}\left(3,G\right)$
 ${P}{≔}{\mathrm{< a Cayley table group with 3 elements >}}$ (8)
 > $N≔\mathrm{Normaliser}\left(P,G\right)$
 ${N}{≔}{{N}}_{{\mathrm{< a Cayley table group with 24 elements >}}}{}\left({\mathrm{< a Cayley table group with 3 elements >}}\right)$ (9)
 > $Q≔\mathrm{SylowSubgroup}\left(2,N\right)$
 ${Q}{≔}{\mathrm{< a Cayley table group with 2 elements >}}$ (10)
 > $\mathrm{GroupOrder}\left(Q\right)$
 ${2}$ (11)

Compatibility

 • The GroupTheory[SylowSubgroup] command was introduced in Maple 17.
 • For more information on Maple 17 changes, see Updates in Maple 17.